Gravitation And Cosmology - From The Hubble Radius To The Planck Scale

Gravitation and Cosmology: From the Hubble Radius to the Planck Scale Proceedings of a Symposium in Honour of the 80" Bi...

Author: Richard L. Amoroso | Geoffrey Hunter | Menas Kafatos and Jean-Pierre Vigier

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Gravitation and Cosmology: From the Hubble Radius to the Planck Scale Proceedings of a Symposium in Honour of the 80" Birthday of Jean-PierreVigier Edited by

Richard L. Amoroso, Geoffrey Hunter, Menas Kafatos and Jean-PierreVigier

Gravitation and Cosmology: From the Hubble Radius to the Planck Scale

Fundamental Theories of Physics An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application

Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A.

Editorial Advisory Board: JAMES T. CUSHING, University of Notre Dame, U.S.A. GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland ASHER PERES, Israel Institute q f Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada TONY SUDBURY, University of York, U.K. HANS-JURGEN TREDER, Zentralinstit foirAstrophysik der Akademie der Wissenschafen., Germany

Volume 126

Gravitation and Cosmology: From the Hubble Radius to the Planck Scale Proceedings of a Symposium in Honour of the 80th Birthday of Jean-Pierre Vigier Edited by

Richard L. Amoroso Noetic Advanced Studies Institute, Orinda, CA, U.S.A.

Geoffrey Hunter York University, Toronto, Canada

Menas Kafatos George Mason University. Fairfax, VA, U.S.A. and

Jean-Pierre Vigier Pierre et Marie Curie Universite, Paris, France

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-48052-2 1-4020-0885-6

c2003 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print 02002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

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Dedicated to Les heretiques de la science The Jean-Pierre Vigier resistant potential struggling within each of us

TABLE OF CONTENTS

Dedication Foreword Preface Group Photos

xi xiii xvii

Part I - Astrophysics & Cosmology 1. From the Cosmological Term to the Planck Constant Jose G. Vargas and D. G. Torr 2. Creation of Matter and Anomalous Redshifts Ja-yantV. Narlikar 3. The Origin of CMBR as Intrinsic Blackbody Cavity-QED Resonance Inherent in the Dynamics ofthe Continuous State Topology of the Dirac Vacuum Richard L. Arrzoroso and Jean-Pierre Vigier 4- Some New Results in Theoretical Cosmology Wolfgang Rindler 5. Whitehead Meets Feynman and the Big Bang Geoffrey Chew 6. Developing the Cosmology of a Continuous State Universe Richard L. Amoroso 7. The Problem of Observation in Cosmology and the Big Bang Menas Kafatos 8. Absorber Theory ofRadiation in Expanding Universes Jayant V. Narlikar 9. Bohm & Vigier Ideas as a Basis for a Fractal Universe Corneliu Ciubotariy Viorel Stancu and Ciprian Ciubotariu 10. A Random Walk in a Flat Universe Fotini Pallikari 11. Multiple Scattering Theory in Wolf's Mechanism and Implications in QSO Redshift Sisir Roy and S. Datta 12. Connections Between Thermodynamics, Statistical Mechanics, Quantum Mechanics, and Special Astrophysical Processes Daniel C. Cole

TABLE OF CONTENTS

Part I1 - Extended Electromagnetic Theory 13. New Developments in Electromagnetic Field Theory Bo Lehnert 14. Comparison of near and Far Field Double-slit Interferometry for Dispersion in Propagation of the Photon Wave-packet Richard L. Amoroso, Jean-Pierre Vigier, Menas Kafatos & Geofrey Hunter 15. Photon Diameter Measurements G. Hunter, M. Kowalski, R. Mani, L.P. Wadlinger, F. Engler & T. Richardson 16. What Is the Evans-Vigier Field? Valeri V. Dvoeglazov 17. Non-Abelian Gauge Groups for Real and Complex Amended Maxwell's Equations Elizabeth Rauscher 18. Experimental Evidence of Near-Field Superluminally Propagating Electromagnetic Fields William D. Walker 19. The Photon Spin and Other Topological Features of Classical Electromagnetism Robert M. Kiehn 20. The Process of Photon Emission from Atomic Hydrogen Marian Kowalski 21. Holographic Mind - Overview: The Integration of Seer, Seeing, and Seen Edmond Chouinard 22. Photons from the Future Ralph G. Beil

Part III- Gravitation Theory 23. Can One Unify Gravity and Electromagnetic Fields? Jean-Pierre Vigier & Richard L, Amoroso 24. The Dipolar Zero-Modes of Einstein Action Giovanni Modanese 25. Theoretical and Experimental Progress on the GEM (Gravity-Electro-Magnetism) Theory of Field Unification John Brandenburg, J- F. Kline and Vincent Di Pietro 26. Can Gravity Be Included in Grand Unification ? Peter Rowlands and John P. Cullerne 27. Gravitational Energy-Momentum in the Tetrad and Quadratic Spinor Representation of General Relativity Roh S. Tung and James M. Nester

GRAVITATION AND COSMOLOGY 28. Spinors in Affine Theory of Gravity Horst V. Borzeszkowski and Hans- J. Treder 29. A New Approach to Quantum Gravity, An Overview Sarah B. Bell, John P. Cullerne and Bernard M. Diaz 30. Multidimensional Gravity and Cosmology and Problems of G M.A. Grebeniuk and Vitaly N. Melnikov 31. Quantum Gravity Operators and Nascent Cosmologies Lawrence B. Crowell 32. Gravitational Magnetism: An Update Saul-Paul Sirag

Part IV - Quantum Theory 33. Quantum Hall Enigmas Malcolm H. Macgreggor 34. On the Possible Existence of Tight Bound States in Quantum Mechanics A. Dragic, 2. Marie & J-P Vigier 35.A Chaotic-stochastic Model of An Atom Corneliu Ciubotariy Viorel Stancu & Ciprian Ciubotariu 36. Syncronization Versus Simultaneity Relations, with Implications for Interpretations of Quantum Measurements Jose G. Vargas and Douglas G. Torr 37. Can Non-local Interferometry Experiments Reveal a Local Model of Matter? Joao Marto and J. R. Croca 38. Beyond Heisenberg's Uncertainty Limits Josee R. Croca 39. Towards a Classical Re-interpretation of the Schrodinger Equation According to Stochastic Electrodynamics K. Dechoum, Humberto Franca and C. P. Malta 40. The Philosophy of the Trajectory Representation of Quantum Mechanics Edward R. Floyd 41. Some Physical and Philosophical Problems of Causality in the Interpretation of Quantum Mechanics Bogdan Lange 42. The Force C4/G , the Power C5 /G and the Basic Equations of Quantum Mechanics Ludwik Kostro 43. Progress in Post-Quantum Physics and Unified Field Theory Jack Sarfatti

TABLE OF CONTENTS

Part V -Vacuum Dynamics & Spacetime 44. Polarizable-vacuum Approach to General Relativity Harold E. Puthoff 45. The Inertia Reaction Force and its Vacuum Origin Alfonso Rueda and Bernard Haisch 46. Engineering the Vacuum Trevor Marshall 47. The Photon as a Charge-Neutral and Mass-Neutral Composite Particle Hector A. Munera 48. Pregeometry Via Uniform Spaces Mark Stuckey and Wyeth Raws 49. A ZPF-Mediated Cosmological Origin of Electron Inertia Michael Ibison 50. Vacuum Radiation, Entropy and the Arrow of Time Jean Bums 5 1. Quaternions, Torsion and the Physical Vacuum: Theories of M. Sachs and G. Shipov Compared David Cyganski and William S. Page 5 2 . Homoloidal Webs, Space Cremona Transformations and the Dimensionality and Signature of Macro-spacetime Metod Saniga 5 3,Pulse Interaction in Nonlinear Vacuum Electrodynamics A. M. Ignator & Vladimir Poponin 54. Proposal for Teleportation by Help of Vacuum Holes Constantin Leshan, S. Octeabrisoe and R. L. Singerei 55. Cosmology, the Quantum Universe, and Electron Spin Milo Wolf 56. On Some Implications of the Local Theory Th(G) and of Popper's Experiment Thomas D. Angelidis Index

FOREWORD

Jean-Pierre Vigier continually labeled one of les heretiques de la science, l'eternel resistant et le patriarche is yet a pillar of modern physics and mathematics, with one leg firmly planted in theory and the other in empiricism spanning a career of nearly 60 years with a publication vitae quickly approaching 400! He wrote of his mentor Louis de Broglie "Great physicists fight great battles", which perhaps applies even more so to Jean-Pierre Vigier himself'. If fortune allows a visit to Paris, reported to be the city of love, and certainly one of the most beautiful and interesting cities in the world; one has been treated to a visual and cultural feast. For example a leisurely stroll from the Musee du Louvre along the Champs-Elysees to the Arc de Triomphe would instill even the least creative soul with the entelechies of a poets muse. It is perhaps open to theoretical interpretation, but if causal conditions have allowed one to be a physicist, visiting Paris, one may have taken opportunity to visit the portion of the old Latin quarter in place Jussieu where Pierre et Marie Curie Universite, reported to be 'the best university in France', is stationed. While there at Paris - VI you might have been even more fortunate still to visit professor Jean-Pierre Vigier and meet with him in his office near the department of Gravitation et Cosmologie Relativistes (GCR) where he has an emeritus position. The probability in the relativistic approximation has now approached unity that your de Broglie wave has entered superposition with a certain chair in Vigier's office; and now causal conditions are such that it is immanent that you will be fortunate enough to settle into de Broglie's revered chair saved from the years at the Institut Henri Poincare for your conversation with Jean-Pierre Vigier, currently the only living scientist who was a student of de Broglie. Now if all this isn't too distracting to the heart and soul of the physicist whose daydream vision clears sufficiently during a pensive gaze out the window of this office at the top of the university, one beholds Notre Dame, the Eiffel tower, the Sorbonne and numerous other awe inspiring Parisian landmarks. Habituation to such a panorama would take a concerted effort even for a regular visitor as scrutiny often reveals something unnoticed before. This December we were working a little late in the excitement of having just perused a videotape of cold fusion research arriving that afternoon fi-om a laboratory Belgrade. While we were taking turns making phone calls to Belgium, San Francisco. Belgrade and Texas; I stood gazing out that window again. The Eiffel tower was dressed as a Christmas tree with a flashing light display which will continue every evening for 10 minutes on the hour until the start of the new millennium- Also a fitting tribute to the year of Jean-Pierre's 80thbirthday.

'

A brief biography ofVigier by Stan Jeffers is found in S. Jeffas, B. Lehnert, N. Abramson & L.Chebotarev (eds.) Jean-Pierre Vigier and the Stocl~asticInterpretation of Quantum Meclmnics, 2000, Montreal: Apeiron.

xii

GRAVITATION AND COSMOLOGY

You may have noticed I have said little so far about J-P the man himself, but only a few things about 'the emperor's shimmering clothes'. Oh that I had a thousand pages of ultafine print! If you are interested in similar areas of physical research Professor Vigier's office is a treasure trove, a tumble of piles of papers and books Lie Einstein's office was reported to be. He is at the point in his noted career where everyone sends him 'stuff, for PLA, for gifts, for his information, for his critique, to support his life's work etc. etc... Strolling along with him during his daily trip to his mail box in GCR is also an adventure because every day papers, books, videos arrive - lots of them, 'things' on the forefront of physics and cosmology. This recursive loop having justifiably achieved an 'ideal state' of self-organized superposition is one of the factors helping to maintain Jean-Pierre as an advanced guard soldiering in the pursuit of knowledge of the physical world. At 80+ he is still immeasurably prolific, probably more prolific than most men half his years. 1 hope his full biography is written as there are a thousand thousand stories that should be told and preserved from his personal, political and scientific life; but he says he wont stand for it, wont use the extensive time required for such a project as long as he can still work effectively on Physics. For example, the well-known incident when Vigier was a young student of de Broglie around age 25. Vigier arriving to meet with de Broglie found the prime minister of France already waiting for some time to discuss the possibility of his membership in the French academy. De Broglie called for Vigier to come in for his usual discussions and proclaimed loudly "as for the Prime Minister, tell him to come back next week!" Finally if one has been most fortunate of all to have had opportunity to work and collaborate with this great man of science, as perhaps most of the authors in this volume have to one degree or another, it's possible to learn more physics in an hour or a day than gleaned from many months or years past in ambles of ones career. As an octogenarian Vigier is still as sharp and focused as the Einstein Nadelstrahlung that emanate from his penetrating eyes. He must be a genius, not only as evidenced from the quality of the quantity of his published lore; but pose a question and he prattles off authors, dates, books with little pause... All of us thank you Jean-Pierre Vigier for the opportunity to hold this symposium in honor of your 8othbirthday and 60 years of physics with 50 years of that time devoted to work on the nature of the photon and aspects of quantum theory. Finally now after such a duration, your work begins to find acceptance in the general physics community. It is a grand inspiration to us all! If [all physicists] follow the same current fashion in expressing and thinking about electrodynamics or field theory, then the variety of hypotheses being generated... is limited. Perhaps rightly so, for possibly the chance is high that the truth lies in the fashionable direction. But, on the off chance that it is in another direction - a direction obvious from an unfashionable view of field theory - who will find it? Only someone who sacrifices himself-.. from a peculiar and unusual point of view, one may have to invent for himself-RichardFeyn~tzan,from the Nobel Prize lecture.

R L-Atrloroso, Paris December, 2000

PREFACE

The Physics of the twentieth century has been dominated by two ideas: the relativity of space and time and the quantization of physical interactions. A key aspect of Relativity is the Principle of Causality, which says that one event (a point in the Qdimensions of space-time) can only possibly cause another event if their separation in space is not greater than their separation in time (measured as -kt, with c the velocity of light), or in other words that physical interactions cannot travel faster than the speed of light. Relativity is a classical theory in the sense that physical interactions are thought to be essentially localized at points in space and time; i.e. that there is no action at a distance. This theory thus embodies the idea of locality as an essential condition for physical interactions to take place. Yet quantum theory is diametrically opposed to this concept of local realism because of the Heisenberg Uncertainty Principle (HUP) by which the precision with which we can simultaneously determine the position and velocity (momentum) of a particle is limited by the finite value of Planck's constant of action h. Whether this Principle is merely a limitation on experiments involving interactions, or whether it is intrinsic to Nature has been the subject ofan ongoing controversy, with Niels Bohr as the founding proponent of the Copenhagen Interpretation, and such notables as Einstein, de Broglie, Schrodinger, Bohm and Vigier as its opponents. Experiments involving observation of particles that have not interacted with the experimental apparatus, may resolve this question. The theory of relativity grew out of the integration ofthe theory of electricity and magnetism into Maxwell's equations. culminating in the latter half of the 19th century with the notable inference (now widely applied in modern technology) that light is simply electromagnetic radiation, which travels at a universal speed. the velocity of light. The concept of particle (as energy concentrated at or in the vicinity of a point in space-time) is ingrained in the thinking of almost all physicists perhaps because it is implicit in that foundation ofelementary physics, classical mechanics (Newton's equations ofmotion and their ramifications in Hamiltonian and Langrangian mechanics). The non-classical aspect ofQuantum theory is its formulation in terms ofwave-like amplitudes for physical processes which can interfere with each other to produce a net intensity, intensities being measurable whereas amplitudes are not. Since quantum theory is essentially wave-like (characterized by interference phenomena), when it is applied to the mechanics of particles it leads to absurdities, the most notable examples being 2-path experiments, whose results are consistent with the idea that the particle travels along both paths simultaneously with the quantum mechanical amplitudes for these alternative routes interfering with each other to produce the observed intensity pattern like that observed in

xiv

PREFACE

Young's seminal double slit experiment. Such experiments with Photons (Aspect in the 1970s) are not too surprising, because photons are not classical particles (you cannot bring them to rest in the laboratory to measure their intrinsic properties such as mass), and indeed some physicists subscribe to the idea that the very concept of photons is fictitious. and that all photon-like phenomena can be explained by a statistical (stochastic) model of electromagnetic interactions, the quantization being a result ofthe quantum mechanics of electrons in atoms and molecules, rather than ofthe light itself. This resistance to the photon concept was present from its inception; it was 18 years after Einstein wrote his 1905 paper proposing" light quanta" as the explanationfor the photoelectric effect before the photon concept became accepted by "respectable" physicists, and it took Arthur Compton's discovery of the effect that now bears his name to effect this acceptance. However, 2-path experiments with real particles (i.e. having a rest mass) are not so easily explained away; Zeilinger and his associates working in Viennahave observed singleparticle interference phenomena with neutrons (circa 2985)and most recently (1999) with C60 (Buckyball) molecules. The latter especially are almost macroscopic (the 60 individual atoms can been seen in high-power (electron and scanning-tunneling)microscopes. The idea that a Buckyball molecule can go through 2 slits in a screen simultaneously, and then interfere with itselfto produce the observed intensity pattern. is so bizarre as to be ridiculed by any chemist who works with molecular beams (gas-phase chemistry). Yet it is the conventional interpretation of the Zeilinger C60 experiments. This is the great mystery, puzzle and paradox of the quantum mechanicsof particles. The majority ofprofessional physicists simply accept the phenomena using quantum theory to predict the results of experiments without being concerned with the logical inconsistency of the concept of a point-particle with the quantum interference phenomena. The discreteness of physical interactions is quantified by the value of Planck's constant, and the physical origin and nature of this discreteness, and what in nature determines the value of Planck's constant, remain elusive questions. The question of whether the fundamental constants (Planck's h, the electric charge on the electron e, the velocity of light c, and the gravitational constant G) may differ in different regions of space and time, must await an answer in terms of our future understanding of their physical origin. Specifically, whether the electron charge is truly fundamental has been brought into question by the quark theory of elementary particle structure and Wheeler's spacetime wormholes. One physical phenomenon is apparently inconsistent with both the Relativity and Quantum theories; Gravity (regarded as a physical interaction) appears to travel at infinite speed (corroborated by classical mechanical calculations of the motions of celestial bodies assuming that the gravitational interaction is instantaneous over astronomical distances), and while attempts have been made to develop a quantized theory ofgravitation, no quantization of gravity has yet been observed. One way out of this dilemma has been to regard gravity as simply the creation of the physical framework (curved space-time) within which all (other) physical phenomena take

GRAVITATION AND COSMOLOGY

xv

place. This viewpoint that gravity is not a physical interaction allows what are classically regarded as gravitational forces to be consistent with the relativistic principle of causality (that no interaction can travel faster than the speed of light), and it leads to the possibility of an infinite universe, our own observable univererse being limited to the part that is receding from us at 'observed' speeds less than the velocity of light. However, the observation of red-shifts not attributable to the receding velocity of the source (non-velocity red-shifts) in recent years. calls into question the cosmological model of an expanding universe, and it may eventually throw light on the nature of gravity. The Physicists who have attended the three Vigier Symposia (at York University in 1995 and 1997, and at the Berkeley Campus of the University of California in 2000) (Planned for Paris in 2003) are, like Jean-Pierre Vigier himself and his mentor Louis de Broglie, committed to the intrinsic Logicality of Nature in terms of determinism; as Einstein remarked: "God does not play dice". The lectures presented at the Symposia (and at similar conferences over a period of several decades) are a variety ofattempts to resolve the intrinsic paradox of the quantum mechanics of particles and to reconcile the locality intrinsic to Relativity theory with the manifest non-local realism of quantum interference phenomena. This rigorous investigation continues to proceed from the Hubble radius to the Planck scale; and ifthere ever should be a demise ofthe bigbang or a handle on the Dirac's polarizable vacuum- perhaps these investigations will lead us infinitely beyond... Geofiey Hunter York University, Toronto, Canada Richard L. Amoroso Noetic Advanced Studies Institute, Orinda, USA March, 2001

The organizers gratefully acknowledge generous financial support from: The California Institute ofphysics and Astrophysics (CIPA) International Space Sciences Organization (ISSO) The Noetic Advanced Studies Institute

and extend special thanks to:

Edmond Chouinard of Measurements Research, Inc. Roh Tung of CIPA for help in preparing the final manuscript!

PREFACE

xvi

The International Organizing Committee R.L. Amoroso, The Noetic Advanced Studies Institute Physics Lab, USA Chairman. B. Haisch,Solar & Astrophysics Laboratory,Lockheed Martin,Palo Alto,USA B. Hiley,Theoretical Physics Research Unit, Birkbeck College, London, UK. G. Hunter, Chemistry Department, York University, Canada - Coorganizer. S. Jeffers, Department of Physics & Astronomy, York University, Canada Coorganizer. M. Kafatos, Centel- for Earth Observation & Space Research,George Mason Univ.,USA - Coorganizer. C. Levit, NASA Ames Research Center, Molecular Nanotechnology Group, Moffet Field, CA USA. M. Moles, Instituto de Mathematicas y Fisica Spain . J. Narlikar, Inter-University Center for Astronomy and Astrophysics. India. S. Roy, Indian Statistical Institute, Calculta. India. A. Rueda, California State University, Dept. of Electrical Engineering, USA. A. van der Merwe, Physics Department, University of Denver, USA. J-P. Vigier, Gravitation et Cosmologie Relativistes, Pierre et Marie Curie Universite, Paris VI, France

Program Committee F. Pallikari, University of Athens, Physics Department, Greece. A. Rueda, California State University, Dept. of Electrical Engineering, USA. R. Amoroso, The Noetic Advanced Studies Institute - Physics Lab, USA. B. Haisch, CIPA, USA. G. Hunter, Department of Chemistry, York Univ, Canada.

Keynote Speakers 1. Jean-Pierre Vigier - France 2. Jayanl Narlikar - India 3. Wolfgang Rindler - USA 4. Bo Lehnert - Sweden

Photopraph I . Left, Professor Jean-Pierre Vigier, right, chairman Professor Richard L. Amoroso. Photo courtesy of Professor Geoffrey Hunter.

Photopraph 2. Many attendees of the Vigier-I11 symposium, taken on terrace at UCB Faculty Club; the symposium venue. 2"6row center, J-P Vigier. Photo taken courtesy Dr. Metod Saniga

FROM THE COSMOLOGICAL TERM TO THE PLANCK CONSTANT J. G. VARGAS'.~AND D. G. TORR~ 'center for Science Education 2 Department of Physics, University of South Carolina, Columbia, SC 29210

1.

Introduction

h this paper, we show the potential of classical diferential geometry to unify gravity with the other interactions and, specially, quantum mechanics. Consider Cartan's comments[ 1]:

...a Riemannian space... in the iinnlediate neighborhood of a given point, it can be assinlilated to a Euclidean space. " And also: "A general space with a Euclidean connection nluy be viewed us rnade of an infinite unzount of infinitesimally smallpieces of Euclidean space, endowed with a connecting law pen~littingto integrate two contiguous pieces into one and the sanle Euclidean space. " And finally: "...collections of sr?lall pieces of Euclidean spaces, oriented relative to the neighboring pieces." 6'

Hence, had general relativity not been born by 1925, ~ e at . a time when the present form of Quantum Mechanics was not yet known, one might have expected that Riemannian geometry (RG) would have to do, if anything, with the realm of the very small, not of the very large. Why are the mathematics of the quantum world and of the very small (as defined above) so different'? Enough mathematical knowledge exists to solve the puzzle. The key to the solution of the problems that we have posed lies in the affine connection of spacetime. At the end of section 2, we sumnlarize the topics to be considered, other than the connection itself.

2.

The AEne Connection Of Spacetime

The field equations of General Relativity (GR) concern the Einstein tensor. which is a piece of the curvature. A modern lecturer in GR would likely explain curvature through the transport of vectors around closed paths. But Einstein himself could not, at the birth of GR in 1915, have explained curvature in this way; this concept of curvature through transport of vectors was foreign to RG until the introduction in 2917 by Levi-Civita[2] (LC) of a rule for parallel transport, i.e. an affine connection, in a Riemannian space. The 1

R.L. Amoroso eta1 (eds.). Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 1-10. O 2002 Klwer Academic Publishers. Printed in the Netherlands

2

J. G. VARGAS & D. G. TORR

quantities known as Christoffel symbols then became also the quantities for the LC affine connection. The set of quantities known as Riemann's curvature then started to represent both the original or metric curvature and the new or affine curvature, a property of affine connections. In 1917, RG ceased to be a purely metric theory and became an affine cum metric theory. As Cartan put it[3]: "With the introduction of his definition ofparallelism, Levi-Civita was the first one to make thefalse metric spaces of Kemann become (not true Euclidean spaces, which is impassible. but) at least spaces with Euclidean connection ... (emphasis in the original). To our knowledge, general relativists never questioned the physical correctness of attaching an affme meaning to the geometric objects of 1915-GR, i.e. those of 1915-RG. By the time that general affine connections were created and understood, the LC connection (younger than GR!) had become part of the foundations of Einstein's theory of gravitation. This happened by default, as physicists did not know about other possibilities. The obvious alternative to a LC connection is a teleparallel connection. They are the canonical connections of the pair constituted by a trletric and a preferred fratrle .field, namely connections which are zero in that fi-amefield. The affine curvature is then zero in any frame field. All geometric quantities (torsion, affine curvature, etc.) again derive from the metric only, as in RG, though there is now a fkme field that plays a special role in the derivation. The earth, punctured at the poles and endowed with the connection where the rhumb lines rather than the maximum circles are the lines of constant direction, has zero affine curvature. This connection is so natural that Chriitopher Columbus and his sailors entrusted their lives to it: they maintained "a constant direction" by staying on the same parallel when they sailed to the New World. Since Columbus' earth also is round, it has non-zero metric curvature. The terms teleparallelism (TP) and parallelism at a distance refer to these connections, Eistein [3] tried to replace RG (cum LC connection) with TP. His motivation was "Given points A and B separated b j ~afinite distance, the lengths qf two linear elenzentsplaced a t A and B can be cornpared, but one cannot do the sarrle with their directions; in RG there is no parallelisrll at a distalzce. " (Emphasis in original).

Einstein then proceeded to develop TP. Notice from the quote that it was self-evident for him that physics must be based on TP: he objected to RG (endowed with the LC connection) for no other reason than for not being teleparallel! We now show that replacing the 1917 LC extension of GR with a TP affine extension one obtains a physical theory with the same metric relations as GR but which enriches Einstein's theory by producing additional relations with profound implications. Eistein tried to develop physical TP and failed. The mathematics of the time was not ripe. Whereas in Riemannian geometry the only independent differential invariants are the d ' s (defined by ds2=z(d)'), we now have as independent differential invariants d a n d (u? But even this is not rich enough for a unified theory of the interactions. When one

COSMOLOGICAL TERM TO PLANCK CONSTANT

3

knows the mathematics, it is also a self-evident truth that the Lorentzian signature is the canonical, preferred signature of Finsler geometry. In the modern view of differential geometry, Finsler geometry must be viewed as pertaining to Finsler bundles, not to Finsler metrics, the metric origins of this geometry not withstanding. The view that Finslerian connections are the connections determined by Finslerian metrics is as incorrect as the view that the theory of affine connections is about connections determined by Riemannian metrics. In Finsler geometry, the differential invariants d' and wLvmay be split into the The d,t d mi, which are the components of the following pieces: id, d,mi and vector-valued 1-forms dP (=d'eJ and du (=miel), span the Finslerian base space or phase-spacetime. In Finslerian TP there are cross sections where qk (but not &) are zero. The torsion, which will represent the non-gravitational interactions in Finslerian TP, is simply the exterior covariant derivative of the differential invariant dP. TP thus is Aharonov-Bohm compliant, and there is a new philosophy. The geometric expectations that we have raised might prompt readers to expect that, in the same way as the metric represents the gravitational interaction. some other differential invariant might represent the other interactions and, still some other, quantum mechanics. The new philosophy, however, is that different members of the set (a o , oi,ml, of differential invariants combine in different ways to give rise to different physical concepts, interactions, representative equations, etc. Thus the Finsler-invariant quantity ( wo, ~lzodulod)contains all the metric relations of the manifold. It satisfies Ids=ld, nzod i 0'. The Riemannian case corresponds to when ( d , trlod w ) is a quadratic form on the velocity coordinates. The set (wo, m', mi) contains all the information about the classical description of motion for all the interactions, to the extent that each individual interaction admits such a description. It also constitutes the input for Dirac equations that exclude the weak and "a combined strong-weal? interactions, etc. The understanding of how Dirac equations enter the geometric picture is the main issue. The beginnings of the solution were provided by Kiihler [4] through the construction of a calculus that, by combining the exterior and interior derivatives, generalizes Cartan's calculus and gives rise to a theory of "KZhler-Dirac" equations, the standard Dirac equation being a particular case. Although K i l e r confined himself to spaces endowed with the LC connection, his work is easily reformulated to apply to TP [5]. The reformulation does not, however, eliminate some peculiar features of his calculus. These disappear, however, when, in order to remove from the total set of differential invariants (of, o',mi, w/k) the invariants t q k that embody the macroscopic rotations, Finslerian TP is further reformulated as a Kaluza-Klein (KK) theory based on ( wo,mi, w d [q. A final feature of the resulting picture is that the connection has to be stochastic: without it, there is no gravitation of neutral matter. This is known as the Sakharov-Puthoff conjecture, which is a consequence here. To conclude, the LC affine connection (and curvature) entered physics surreptitiously. In case it is the wrong affine connection of spacetime, one should try TP.

4.

4)

4


3. The Gravitational Sector Of Teleparallelism Given the structural richness of TP, there is no need for ad hoc introductions of additional structure, at least not until one has developed the consequences of TP as a postulate. Let upV be the affine connection and let %"be LC object (no longer the affine connection). We have: wpV= apv -t PpV (1) where PpVrepresentsa 2-tensor-valued I-forrn. Its components are linear combinations of the components of the torsion and have three indices, the form index being hidden in Ppv. Users of the tensor calculus should think of upvas representing the rpV', and of a " as !' representing the Christoffel symbols. The metric curvature, or Riemannian curvature, is V a OpV = dap - a;, A aav

(2)

The R, VAp known from the tensor calculus, are components of the 2-tensor-valued 2form qV, the form indices being hidden. The affine curvature of the space, or curvature ofthe rule that compares vectors at different points A and B, is G V = d w v, - upA A onv Substituting (1) in (2):

~ e ~ u a ~ Expanding (4), or its tensor calculus equivalent, and setting d ~ - u ~ ~ o to~zero:

The contents of the parenthesis is a tensor. and so is P/A A". The term "Einstein contraction" will refer to the process that takes us from the curvature 2-form dol,V-ak~aAvto the so-called Einstein tensor: get the components, contract them to obtain Ricci, etc. One obtains a completely geometric Einstein equation by applying this process to both sides of Eq. (5). If we deal with point masses and symmetry arguments (binary pulsar), one cannot distinguish between the Einstein contraction of (5) and Einstein's equations, since the torsion (and its derivatives) for these masses will be non zero only at the point masses positions. Let us now show that the equivalence principle is part ofthe rich contents of Eq. (5). For a homogeneous and time independent contiguration of all fields, dPtp would be zero. to be zero. the whole contents of the parenthesis (and. with it, Since we can choose upV its Einstein contraction) can be made to disappear. One then interprets this term as gravitational energy. For confirmation observe that the equivalence of an accelerated frame and a gravitational field is only as large as homogeneity allows (Einstein elevator). Hence it is only as large as the differentials dPaPremain negligible. The gravitational energy contained in the Einstein contraction of the contents of the parenthesis thus exhibits the equivalence principle and that it is valid only locally (physical sense of


5

L'l~dlly").The contracted quadratic term, -P BAPA", then represents non-gravitational energy. The term d w is the key to controlling gravity through inhomogeneous and/or time dependent electromagnetic fields. It causes a variation in the weight of a body by an inhomogeneous electric field. The experimental detection of this effect at the University of South Carolina by M. Yin and T. Datta will soon be submitted for publication. Previous work was reported by Dimofte [7]. Our present interest. however, is in establishing the connection between the very large and the very small. Since the same differential invariants generate the macro and microscopic sectors, implications of one sector for the other will propagate through these differential invariants.

4. Completion Of The Classical Sector Of Teleparallelism The torsion is the formal exterior derivative of dP: f i ~ ~ ~ e ~ = d ( d = ~ ~ d ~ d e , + $ ' , w p ~ w i e v(notice the Finslerian generalization). The "d" in dP does not mean the differential of anything. dP is not an exact form but notation for the vector-valued 1-form that gives the translation vector assigned to a path by integration on it of d e p One differentiates dP using the definition wfep of det, which relates bases at x+dx and x. The first Bianchi identity for TP reads d+O, i.e. that the exterior covariant derivative of rR is 0. The "K2hler complement" to this system is constituted by the specification of the interior covariant derivative of the torsion, b@J, where J i s a vector-valued 3-form and d is the interior covariant derivative defined in the TP K i l e r calculus [5]. The system (drR =O, contains Maxwell equations in vacuum (charges but no material media) up to a constant. The zero-current equations are not selfdual, in general, since 6 i s not the usual *d* of the zero torsion, except as an approximation [8]. The structure of Finsler geometry allows one to perform an identification (within the limits imposed by the state of the theory) between pieces ofthe torsion and the different interactions [9]. Equations of motion in the form of autoparallels (up to particledependent "dressing constants'? have been obtained [9]. It has been found that some pieces of the Finslerian torsion do not contribute to the classical motion (meaning that they have zero classical range), except through the gravitational effects produced by their energy-momentum tensors [9]. This fact, together with the O(3) symmetry they enjoy (which becomes SU(2) in the Dirac sector) suggests their association with the weak and weak-cum-strong interactions of this theory. It has further been argued that the corresponding particles constitute "a second cosmological fluid" of dark matter whose pressure term is "the" cosmological term [9].

5. The Quantum Sector of Teleparallelism The Kahler generalization of the Dirac equation refers to any equation of the type [4]:

d u = avu

(6)

6


where a and u constitute the input and output differentialforms. d i s the sum of the interior and exterior derivatives, defined as in K i l e r , but with generalized connection instead of the LC connection. The free torsion equations dn=O,dn=Ocan be written as:

and thus are Kier-Dirac equations where a=O. Kahler showed that. if dv=O, then

i.e. another solution for the input a. A conjugate Dirac equation can be defined for the same form a. From the solutions for the direct and conjugate equations for given a, a conserved current follows, at least for the LC connection (we have not yet dealt with all these issues in TP). With the form aFp(271;/hc)(-m+ieA), Kahler solved the hydrogen atom [lo]. Note the unusual location of "i". Also, the existence of different Clifford algebra representations of Dirac's equation speaks of the fact that the geometric meaning of the equation has not yet been understood. The form a r is~ scalar-valued. Assume we had a vector-valued a. Since d has zero-valuedness, u must be of inhomogeneous valuedness, going, in principle, up to infinity, The non-ad-hoc way to control this explosion of valuedness is to assume a tangent Clifford algebra, not a tensor algebra. This provides the Kahler equation with additional structural richness and solves structural problems of present day Finsler geometry [6,11]. One defines a vector-valued 1-form in a canonical KK space, namely dp=dP+dar. The interpretation of dr (ds in previous publications) is proper time. u is a unit vector spanning the "fifth" dimension of the (4+1)-KK space and corresponds to the velocity. In this structure, one The differentiation of d g involves both d and ui but not the 9'. encounters U(1) as a spacetime symmetry [ll]. We have not yet developed the algebraic details of this theory, so as to give the opportunity to SU(3) to show up as an external symmetry. We expect that its appearance is just a matter of time, since Schmeikal has already shown that the SU(3) symmetry is contained in basic spin representations of the octahedral space group Oh,thus linking this inner symmetry with spatial geometry [12]. One can then think of dY=dgvY (9) as the aforementioned, extremely rich "canonical" equation where Y has Cliffordvaluedness and not just scalar-valuedness, like the electromagnetic field, or even vectorvaluedness.

6.

Relation Between Gravitation And Quantum Physics In Teleparallelism

We now show deep connections between the very large and the very small in TP. To start with, we emphasize the significance of Kahler's theory of D h c equations (the K i l e r who generalized Cartan's theory of exterior differential systems). With the humility of


7

great mathematicians when they touch on matters of physics, fihler stated: "So that the interior calculus may have its confirmation test in the quantum and relativity theories,..." [13] (he used the term interior calculus for what is nowadays called exterior-interior calculus). When the humble dressing of this subordinate statement is removed, K2hler has claimed the creation of a language suitable for two sectors ofphysics which are advertised as seemingly irreconcilable. This language should be studied by those prominent and, therefore, influential physicists who dare to pass negative judgment on these issues from a position of ignorance of the mathematics involved, or future generations will taint their memories for such serious oversight. Indeed, unless the God that Einstein rhetorically invoked in a now famous statement is not utterly malicious, one wonders why nature would ignore such a simple and rich option to create a most sophisticated and elegant world in the main or tangent bundle, rather than in the auxiliary bundles of gauge geometry.. The key to understanding the relation between TP gravitation and quantum physics is the realization that, whereas Einstein's equations constitute 10 equations for 20 unknowns (the 20 independent components of the curvature tensor), the gravitational equations are now 20 for the same 20 unknowns. Of course, the torsion components also appear in the same equation, but the torsion has to be considered as a given in this part of the argument, like the energy-momentum tensor. Because there are now 20 field equations, the Cauchy problem involves the specification of the initial condition just at a point ofspace (not on a hypersurface) at an instant of time. To make the argument clear. imagine the sudden switching-on of sources of torsion all over the universe. The initial conditions that those sources create will start to arrive at A from within the past light cone of any given spacetime point A. Of course. the geomett'lc fields that the infinity of sources of torsion produce do not match at A. What this implies is that, the "exact" solution for the basic differential invariants (or connection, wO,wi, of our closed system of differential equations does not even make sense except as a stochastic solution, though allowing for a dominant non-stochastic component of the stochastic solution. We would have a background of torsion in the universe as we now have the electromagnetic background. In fact, it will look electrodynamic away from matter, but, close to it, it will have the dynamical richness of a vacuum whose governing differential equations are non-linear, with all the concomitants of such non-linearity. For instance, we cannotjust say that the background torsion field (or, better, the background dP and/or d g field) is very small. This may be the case here (i-e. at some point) and now (at some instant of time), But the derivatives may be very large and virtually cancel among themselves. Large fields may resurrect further down the line (in space and or time) in the form of solitons, as for Muraskin's equations [14]:

These equations are being considered here because different types of computer-generated solutions of them exist [14] and because their quadratic terms are highly similar to those that appear in the equations for the electromagnetic vacuum of o w own system [8] (We have also provided these equations with an interpretation within TP [15]). Notice that,

8


when the quadratic terms are eliminated, these equations reduce to drpV'=O, whose where the CpvA are integration constants. solution obviously is TpVA=CpVa, It is worth noting the randomization of the spacetime structure (i.e. of the basic differential invariants of the geometry) that would take place in a radiation-dominated spacetime where the background could not be treated linearly. Imagine integrating the field equations and finding all the geometric quantities at a spacetime point under the assumption. for simplicity, that there were just two sources in the universe. namely at points B and C. The field equations being quadratic, the signals coming from B and C become badly scrambled. Imagine now that this launching of initial conditions happens for the signals coming from any source in the universe (ferrnionic fields), not to mention the fact that even the torsion of the background field (and of any field for that matter) is itself a source of the metric structure of spacetime, as exhibited by equation 5. This is not unlike the fact that any tiny piece of energy contributes to the Einstein equations. The difference between this stochastic picture and the classical picture of GR arises with their differing Cauchy problems! By increasing the sought precision of the solution, one eventually reaches the stochastic background. The latter may be endowed with little energy and yet have huge effects, the fluctuations arising when the derivatives of the field do not cancel among themselves. In our geometric picture, these fluctuations are to be associated with the stochasticity of the solutions for the field equations satisfied by the differential invariants of the geometry. The preceding argument referred to the bosonic fields (in the absence of similar studies for our own equations, we argued by analogy with the closely related Muraskin equations). Through the sharing of common differential invariants, whose basic equations constitute a closed system of equations when a Kiler-Dirac equation is included. the stochasticity of the connection has to be associated with the value of h, since the quantum effects, and the vacuum fluctuations with it, vanish as h goes to zero (Notice that, although one needs to include the canonical Kiihler-Dirac equation in the geometrically closed system of field equations, there is no need to include Y among the basic differential invariants, since a knowledge of dP and du allows one to obtain Y;3. Without such stochasticity, there are no vacuum fluctuations and no quantum physics as we know it. It is the nature of the gravitational sector of the system of field equations for TP that causes the necessarily stochastic nature of the vacuum of TP.

7. The Sakharov-Puthoff Conjecture As An Integral Part Of Teleparallel Physics Let us finally deal with the Sakharov-Puthoff conjecture [16, 171. In a nutshell, it states that the gravitational interaction is wiped out when one switches the other interactions off. Specifically, gravitation is an effect caused by the vacuum fields of the other interactions. This is a consequence of TP in the following way. Without stochasticity, the vacuum would have to be considered as empty space in a true sense. The torsion field would then be zero. Since, by postulate, the affine curvature also is zero, the spacetime is flat. In other words, it becomes affine space [Only the flat metrics (Euclidean, Minkowskian, etc.) are consistent with affine space, or else the affine connection would not be metric-

COSMOLOGICALTERM TO PLANCK CONSTANT

9

compatible. Equation (5) indeed shows that making the torsion (and, therefore, P) equal to zero, annuls the metric curvature] One is not claiming that the energy of the vacuum @sons) causes the metric curvature of spacetime. This may be the relativistic way of thinking, but is not what TP dictates. The gravitational energy to be associated with the gravitational force in TP is the Einstein contraction of dPap (See Eq. 5). This term becomes significant next to matter. where the torsion field becomes associated with fermions ( Yof type u, Eq. 6). As for the nature of the background field, the solutions of the Muraskin equations [14] suggest that the magnitude ofthe fields which are solutions of the geometric equations of this theory may be very small at some point and very large at some other point. It is conceivable in principle, that the zero point field will become, far awayfrorn matter, a linear-looking non-linear version of the cosmic background field. The problem, to the different extents that it is perceived to be one, of the infinite energy density of the zero point field of QED and stochastic electrodynamics (and concomitant infinite curvature of spacetime) does not even arise in this theory. Very close to particles, phenomena like vacuum polarization would show up. For inutter in bulk, regular gravitation would then occur, as intimated by the work of Haish-Rueda-Puthoff [18]. If we want to alter gravitation. i.e. how the background field appears to matter in bulk, we would have to put very large gradients where matter and background fields meet, i.e. very large inhomogeneous fields. Most of the statements in the previous paragraph are not theory but speculation, as the mathematical proof is not there yet. It took many of the very best physicists of the beginning ofthe twentieth century to develop the paradigm of Quantum Mechanics. Each of them provided a piece to the puzzle, piece which eluded everybody else in such distinguished group. Hence, the solution that TP may provide to the grave problems that afflict present day theoretical physics will remain largely speculative until a significant amount of theorists join this effort.

8. Acknowledgments

One of us (J.G.V.) deeply acknowledges generous funding from the Offices of the ViceProvost for Research and of the Dean of the School of Science and Mathematics of the University of South Carolina at Columbia.

References 1. Cartan, E : Oeuvres Complktes, Editions du C.N.R.S., Paris, 1983. 2. Levi-Civita. T.: Nozione di parallelismo in una v a r i e qualunque ~ e conbequente specificazione geometrica della curvatura Riemanniana, Rendiconti di Palenno 42 (19 17), 173-205. 3. Einstein. A.: Thkrie Unitaire du Champ Physique".Ann. Inst. Henri Poinc-crre' 1 (1930). 1-24. 4. Kiler, E.: Innerer undausserer Differentialkalkiil.Abk.Dt.~ch.Akad. Wiss. Berlin, Kl. Math., Pky. Tech., 4 (1960). 1-32. 5. Vargas, J. G. and Torr, D. G.: Teleparallel Kahler calculus for spacetime, Fmrd. Phys 28 (1998), 931 -958. 6. Vargas, J. G. and Torr, D. G.: Clifford-valued clifforms: a geometric language for Dirac equations, in R Ablamowicz and B. Fauser (eds.), CliffordAlgebms and tkeirApp1ication.s in Matkernatical Pkysics.

J. G. VARGAS & D. G. TORR Birkhauser Boston, 2000, pp. 135- 154. 7. Dimofte, A.: An experirrzem cmcenling elecmkally inducedgravitation, Master's Thesis, University of South Carolina, Columbia, 1999. 8. Vargas. J. G. and Torr. D. G.: The Cartan-Einstein unification with teleparallelism and the discrepant measurements ofNewton's constant G, Found. Phys 29 (1999), 145-200. 9. Vargas, J. G. and Torr, D. G.: The theory of acceleration within it 0 and k < 0 universes; between A > 0 and A < 0 universes: and between oscillatory and nonoscillatory universes; the latter boundary consists of the A = 0 line €or Sd, l1, and the dotted line for Go >1. Present observations suggest Go 0.3 and go = -1. Augmented by realistic estimates for Ho and t o , these numbers strongly favor a non-oscillating universe: but it is too early to decide between k > 0 and k < 0. The inflationary hypothesis favors k = 0.

-

3. Milne's Model And The Point-Like Big Bang The total volume of a positively curved universe (a 3-sphere) is finite and the big bang presents no topological problems. It is a singular point-event, before which neither space nor time existed. If, on the other hand, the universe is negatively curved (and we assume no "funny" artificial topology), its volume as well as its matter content is infmite. And this is true at every cosmic moment, no matter how close to the big bang. So can the big bang be point-like in this case, or is it infinitely extended? To understand this problem, I like to make propaganda for an old "toy" model-universe discussed by Milne in 1934, where the topology is quite transparent. There is a solution of Friedman's differential equation (2) that has A = 0 and C = 0 (which corresponds to G = 0 and thus to gravity being "switched off"). Under these conditions the model must live in special-relativistic Minkowski space! Its Friedman description, as we see at once € o m eq. (2), must have k = -1 and R = ct; regarded at successive cosmic moments it is thus infinitely extended and expands at a constant rate. Milne found that by slicing differently through the fundamental-observer worldlines, he could describe the model as an expanding finite ball in an ordinary Euclidean inertial frame, Any one of the fundamental observers can be the center of the ball, and the others move radially away from that one at all speeds short of the speed of light. The time T used in this view is the "ordinary" inertial t i e of the central observer. Cosmic tints t is indicated on the standard clocks carried by all the other fundamental observers, starting from zero at the big bang. At a given T-instant these clocks read more and more behind T (by time dilation) as we go towards the outer unattained edge of the ball, which moves at the speed of light. The infinite cosmic-time sections consist of infinitely many identical origin-neighborhoods of Milne's ball.

NEW RESULTS IN COSMOLOGY

43

I have recently demonstrated (in a paper to be published elsewhere) that even in the general k = -1 case, the universe can be sliced into finite sections and thus be regarded as an expanding ball -- though not in flat 3-space -- starting with a point-like big bang. Here the question arises how an infinite amount of matter can be contained in a finite volume. (This question has no answer in Milne's model, where the "matter" can only consist of geometric points.) In the general case the answer hinges on the phenomenon of gravitational collapse. Just as in the case of a black hole, there comes a stage in the collapse of a collapsing universe where the collapse cannot be halted and the concentration of matter must increase without limit. Playing such a collapse in reverse, we get a big bang near which the matter concentration becomes infinite. In the picture of the universe as an expanding ball, this infinite concentration occurs towards the edge, and integrating the matter over the volume yields infinity.

4. Particle Horizons The last topic I want to examine in some detail is horizons. As we look with ever better telescopes into the night sky, we see ever farther and thus fainter galaxies. And since light travels at finite speed. we see the distant galaxies as they were at ever-eatlier times. In principle. there are two barriers to this looking backwards in time. The first we reach when the age of the matter we see has dropped about 500.000 years after the big bang, the so-called recombination time. Earlier than that. the universe was not transparent to light: ionized matter interacting with radiation formed and opaque fluid. (Later on the ions and electrons combined into atoms and no longer interacted strongly with photons.) However, if we could "see" with neutrinos instead of photons, we could see through this barrier -- all the way to the big bang: in principle we could receive neutrinos emitted at the big bang itself. So as we look in any direction (say with neutrinos -- these are purely theoretical considerations) as far as the big bang, we may well believe that we have surveyed all the matter of the universe. But this is far from being the case. We have merely reached the second barrier, our "particle horizon", which we shall now discuss. A very good model of a closed universe (k = 1) is provided by an ordinary balloon that is being blown up. Galaxies are ink dots uniformly distributed over the balloon. Photons are little bugs crawling over the balloon along great circles, always at the same speed. Such "balloons" ofradius R (t) are actually sub universes of the full 3-D universe. (In the cases k = 0 and k = -1 the balloon becomes a plane rubber sheet or a rubber saddle, respectively.) Now cover the balloon with silver dollars. Their rims momentarily are the light fronts (or neutrino fronts) emitted at their center at the big bang. (At the big bang space itself expanded much faster around each particle than the first light front emitted at that particle!) As my creation light front (read: my particle horizon) passes you, you see me for the very first time, at my creation. By symmetry, at that same cosmic instant I see you at creation. So my particle horizon at any instant is as far back into the past as I can possibly see at that instant. But that is only a very small fraction of the entire universe. And thereby hangs a puzzle. How can we explain the observed homogeneity of the universe -- since the influence from each particle cannot have spread further than its

44

WOLFGANG RINDLER

particle horizon? So the universe cannot have homogenized itself. Of course. in the description we gave (the nice spherical balloon the nice circular horizons). the big bang itself is assumed a priori to be so delicately designed as to produce homogeneity and isotropy from the start. If there were some flaw in the execution, the universe could not correct itself. Thus if among all the natural miracles you find such delicately designed initial conditions unacceptable. you have the "horizon problem" or "homogeneity problem". Inflationists claim to have cured the homogeneity problem by producing vastly larger horizons -- though it is still not clear how this would lead to isotropy. The argument usually given is that during the inflationary period, when the universe is (in seconds!!), the horizons are also stretched by stretched by a factor of that factor, and that is what makes them so big. Let us look at the situation in some detail. Even inflationists agree that today the observable universe is of FRW type. So in principle we can determine the best-fitting FRW dust model. We can follow this model back in time. Somewhere around recombination time, for increased accuracy, we can replace the dustdominated model by a radiation-dominated model -- and inflationists would have no quarrel with that. Only after we come down to a radius R w do the continuations backwards in time diverge. (We are now within lO-3O seconds of the big bang!) Whereas the FRW model just keeps on going according to general-relativistic dynamics, inflation has a sudden exponential drop. In the FRW model all the matter of the universe is created at the big bang. In the inflationary model, by contrast, the big bang is a mere "big whimper", where only a minute fraction ofthe matter is created -- the bulk being created during the inflation, when the density remains constant! ("The ultimate free lunch" according to Alan Guth, who invented inflation.) Now is it really the stretching that makes the inflationary horizons big? Ro, only not Evidently not, since the standard model must stretch equally to reach quite so fast. The real reason why inflation produces larger horizons is that a whimper produces larger horizons than a bang. The advantage of the inflationary horizon over the standard horizon was present even before inflation! For it can be shown that in radiative FRW models (and all models are radiative near the big bang) the radius of the particle horizon near the big bang is given by

-

-

-

and is thus proportional to p-'I2. Since after R = 10-a Ro, inflationary and noninflationary FRW models have the same density. the density of the standard model at Ro is the same as that of the inflationary model at the onset of inflation. i.e. at R w 10-68Ro. k t us assume that the whimper and its development until the onset of inflation is strictly FRW. Then the horizon size in the whimper universe at R =

.

is identical to that in the standard universe at R w What inflation then does to the whimper horizon is pure gain: it expands it by more than a factor of 1043 ! By that factor, owing to the space expansion, and by more, since the horizon light front does not


45

stand still during inflation. So at the end of inflation the ratio of the horizon radii is better : 1 in favor of the inflationary model. Thereafter that ratio decreases owing to than the equal motion of the horizon light fronts through space, but the advantage is permanent. Even if the whimper is chaotic (and inflationists clearly do not posit perfectly Friedmannian initial conditions), the Friedmannian horizon-density relation may well be indicative of a similar relation holding locally when the density is not homogeneous. The same horizon enhancing argument would then still apply to inflation. Part 2

In this second part, added only for the printed version of this lecture. I provide some of the mathematics that I omitted before. There is little point in repeating the rigorous derivation of the Friedman-Robertson-Walker metric (1) for homogeneous-isotropic universes, since that can be found in all the textbooks (1,2,3,4]. Equally standard is the application of Einstein's field equations to this metric, which yields, first, the conservation equation pR3 = const for the later, matter-dominated universe (zero pressure, t 2 500.000 y and

p ~ '=const

for the early, radiation-dominated universe

( p = f C' p , r 5 500,000y). It also yields the evolution equation (2), often called Friedman's differential equation. when matter dominates. and a similar equation with the

fmt term on the right side replaced by a 2 / R 2, where a2 = f 3GpR4,when radiation dominates. The very interesting phase diagram. Figure 1. is obtained by integrating the equation for the age of the universe -- for which the first -500.000 years can be safely fudged and the entire universe treated as matterdominated. One defines the Hubble parameter H o , the density parameter 8, and the deceleration parameter go as we did after eq. (2) in Part 1, whereupon one finds, directly from the Friedman eq. (2) and its time-derivative, the following important identities:

WOLFGANG RINDLER

Figure I . Phase diagram of the big bang universe. With these one goes back into eq. (2), rewriting it in the form y = RIG

(6)

This can at once be integrated. Since we are interested in the present age models, we assume R(0) = 0 and perform a definite integration:

toof big-bang

y 2 = H ; ( a y - l +(+Qo

-qo)y2 + l + q , -+&I,


the empty brace denotes the braced expression in (6), and f (Qo,qo) is defined by the last equation. This function can be machine integrated, whereupon we can tabulate and graph corresponding values of Qo, %, and Hoto.

The line A = 0 by (4) corresponds to +Qo -a= 0 , and separates universes with A > 0 from those with A c 0 . The line k = 0 by (5) corresponds to 3Q0 -go - 1 = 0 and separates positively from negatively curved universes. Solutions of Friedman's equation for which R has a zero at a finite time are necessarily "oscillatory". For this to happen, we need the right side of (2), regarded as a function of R, to have a zero, which is certainly the case when A c 0 , but it also happens when that right side has a minimum less than zero. The condition for that is easily seen to be 27C2 < 4k(3/A), which by (3) - (5), corresponds to the region below the locus

shown as a dotted line in Figure 1. Now for the mathematics of the Milne model. Looking at eq. (2). we see that when C = 0 (no gravity) and A =0 , there is only one non-static solution: k = - 1. R = ct. This is a uniformly expanding, negatively curved, infinite universe, with FRW metric (1) specialized to

where dm2 = dB2 + sin28dp2 is the meuic of the unit sphere. Milne realized that because of the absence of gravity and of A this model must live in Minkowski space. Indeed, if we set ?=tcoshy, r=ctsinhy, (10) the metric (9) becomes Minkowskian:

The worldlines of the substratum -- y = const -- now become the bundle of straight lines

48

WOLFGANG RINDLER

filling the forward light cone r = d&

with vertex at the big bang r = t^ = 0 (R = 0).

Every time-cut through this bundle, t" = const , yields a 3-sphere of radius ct^: Milne's "infiiite" universe now appears as a finite ball of matter expanding uniformly from zero radius at t" = 0! It is my contention that even the most general "infinite" FRW universes can be similarly regarded as expanding finite balls -- though this time not with flat interiors -ie,, they all pennit, l i e the Milne universe, a space l i e foliation into finite ball-lie sections. For this purpose 1 consider a typical k = -1. A = 0,nonempty FRW universe (2), choosing the radiative equation of state, since our main interest lies in the vicinity of the big bang:

k2= l = a 2 / R 2 ,

a2 =tnGpR4,

R-6

(13)

in units making c = 1. This universe I then Embed in & f dimensional Minkowski space M5,essentially following a method originally developed by Robertson [5]. The usual embedding procedure is to introduce a redundant fifth variable (here denoted by 0)into the metric, in hopes of making it five-dimensionally flat (M5).The relation of that fifth variable to the others determines a hyper surface, which represents the original space as a subspace of M*. Here our FRW universe has the metric

with R given by (13). A useful trick (due to Schiicking) is to replace t by R as coordinate, using (13), whereupon (14) becomes

This suggests defining

which has the effect of replacing the last term in (15) by -du2 . Finally we follow Milne's procedure (lo), anddefine t^ and r by

i=RcoshY, r=Rsinhy thus converting (15) into

which evidently represents MS. Our FRW universe is now embedded therein as the conoidal hyyersurface


49

which osculates tile big-bang light cone iZ - rZ = uZ, just as does the Milne universe. However, we have here mapped the FRW universe twice: once into the "front" half (u > 0) of the conoid. once into its "back half (u < 0). The null lines (u= 0) do not belong to the map. And now it is seen at once that successive sections t = const represent successively bigger balls. of surface area 4zRZ, and that the big bang was a point-event. As for the last topic of my lecture, the horizon enhancement in inflationary universes, the only mathematics 1 omitted earlier was the proof that the particle horizon in FRW universes near the big bang is given by the formula

now again working in full units. Consider the FRW metric in the form (1 4), where the sin h term -- characterizing k = -1 universes -- is irrelevant for our present purposes (sinha is replaced by sin@if k = 1, and by yl if k = 0 ). Radial light signals in any case satisfy d~ = kcdtlR . A "creation" light signal, emitted forward at the big bang (i.e., at t = 0) will therefore satisfy

If this integral converges, i-e., if it yields a finite value for w-that light signal is still within the substratum at time t and determines, in fact, the particle horizon. For its proper distance r,, = Ryl from the origin at an early time t, we have, utilizing (13),

'1 cdt rpH= R I J - = R , O R

R~ cdt

J-=R, oRR

cdR /-=cR,2/a o

a

Dropping the index 1 and substituting for a we then obtain (19).

WOLFGANG RINDLER Acknowledgements

I dedicate this paper to Jean-Pierre Vigier on the occasion of his eightieth birthday. And special thanks to Edmond Chouinard for typesetting my manuscript. References 1. C.W.Misner, K.S.Thorne, J-A-Wheeler, Gravitation, Freeman, San Francisco, 1973 2. H.C.Ohanian. R.Ruffini. Gravitation and Spacetime. 2nd. ed.. Norton. New York 1994 (This reference also deals with inflation.) 3. W.Rindler, Essential Relativity, 2nd.ed.. Springer. New York, 1977 (This reference describes the Milne model in detail. Also the phase diagram.) 4. S.Weinberg, Gravitation and Cosmology, Wiley, New York, 1W2 5. H.P.Robertson, Proc. Nat. Acad. Sci. 15(1929) 822.

WHITEHEAD MEETS FEYNMAN AND THE BIG BANG GEOFFREY F. CHEW Theoretical Physics G r o q , Physics Division Lawrence Berkeley National Laboratory Berkeley, California 94720, U.S.A.

Abstract. Historical quantum cosmology (HQC) is based not on matter but on a chain of local history--a chain lengthened by many local steps in each global step that expands a doublecone spacetime. The universe's forward-lightcone lower bound corresponds to the big bang while its backward-lightcone upper bound corresponds to the present. (All history occurs after the big bang and before the present.) HQC adapts continuous string-theoretical and Feynman-graphical notions to discrete Whiteheadian process. While standard physics-cosmology posits a spatially-unbounded universe of matter that (continuously) carries conserved energy-momentum, angular momentum and electric charge. in HQC only a tiny "rigid" component ("enduring process" in Whitehead's terminology) of a discrete and finite history corresponds to matter. The huge majority of history is "nonrigidly meandering" in time as well as in space and carries none of the above conserved quantities. Dense "vacuonic" history. unobservable by the scientific method. nevertheless carries conserved magnetic charge, contacts material history and participates in magnetodynamic action at a distance. One outcome is zitterbewegung for most standard-model elementary particles, leading to rest mass and collapse ofmaterial wave function. Another is probabilistic nature of predictions based solely on past material history. There is prospect of understanding gravity as outcome of interplay between magnetically polarized vacuum and matter. Origin of the standard model's 3 colors, 3 generations and (approximately) 30-degree Weinberg angle will be sketched and related to the internal structure ofthe photon and other "elementary" particles.

1. Introduction Alfred North Whitehead is celebrated among philosophers for representing reality not in terms of matter but through process [I]. Whitehead saw matter not as fundamental but rather as a very-special "enduring" type of process. His approach has seemed useless to the practice of a science founded on the "reproducible measurement" notion that posits matter as a priori However science within the last century has come (reluctantly) to recognize that no measurements are exactly reproducible-that none of its matter-based descriptions ofthe universe can be more than approximations appropriate to some selected scale. The huge ratio between different scales displayed by our universe accounts for the high accuracy of certain scientific descriptions. Once unavoidability of scale-based approximation in science is acknowledged, it becomes tempting to use process as basis for a mathematical model of reality more general than any scientific model. A process basis can represent non-reproducible phenomena while explaining the high accuracy exhibited by the reproducibility principle known to physicists as ''Poincark invariance". If process

51 R.L. Attloro.so et a1 (eds.),Gravitation and Co.stnologr Frmtl the H~ibbleRadius to the Plunck Scale, 51-58. O 2002 Kluwer Acadettlic Publishers. Printed in the Netherlands

G. F. CHEW patterns interpretable as "material" and others interpretable as "measurement" can be found, and if these patterns conform to the (approximate) scale-based scientific "knowledge" accumulated by mankind, then the mathematical model enjoys not only philosophical value but also scientific. I shall here describe a process model that is illuminating mysteriously arbitrary features of the standard particle-physics model. Application to wider mysteries is looming. I call the model "historical quantum cosmology" [2].HQC employs lattice coherent states (a Von Neumann invention [3]) within the direct product of 16 simple Fock spaces. [4] This basis, which exploits isomorphism between the complex-conformal group for 3+1 spacetime and the group GL(4,c) minus center, was serendipitously suggested 16 years ago by topological twistor-related remarks of V. Poenaru that dovetailed with Hilbert-space considerations known to my Berkeley colleague Eyvind Wichmann and generously passed along. A "history coherent state" is labeled by the discrete (global) age spacing between the big bang and the present together with a long but frnite chain of (local) "pre-events" each of which carries a 16-valued label comprising a complex 4-vector of inverse-time dimension and a dual (right-handed and left-handed) pair of real time-dimension 4-vectors. A pre-event's complex 4-vector label not only prescribes an impulse but locates it within a spacetime doublecone whose forward-lightcone lower bound corresponds to big bang and whose backward-lightconeupper bound corresponds to the present. The age of a pre-event is its Minkowski distance from the big-bang vertex. The Minkowski distance between big-bang and present vertices sets doublecone "size", both spatial and temporal. This size exceeds Hubble time (or length), H-' by a huge although frnite factor that reflects our distance from doublecone center. Existence of a center violates the homogeneity principle underlying standard cosmology, but mankind locates so far from the center that within our observation-accessibleneighborhood- a redshift interval of order 1o3-- homogeneity is an excellent approximation. The impulse means that at each pre-event "something happens", even though not to an "object". Impulse together with location attaches to each pre-event a phase-i.e., an "action". The impulse at a pre-event is determined through electro-magnetodynamic '-action at a distance" by the intersections of the history chain with the backward lightcone of this pre-event 151. Any history chain comprises 3 successive closed loops, each with a distinct electric-charge label (+, 0, -). A loop begins at the big-bang vertex and meanders, except for rare rigid segments interpretable as matter, throughout the doublecone interior, contacting the present boundary in "ongoing" material segments but not contacting the big-bang boundary, before returning to origin. Chain lengthening in doublecone expansion occurs at the present boundary. The dual time-dimension 4-vector label pair assigns to each pre-event electric and magnetic "pre-currents" constrained by charge labels. The magnetic-charge label maintains a single value (+) throughout all 3 loops. (The magnetic-electric asymmetry generates parity asymmetry.) By Feynman's rule, which ensures charge conservation, physical charges reverse sign during those historychain segments that retreat in age. Feynman, through his graphical representation of quantum electrodynamics, recognized that a single electron line, meandering in spacetime, can represent all electrons and positrons throughout the history of the universe. A discrete generalization of Feynman's continuous idea is a cornerstone of historical quantum cosmology. Our generalization represents by a single history chain not only all matter but also a dynamic vacuum that profoundly influences the behavior of matter. Only a tiny fraction of the history chain exhibits the rigid (straight-line) structure interpretable as matter. (Whitehead's concept of "matter" is equivalent to "inertia".) The vast majority of history, meandering in age (as well as in space) almost randomly from one pre-event to the next, is interpreted as "vacuum". Each step along the history chain is light-like and the magnitude of localage step has a fixed (scale-setting) value believed somewhat below Planck scale, but the sign of localage step may be either positive or negative. One consequence of meandering is a (spatial) density of pre-events at the scale of local step and, at much larger scales, approximate homogeneity for a

WHITEHEAD, FEYNMAN AND BIG BANG

53

"dynamical vacuum". Homogeneity of dense HQC vacuum underpins approximate Poincak invariance for dilute rigid material history. The density of material history may not exceed a limit corresponding to a spatial "parton scale" (hugely larger than the local step) that characterizes the very special rigid patterns of local history corresponding to "creation, propagation and annihilation of an elementary particle". Already at parton scale the HQC vacuum is extremely homogeneous.

2. Elementary-ParticlePropagation The special 'lower" history pattern corresponding to propagation of a standard-model zero-rest-mass elementary particle (lepton, quark, gluon or electroweak boson) comprises 4 distinct parallel segments ofthe history chain, two segments advancing in age and two retreating so net magnetic charge vanishes [6]. The tower may be described as a "4-beaded closed string" moving in discrete steps. "String tension" is provided by magnetic coulomb "attraction" between adjacent beads, of opposite-sign magnetic charge, around the closed string. Two of the 4 pre-events building a "string quartet" share the same age and same magnetic charge and are "retarded" with respect to the other pre-event pair, which carries the opposite magnetic charge and a larger age. Each of the two retarded pre-events within a string quartet lies on the backward lightcones of the two advanced preevents. The age difference between advanced and retarded quartet members defines parton scale. Transverse tower extension ("string radius") is determined by the value of elementary magnetic charge together with local-step magnitude and is believed to be in the neighborhood of Planck scale. A tower thus displays 3 distinct scales: (1) The longitudinal displacement between successive quartets-the universal local-age step-- is believed to be below Planck scale. (2) The longitudinal displacement between communicating advanced and retarded "halves" of the tower-defines a parton scale that sets an upper limit to elementary-particle rest masses. (3) The transverse (spatial) extension-"radius'-is believed to be near Planck scale. The ratio between parton scale and local step has been tentatively located in the logarithmic neighborhwd of 10" while (using Dim's relation between elementary electric and magnetic charges 171) the ratio between radius and local step is

lo2

. estimated to be near Standard-model chiil-fermion propagation is represented by a "pinched" tower-pre-events along the two retarded chain segments sharing spacetime locations along tower central axis, whereas vector-boson towers exhibit ("unpinched") advanced-retarded symmetry. Within a "tower half' (advanced or retarded) the wave function is either symmetric or antisymmetric under interchange of the two constituent chain segments 181. The half wave functions of a vector-boson tower are both antisymmetric while for chiral-fermion towers the retarded (pinched) half is symmetric. Fermion advanced-half wave functions match the antisymmetric half wave functions of vector bosom. An advanced quark half matches a gluon half while an advanced lepton half matches an electroweak vector-boson half. A pre-event label related to the complex 4-vector. on the chain segments building a retarded symmetric fermion half, represents the 3-valued standard-model attribute that has been called "generation". This same label within an antisymmetric half represents color. How does a complex 4vector manage to represent both color and generation for elementary particles. as well as spacetime location and impulse for general pre-events? 3. Cosmospin A complex 4-vector of inverse-time dimension is equivalent to a complex 2x2 matrix, which may be written as the product ofageneral (dimensionless) unitary matrix and a (dimension-carrying)hermitian

G. F. CHEW impulse. The former may be factored into a ("special") unitary matrix of unit determinant, an element of the group SU(2), multiplied by a unit-modulus complex number - a "phase factor", an element of the group U(1). The 4dimensional parameter space for a general 2 x 2 unitary-matrix would be the product of a 3-sphere (unit sphere in 4 dimensions) and a circle, except that "doubly-antipodal" points (antipodal on both 3-sphere and circle) are equivalent+orresponding to the same unitary matrix. The entire circle may be used but only half of the 3-sphere [4]. The compact 4-space spanned by a general unitary 2%2matrix has been called "gauge space". Requiring pre-event action to be given by a Lorentz inner product of impulse 4-vector with spacetime-location 4-vector, establishes a unique 4 to 1 mapping of gauge space onto the interior of a spacetime doublecone [4]. Four distinct gauge-space sectors map separately onto this spacetime. Each closed loop of the history chain passes once around the circle, beginning and ending at a special point in gauge space that maps onto the vertex of the big-bang (forward) lightcone. All four gauge sectors are encountered in each loop. History-chain direction distinguishes "first half' of circle, which we label "cosmospin up ', from "second half' which we label "cosmospin down". (Two of the four sectors occupy the fmt circle-half, the remaining two the second.) Mapping between gauge space and spacetime is singular at circle midpoint; a 3-dimensional gauge subspace projects there onto a single spatial location that has been called "center of universe". History-chain passage between cosmospin up and down is "passage through universe center". The special locally-enduring character of material history is incompatible with such passage, but that portion of history described by human physics locates within a doublecone region far from universe center (from which universe center is invisible). In our region the 2-valued cosmospin label on a history-chain segment like the 3-valued (+, 0, -) electric-charge label, does not change. Elementary-matter in our neighborhood, built from patterns of 3 rigidly-correlated historychain segments close to each other in spacetime and each carrying a 6-valued label, reflects distinction between gauge-space sectors that connect through universe center. Tower-half wave functions (in a 36-dimensional space) are products of cosmospin wave functions and electric-charge wave functions. Their symmetry is correspondingly the product of cosmospin symmetry and electriccharge symmetry. Gluon-half wave functions are symmetric in cosmospin and antisymmetric in electric charge while the converse is true for electroweak-boson half wave functions. The antisymmetric cosmospin wave function carries zero cosmospin and the antisymmetric electric-charge wave function carries zero electric charge. A triplet of symmetric cosmospin wave functions is responsible in antisymmetric tower halves for the particle feature called "color" and, in symmetric halves, for "generation". Symmetric electric-charge wave functions represent chiral isospin (chiral asymmetry stemming from magnetic-electric asymmetry). Irrelevant to particle physics. even though important. is the second gauge-space doubling (with respect to spacetimeta doubling that relates to meaning for "the present". The boundary separating the two cosmospin-upgauge-space sectors maps onto the present doublecone boundary, as also does the boundary between the two cosmospin-down sectors. Subtle issues ignored by science (except at the 1927 Solvay Conference) surround the concept of "present". With the excuse that we are today meeting as scientists, nothing will here be reported in this regard. If HQC survives, other meetings will hear surprising ideas about the meaning of time. 7

4. Structure of the HQC Photon Although a photon propagator tower has the general characteristics of any electroweak vector boson (each half odd in cosmospin symmetry and even in electric- charge symmetry), the photon elecuiccharge wave function has a special character allowing photon coupling to the net electric charge carried by a tower regardless of color, generation and chirality [8]. Orthogonality, between photon


55

carried by a tower regardless of color, generation and chirality [8]. Orthogonality, between photon and the other weakly-coupled (massless) elementary neutral boson that overlaps with the vector-boson isotriplet coupled to left-handed fermions, requires unambiguously a 30-degree Weinberg angle. (Acquisition of rest mass by weak bosons may change this angle.) Within each photon-tower half, one history-chain segment carries electric charge while the other is neutral. (In photon creation or annihilation, the 2 electric-charge -carrying segments transfer to other towers, while the 2 electrically-neutral segments connect to each other.) Because preevent labels on the two halves of a photon are the same, net photon charge vanishes. but the presence of electric charge as well as magnetic charge "inside a photon" is essential. Its internal magnetic charge "stabilizes" a propagating photon, allowing it to "endure'- (Whitehead's terminology), while its internal electric charge couples the photon to any matter that carries (net) electric charge. The material component of the universe thereby becomes "self observable". Direction difference between magnetic and electric pre-currents precludes electriccharge contribution to the endurance of elementary matter. Magnetic charge is responsible for the "existence" of matter while electric charge renders matter "observable". Although history patterns describable as "observations" remain to be worked out in detail, massless-photon emission, propagation and absorption are necessary ingredients, together with "classical electromagnetic fields" from electriccharge sources. Ideas developed two decades ago by Henry Stapp, [13] on the basis of Feynman's formulation of quantum electrodynamics, together with the magnetoelectrodynamicsof Reference [5], show how electric charges on the material segments of history associate to each history chain a classical electromagnetic field. I t is presumed that, in historical quantum cosmology as well as in standard physics, zero photon rest mass will be maintained by the photon's coupling to conserved electric charge.

5. Vacuons I now turn attention to a nonrigid local-history pattern very different fromatower or even from a tower h a l f -a pattern that builds the great bulk of nonmaterial history. The pattern, called "vacuon", is a single pair of history-chain segments, carrying "opposite" cosmospin and electric-chargeindices, along which successive pre-event pairs occupy the same spacetime locations (as in the pinched retarded half of a fermion propagator)[9]. Along a vacuon (in contrast to any propagator half), age advance and age retreat, together with change of spatial-step direction, occur incoherently. "Opposite" cosmospin and electric charge indices, furthermore, mean vanishing ofnet cosmospin and electric charge at each pre-event pair along a vacuon. Magnetic charge need not vanish locally, although age meandering yields zero average magnetic charge at scales large compared to local step. In an "elementary material event '- a parton-scale pattern of -10 preevents that represents a standard-model Feynrnan-graph vertex-- a vacuon may be "absorbed" or "emitted" by a material history-chain segment, "transmuting" that segment to another of opposite sense but nevertheless carrying the same observable material quantum numbers. Rigid material history provides sources and sinks of meandering vacuum history. Although, as the doublecone universe expands, the rigidity of material-history ageing requires diminishing matter density, vacuum history by meandering in age can maintain constant spatial density. Beyond direct contact with material history, vacuum history "at a distance" magnetodynamically affects material action (as well as vacuum action) through those vacuon preevent pairs at which magnetic charge is nonzero. It is believed that rest mass and material wave-function collapse result from direct contact between vacuum history and matter, while gravity results from vacuum action at a distance. 7

G . F. CHEW 6. Rest Mass from Vacuum-Induced Zitterbewegung The simplest material event within a history is "trivial"invo1ving a single (zero-rest-mass) elementary particle whose spatial-direction ofpropagation is reversed through a direct vacuum-history contact that endows the material wave function with a phase decrement (MED action) of order unity [9]. Accumulation of negative phase through a succession of many velocity reversals is presumed, following Dirac,[lO] to be phenomenologically equivalent (at scales far above parton scale) to rest mass. Dirac, through his celebrated equation for spin-112 propagation (where the velocity operator is

y

interpreted chirality reversal ( Y

7 -) as (maximum-velocity) spatial-direction

reversal at fixed momentum and spin that with repetition amounts (at "large scale") through quantum superposition to propagation with nonzero rest mass. Historical quantum cosmology accepts, not only for chiral fermions but also for vector bosons, such "zitterbewegung" meaning for "elementary" rest mass, attributing the "source" of zitterbewegung to direct contact between matter and vacuum history. Rest-mass magnitude (which must lie below the inverse of parton scale) is expected to depend on particle "structure" partly through trivial-event phase and partly through interplay between particle structure and vacuum structure. The phenomenological wave-function collapse model publicized by John Bell indicates correlation between rest mass and rate of collapse [l 11. It is anticipated that quantum superposition of different history chains--a superposition providing meaning for material energy-momentum and angular momentum--will not only confirm Dirac's meaning for rest mass but show that contact between vacuum history and matter is a source of material-wave-function collapse. Dense dynamicalvacuum history is a promising "environment" to induce material decoherence.

7. Gravity Although the nonrigidity of vacuum history precludes it carrying energy-momentum in a material sense, there is vacuum-vacuum and vacuum-matter magnetic-coulombaction at a distance. (Material source of magnetic coulomb action is screened outside tower radius.) Vacuum history that is homogeneous in absence ofmatter (in the sense that magnetic charge fluctuates randomly at the scale of local step) may be "polarized" by presence of matter. Because large-scale inhomogeneity of vacuum would affect the large-scale behavior of matter, speculation is irresistible that gravity is a manifestation of large-scale magnetic-charge interaction between matter and polarized vacuum. Before verification of such an origin for gravity, precise meaning for material energymomentum and angular momentum must be given through ("quantum'? superposition of different rigid history patterns according tounitary infinite-dimensionalLorentz-grouprepresentations [12]. Although meaning is apparent in outline, precision remains to be achieved. One obstacle is the coordination of local step with global step. The global doublecone expands in discrete age steps-"global occasions"-- that, while huge on parton scale, are small on the scale of human consciousness and relate to "measurement" in Copenhagen quantum mechanics. accommodating the "observable" discrete-process notion of nuclear or atomic "transition". (The scale of such transitions, although large on parton scale, is smaller than the global step.) A sample unresolved question, relevant to superposition of different histories, is

-

whether a parton-scale "elementary material event" (comprising 1 ()I8 pre-events) may overlap the boundary between successive global occasions or must be complete within a single such occasion.


Expsnsion t

Figure I . Doublecone Spacetime (Compact).

Figure 2. "Action at a distance". Impulse at a pre-event is determined by the magnetic and electric precurrents at intersections of the history

"rrdius"(-l~-!~ ? time parton a+ unrts) * a -. -.,

-7

Advanced... .' .

..

.. ..

.

.

.:

parton scale :

"

0.5

(15)

The density ofrandom walk stopovers will be lower than in the case of flat space. The ideas presented here on surface curvature and random walk are inspired by George Garnow [14]. In this Lucid, admirable exposition of curved space geometry he argued that the analysis applicable to two-dimensional space will equally hold true for a three dimensional space, that is in other words for our universe. A random walk and a Brownian motion are Frequently used under equivalent contexts. The scaling properties observed in a one-dimensional random walk having steps determined by the distribution ofrecords in time of the converted Johnson electronic noise (equation 4), were shown above (equations 13 & 15) to also account for three dimensions as in the case of the distribution of matter in the universe. In both processes extending at diverse scales from each other, the world of the very large and the world of the very small, the representation of their Geometries are fabricated by an underlying flat 'space' (H=0.5)

=

4

area ,as if the surface was assumed to be flat.

L

The same letter H is used as in fBm for simplicity of comparison.

A RANDOM WALK IN A FLAT UNIVERSE TABLE 2. universe.

101

Similarities between the geometry of 'space' in (a) the fBm process and (b) the

fBrn

Antipersistent Persistent Ordinary Brownian motion

H exponent

Radius curvature

H 0.5

positive negative

Geometry space spherical hyperbolic

H = 0.5

infinite

flat

of

of

displaying localized regions of curvature ( H + 0.5 ). At the cosmological scale the deviations from a flat space are the consequence of the presence of massive bodies and curve physical space spherically, while in the case of the electronic noise process the deviations result from an inherent chaotic element and curve their representational 'space' (described by equation 4) hyperbolically.

Appendix Consider a circular area Af of radius R=2, see figure 1,

Af = zR2(= 12.566)

(A-1) curved by shrinking into a spherical cap on a sphere of radius r. Let us suppose

that r, be such that the area of the hemisphere, Ah: A 2), equals the original area of the circle:

therefore, related through: will bed, (equation 11):

R=r f i

A = Ah

f

= 2m 2 (= 12.566)

(A-3). The two radii are,

6

The area of the spherical cap As

= X R ~-0.844 (= 10.606)

(A-6), and in this

A, = X R .0.844 ~ (= 10.606) 0 (A-7).

example is equal to:

(A-

( r r =~ 2m ~ 2 ) 0 (A+, then the angle

8= 6 ( m d ) 0 (A-5).

through equation 7, 8, A-3 & A-5 is: A,

h

The extent of

Af -As

= 1 -0.844 = 0.1 56 (A-8), that is, about 16% of Af the original area, due to curving. This could have occurred by a mass of the

shrinking is therefore:

sphere M = 6.55.10Z6kg comparable to the mass of saturn5. Suppose that No=100 stopover points were expected to grow on A by an ordinary random walk. After shrinking, the circle becomes the spherical cap of area A, (< Af). The number of stopovers on the original area d2(now the area of the 4 5

since from the geometry of this example r$ = R a, I $ = ~ f and i $ =f i Assuming that the excess radius due to the bending of the surface Af tO A,

is

R-JR.IR=GM [s]. 3c2

.

102

FOTINI PALLIKAFU

hemisphere) has increased to the number 3:N, =-= loo 0.844

118

N,, as discussed in section

(A-9). Being unaware of the surface curving we would

think that our initial law: N,(= 100) =c R2 ,(R =2) (A-10) (where C is a constant equal to 10014 in this example) has changed into N,(= 118) = C R ~ (A-1 . ~ 1). The ~ number of stopovers grows now faster with

.

range than in the case of the plane, or the range R varies with N as: R oc N 0.45 (A-12), noting that the exponent takes values below what is expected for flat space, (H= 0.5).

Acknowledgements The electronic noise data were generated by Emil Boller and Holger Bosch of IGPP, Germany. Petros Belsis of Athens University created the original Hurst analysis software. A comment in a public e-mail posting by Jack Sarfatti has triggered my interest in the Hurstanalysis. 'This work was funded by the special research account of Athens University.'

References 1.

De Bernandis P et al, (2000) A flat Universe from high-resolution maps of the cosmic microwave background radiation. Nature 404. 955-959 2. Hannany S et al, (2000) MAXIMA-1: A Measurement of the Cosmic Microwave Background Anisotropy on Angular Scales of 10 arcminutes to 5 degrees. Submitted to Astrophys. J. Lett 3. Wittman DM, Tyson JA, Kirkrnan D. Dell'Antonio I, & Bernstein G (2000). Detection of weak gravitational lensing distortions of distant galaxies by cosmic dark matter at large scales. Nature 405, 143-148 4. See for instance, Eddington AS (1920). Report on the Relativity Theory of Gravitation: Fleetway Press Ltd, London, and also Ciufolini I, Wheeler JA (1995). Gravitation and Inertia: University Press. Princeton. Feynman RP: (1963) Lectures on Physics. Prepared for publication in 1997 by Leighton RB and 5. Sands M, under the title: "Six Not So-Easy Pieces, Einstein's Relativity, Symmetry and Spacetime". Addison-Wesley, USA. 6. Mandelbrot, B. B. and Van Ness J. W. (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev.. 10.4: 422-437. 7. Mandelbrot, B.B. (1982). The Fractal Geometry ofNature. (W. H. Freeman, San Francisco). 8. Mandelbrot, B.B. (1999). Multifractals and 11fNoise. Wild Self-Affinity in Physics. (Springer, New York). 9. Pallikari, F. (2000) A Study of the Fractal Character in Electronic Noise Processes Chaos, Solitons & Fractals, 10 (8). in press. 10. HursL I-L A. (1951) Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng., 116:

770-8a. 11. Feder J. (1988) Fractals. Plenum Press New Yo&. 12. Mandelbrot, B.B. (1999). Multifractals and 11fNoise. Wild Self-Affinity in Physics. (Springer. New York). 13. M. Schroeder (1991). Fractals, Chaos, Power Laws. Minutes from an Infinite Paradise: W. H. Freeman and Company, New York. 14. Gamow G (1993). Mr. Tompkins in Paperback: University Press, Cambridge

MULTIPLE SCATTERING THEORY IN WOLF'S MECHANISM AND IMPLICATIONS IN QSO REDSHIET SISlR ROY * & S.DATTA** Physics & Applied Mathenlatics Unit Indian Statistical Institute Calcutta 700035. India *[email protected] **[email protected]

Abstract. The theory of Correlation-induced spectral changes is becoming popular over the past decade and it has established its possibility in the field of QSO redshift. A brief review of its development including multiple scattering, no blueshift condition, effect on spectral width and the correlation between shift and width of a spectral line is presented. 1. Introduction Today one of the most controversial topics of astrophysics is the nature of quasar redshift. Some questions are raised against the well established BBC(Big Bang Cosmology)-black hole school of thoughts as follows: 1. The presence of a black hole, or of dark matter in general, is inferred From the study of the dynamics of the environment and it is expected that a black hole would pull matter, which will fall towards it. But unfortunately, in reality, we have yet to get a spectroscopic evidence of such inward flow. 2. The maximum luminosity that can be sustained by an accreting source is the E d d i i o n luminosity [ Kembhabi & Narlikar 1, given on the assumption that the size of the emitting region is hundred times the Schwarzschild radius: 10 41 ( erg ) sec - I 1 sec1 For time variations observed on the scale of hours we should get z at the order of lo3 and thus arrive at a maximum luminosity of order erg sec-', The black hole in this case has to have a mass lo6 M,, M, being the solar mass. The most luminous quasars are however, are at least hundred times as powerful. Thus if the above scenario is correct, then we should not expect the time variability of the order of hours that is actually observed in some quasars. 3. Basically, the violation of cosmological hypothesis can be demonstrated by showing the existence of two extra-galactic sources in close proximity but with different

-

103 R.L. Amoroso et a1 (eds.), Gravitation und Cosmolog!: From the Hubble Radius to the Planck Scale, 103-1 10. O 2002 Kluwer Academic Publishers. Printed in the Netherlandr.

redshifts. The cosmological hypothesis requires a unique relation between redshift z and distance D.

=f(D), say, and hence if close neighbors have the same value ofD, it is expected that their redshifts will be the same. An example of Quasar-galaxy pair is described below to explain the above. A galaxy (NGC 4319) and a quasar (Markarian 205) are connected by the straight luminous link. However the redshift of the quasar is approximately twelve times greater than that of the galaxy, and consequently, according to Hubbies law the quasar should be twelve times further from Earth than the galaxy [ WolL 19981. To solve this discrepancy, several explanations have been proposed specially to interpret redshift other than due to Doppler effect. Some of them are stated below. 1. Recently a mechanism has been proposed by L. Accardi and his coworkers modeling the interstellar medium by a low density Fermi gas [Accardi et al, 19951. They proposed a general redshift theorem. 2. HArp has shown, over the past two decades, that the quasars appear to cluster near normal galaxies [A rp. 19871, 3. Another unexplained ( perhaps circumstantial ) result is by William Tifft who has suggested that redshifts are quantized and they are whole multiples of 72 km /sec! [Tifit, 19871. But all of the above theories are neither well explained nor have any scope to be tested experimentally. After several such attempts to explain the redshift alternatively, a mechanism was proposed by E. Wolf in the mid-eighties [ WolL 19871 that has no connection with the relative motion and gravitation. The main features of this new mechanism for redshift, called the Wolf Effect are as follows. The Wolf effect is the name given to several closely related phenomena in radiation physics dealing with the modification of the power spectrum of a radiated field due to spatial fluctuations of the source of radiation [ Jarrzes, 19981. It was shown [ Jarnes & Wolf 19901 that if an incident spectral line is of Gaussian profile and the scattering medium is also Gaussian, then the scattered field is also approximately Gaussian, but with different mean frequency w ' and standard deviation aO' , The changes are as

0

follows.

where w and g are the mean frequency and the width ofthe incident spectral line, and 0 0 a, a', p are medium parameters which mainly determine the angle of scattering. Consequently the relative frequency shift z is given by

MULTIPLE SCATTERING IN QSO REDSHIFT z

=

O'o

- a'+i!ToZ (aa'-@) -I

IPI

Let us now assume that the light, in its journey, encounters many scatterers described as in the previous chapter. What we observe at the end is the light scattered many times, with an effect as that stated in the last section of the previous chapter in every individual process. Further we assume that the scatterers are weak and incoherent in nature, i.e., scattered rays do not interfere with the incident ones. This assumption leads to the validity of first order Born approximation in the case of multiple scatterers too. The physical justification of this assumption lies in the fact that the dimension of the scatterers is small enough compared to the cosmological distance scale.

2. Effect Of Multiple Scattering On Broadening Of Spectral Lines Since the z-number due to such effect is Doppler-like (i-e.,does not depend on the central frequency of the incident spectrum), each zi depends on 6. r - 2 only, not on ru.r - 1 . Let us calculate the broadening of the spectral line after N number of scatterings [ Datta et. al., 199CIB 1. From the 2d equation of (1), we can easily write,

and from the 1 equation of (1) --

%+I

=

a' + (aa' - p2)2in2

Let us assume that the redshift per scattering process is very small, i.e., O < E = Z ~ 0. the average energy picked up due to the fluctuating impulses from the radiation should roughly average out to equal the energy loss due to a velocity dependent "drag" force. Einstein and Hopf first investigated these aspects [36]. Boyer noticed, however, that at T = 0, the usual "drag force" is necessarily absent, thereby leaving the effect of the fluctuating radiation impulses due to ZP radiation uncompensated [37-381. An average continual increase of velocity was then predicted, unless collisions with other matter occur to dissipate the increased kinetic energy. Rueda subsequently proposed [33] that these effects might be observable in astrophysical processes. The extensive review in Ref. [34], and the more recent reference [35], contain relevant work and references related to Rueda's investigations. In particular, though, it seems possible that this "secular acceleration" mechanism may contribute, or even be a very important aspect of, phenomena such as cosmic rays, cosmic voids, and the observed X-ray and gamma ray backgrounds. Thus, here we have yet another possible dramatic consequence of the existence of radiation being present at T = 0, i.e., ZP radiation. Naturally such possible phenomena brings up many issues on whether basic thermodynamic concepts and laws are being violated, such as the first and second laws. These questions were investigated in Ref. [31].

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The conclusion was that, no, if ZP radiation does indeed contribute to cosmic ray formation, then the secular acceleration phenomena should not violate either the first or second law of thermodynamics. Rather. the situation of "free" charged particles without a "container," situated in an environment with a near ZP-like spectrum. is really a system out of thermodynamic equilibrium. Indeed. if ZP-like fields are contributing to cosmic ray formation, then the particles being accelerating across the long "empty" (low mass regions) of intergalactic space, are clearly not in any sort of thermodynamic equilibrium with the matter they eventually encounter and strike. If secular acceleration is indeed the main contributor of cosmic rays, then the apparent free energy that has been acquired by these very high velocity particles is due to a finite change in the enormous amount of electromagnetic energy available in space. Energy can certainly still be conserved, as the kinetic energy picked up by the particles is lost by the radiation [27, 311, on average, but can be returned to the radiation upon the particle undergoing a deacceleration and colliding with other matter. As for the second law, at first glance it does indeed appear to be violated if ZP radiation provides secular acceleration effects, as the particles appear to be, roughly speaking, extracting energy from a heat reservoir. Clausius' statement of the second law is [14], "No process is possible whose sole result is the transfer of heat from a cooler to a hotter body." \However, a closer analysis of this phenomena reveals that this statement is not violated by this phenomena, since it is important to consider changes in thermodynamic equilibrium states. If one had a large enough container, so that the particles hit the walls, then it would be necessary to examine many traversals of such a system to really contemplate a system in equilibrium. The secular acceleration effect really only has to do with the average behavior of particles at different points in their trajectory. Upon averaging o v a the trajectories. the particles both pick up and lose energy, via collisions. in a natural stochastic behavior that needs to be considered in its totality. This point and related ones. are discussed in more detail in Ref. [31]. However, one point should be brought out that was not fully emphasized in Ref. [31]. At a certain level, we cannotjust take one system at some temperature T, and another system at the same temperature T, and put them together, and expect there to be "no changes." This is somewhat contradictory to our intuition, since we have long learned the "zeroth law of thermodynamics." namely: "Two systems in thermal equilibrium with a third are in thermal equilibrium with each other." Macroscopically. of course, that is what we see. For example, suppose we have a thermometer. or temperature gauge, and two substances. Suppose we can put the thermometer in either substance and never see a change in the thermometer's reading. We would then say that via the zeroth law, all three are at the same temperature, However, two systems that can effect each other always have an interaction energy. Separate systems, nominally at some temperature, will change in subtle and sometimes not so subtle ways as they are brought into contact with each other. At the very least, a microscopic thermocouple or small thermometer will experience van der Waals or Casimir-like forces as it is brought close to a substance that is nominally at the same temperature. If we were to take an atomic force microscope probe, and have it "walk" around the surface of a cavity, where the probe and the walls of the cavity are all nominally at the same temperature, the probe will experience changes in forces as it moves closer and farther from the walls, and as it examines a wall of one material versus

CONNECTIONS BETWEEN PROCESSES

L21

another. Contrary to what one might intuitively suspect, such a probe can distinguish the different types of material forming the walls of a blackbody cavity. This is a quite different situation than what Kirchhoff, Wien, and Planck [I] originally described in their analysis, where the ra ation and radiation pressure were treated as being independent of the material of the walls of the cavity. My reason for making these points is that it may be critically important to consider the concept of thermodynamic equilibrium as being the equilibrium state of systems in net equilibrium with each other. Probes, particles, radiation, etc., all need to be treated as part of the net system, and need to be analyzed as being in interaction with each other, even when they are in thermodynamic equilibrium with each other. Having a cavity of electromagnetic thermal radiation, without ionized atoms present in the cavity, can in many important ways be quite different than the situation where the ions are di present, as we know from plasma studies, For systems not in thermodynamic equilibrium, the system is yet far more complicated. Extrapolating these ideas to the whole universe requires a careful examination of the basic assumptions, as will be discussed more in the following two sections. 6. Increasing Expansion Of The Universe A fascinating phenomena has been observed and reported in recent years, that has

attracted considerable attention in the astrophysical community. A major turning point occurred in 1998 [39-401, when two teams of astrophysicists reported on new studies involving the luminosity of a particular type of supernova in nearby and distant galaxies. The results provided surprising evidence that not only is the universe not slowing down its rate of expansion since the Big Bang, but the expansion is actually increasing its rate. In ad tion, very recent reports this year on the mappings of tiny fluctuations in the cosmic microwave background from the experiments BOOMERANG (Balloon Observations of Millimeter Extragalactic Radiation and Geophysics) [41] and MAXIMA (Millimeter Anisotropy Experiment Imaging Array) [4243] have provided additional evidence that the universe is flat. Thus, we now have several sets of confuming evidence that the universe is expanding, and apparently at an increasing rate. However, the density of all visible matter and dark matter in the universe appears to be only about one third of what is needed to account for a flat universe, where expansion will continue forever. Many physicists are proposing that vacuum energy is a key component of the puzzle for understanding this phenomena. At a recent meeting of the American Astronomical Society (AAS 196, June 2000) in Rochester, N.Y., a team of astronomers reported they have new evidence for what makes up most of the mass of the universe, based on a survey of the redshifts of 100,000 galaxies. According to their analysis of this mapping of galaxies' redshifts, called the Two Degree Field (2dF) Galaxy Redshift, a universe composed of 213 vacuum energy contribution and 1/3 visible and dark matter contribution. would fit the observed astronomical data well [44].

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7. Concluding Comments On The State Of The Universe And Zero-Point

I will end this article with a few cautionary comments. In addition to well recognized microscopic phenomena, such as van der Waals and Casimir forces and the Lamb shift, which are commonly attributed to the effects of electromagnetic ZP radiation [lo, 171, we have what appear to be very much macroscopic consequences of ZP fields. In particular, it appears that "zero-point" fields may play a role in astrophysical phenomena and cosmology. Without doubt, this comment is certainly speculative at this point, as both the theoretical and experimental aspects still require deeper investigation. However, an increasing number of physicists are beginning to examine the possible role of "ZP fields," particularly in light ofthe experimental data obtained during the past two years. One reason for spealung with caution about the possible role of ZP fields in astrophysics is undoubtedly the one most frequently cited, namely, the famous cosmological constant problem [45]. Formally, the vacuum is supposedly infinite in energy, although most researchers feel there must be some sort of effective cutoff [20]. Nevertheless, even if huge, how one can reconcile the enormous energies to the ones needed to provide some gravitational effects, but not enormous ones, is by no means clear. However, another reason for speaking with caution here, is one that I have not seen mentioned elsewhere, and which brim@ us back full circle to the i n i i l d i m i o n of this article on the thermodynamics of blackbody radiation. As mentioned in Sec. II, in quantum mechanics, "zero-point" is now commonly accepted as meaning the "zeroth," or lowest, quantized energy level of a system. In contrast, "zero-point" historically first referred to the properties of a system at absolute zero temperature. Specifying a temperature. even at T = 0, implies the system under discussion is in thermodynamic equilibrium. Our universe is not in thermodynamic equilibrium: indeed, in many ways. it is far from being in equilibrium. Thus, just as it was important to more carefully examine the meaning of ZP in the early classical analysis of blackbody radiation, so also it may be important to be careful regarding introducing "zero-point" notions in regard to the entire universe. Acknowledgments This work was supported in part by the California Institute for Physics and Astrophysics via grant CIPA-CD2999 to Boston University.

Annotated References [l] M. Planck, The Theory of Heat Radiation (Dover, New York, 1959). This publication is an English translation ofthe second edition of Planck's work entitled Waermestrahlung, published in 1913. A more recent republication of this work is Vol. 11 of the series The History of Modern Physics 180-1950 (AIP, New York, 1988). [2] See, for example, Ref. [3],pp. 246 and 247, or the fwtnote on p. 73 in Ref. [4]. [3] T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894-1912 (Oxford University Press, New York, 1978). [4] L. Pauling and E. B. Wilson, Jr., Introduction to Quantum Mechanics (Dover. New York, 1985). [S] See, for example, pp. 134 and 170 in the undergraduate textbook of Ref. [6] or, p. 69 in the graduate

CONNECTIONS BETWEEN PROCESSES textbook of Ref. m. [q A. P. French and E. F. Taylor. An Introduction to Quantum Physics (Norton. New York. 19783. [7] L. I. Schiff, Quantum Physics, 3rd ed. (McGraw-Hill, New York, 1968). [8] The traditional reasoning for deducing the Stefan-Boltzmann relationship is widely available and can be found, for example, in Part 11, Chap. I1 in Ref. [I], Appendix XXXIII in Ref. 191, pp. 5 4 in Ref. [3], p. 2 in Ref. [lo], or Prob. 9.10 on p. 399 in Ref. [Ill. For a close description of the underlying assumptions, Ref. [llis particularly helpful. [9] .M Born, Atomic Physics, 8th ed. (Dover, New York, 1969). [lo] P. W. Milonni, The Quantum Vacuum. An Introduction to Quantum Electrodynamics. (Academic Press, San Diego, 1994). [l 11 F. Reif. Fundamentals of Statistical and Thermal Physics (McGraw-Hill. New York. 1965). [12] D. C Cole, Phys. Rev. A 45, pp. 8471-8489 (1992). [13] For example, Planck writes on p. 2 in Ref. [I], Only the phenomena of diffraction, so far at least as they take place in space of considerable dimensions, we shall exclude on account of their complicated nature. We are therefore obliged to introduce right at the start a certain restriction with respect to the size of the parts of space to be considaed. Throughout the following discussion it will be assumed that the linear dimensions of all parts of space considered, as well as the radii of curvature of all surfaces under consideration, are large compared with the wave lengths of the rays considered." Other similar statements are made elsewhere in his treatise in his coverage of the early thermodynamic investigations of blackbody radiation, including Kichhoffs law, the St'efan-Boltzmann law, and the Wien displacement law. [14] M. W. Zemansky and R H. Dittman, Heat and Thermodynamics. 6th ed. (McGraw-Hill. New York. 1981). [I51 D. C. Cole, Phys. Rev. A 42, pp. 7006-7024 (1990). [I@ H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, pp. 793-795 (1948). [17]L. de la Pena and A. M. Cetto, The Quantum Dice - An Introduction to Stochastic Electrodynamics (Kluwer, Dordrecht, 1996). A review of this book is given in D. C. Cole and A. Rueda, Found. Phys. 26, pp. 1559-1562 (19%). [I81 J. C. Slater, Microwave Electronics, (D. Van Nostrand Co., Inc., New York, 1950). 1191 H. B. G. Casirnir, Philips Res. Rep. 6, pp. 162-182 (1951). [20]R. P. Feynman and A. R. Hibbs. Quantum Mechanics and Path Integrals (McGraw-Hill. New York 1%5). p. 245. [21lT. H. Boyer, Phys. Rev. A 9, pp. 2078-2084 (1974). 1221 P. W. Milonni, R. J. Cook, and M. E. Goggin. Phys. Rev. A 38. pp. 1621-1623 (1988). [23] D. C. Cole, Found. Phys 30 (ll), pp. 1849-1867 (2000). [24] A similar subtle restriction is made in Planck's analysis in [I] following Eq. (124) on p. 91: this restriction is invalid if ZP radiation is to be considered. [25] D. C. Cole, Phys. Rev. A 42, pp. 1847-1862 (1990). [26] D. C. Cole, Phys. Rev. A 45, pp. 8953-8956 (1992). [27] D. C. Cole, "Reviewing and Extending Some Recent Work on Stochastic Electrodynamics," in Essays on Formal Aspects of Electromagnetic Theory (refereed compendium), ed. by A. Lakhtakia. (World Scientific. Singapore, 1993). [28] D. C. Cole, Found. Phys. 29, pp. 1819-1847 (1999). [29lD- C. Cole, "Relating Work, Change in Internal Energy, and Heat Radiated for Dispersion Force Situations," Proc. Of Space Technology and Applications International Forum -- 2000 (STAIF 2000), AIP 504, ed. by M. S. El-Genk, pp. %0-967 (2000). 1301 C. Cole and H. E. Puthoff, Phys. Rev. E 48 , pp. 1562-1565 (1993). [31] D. C. Cole, Phys. Rev. E 51, pp. 1663-1674 (1995). 1321 D. C. Cole, "Energy and Thermodynamic Considerations Involving Electromagnetic Zero-Point Radiation," Proc. of Space Technology and Applications International Forum -1999 (STAIF 99), AIP 458, ed. by M. S. El-Genk. pp. 960-%7 (1999). [33] A. Rueda, Nuovo Cimento A 48, 155 (1978). 1341 A. Rueda, Space Sci Rev. 53,223 (1990). This review contains an extensive set of relevant references. 1351 A- Rueda, B. Haisch, and D. C Cole, Astrophys. J. 445, pp- 7-16 (1995). 1361 A. Einstein andL. Hopf, Ann. Phys. (Leipzig) 33, 1096 (1910); 33, 1105 (1910). 1371 T. H. Boyer, Phys. Rev. 182, 1374 (1969). 1381 T. H. Boyer, Phys. Rev. A 20, 1246 (1979). [39] S. Perlmutter, G. Aldering, et al., Nature 391, pp. 51-54(1998). 1401 B- P. Schmidt, P- Challis, A.V. Filippenko, d aL, Astrophysical Journal 507 (Nov. 1, 1998). [41] Hu, Nature 404, pp. 939-940 (2000).

DANIEL C. COLE [42] Hanany, et al., "MAXIMA-I: A Measurement of the cosmic microwave background anisotropy on angular scales of 10' to 5 degrees," http://xxx.lanl.gov/absl number 0005 123. [43] A Balbi, et al., "Constraints on cosmological parameten from MAXIMA-1," http://xxxlanl.govlabsl number 0005 124. [44] Additional information about the 2dF Galaxy Redshift Survey is available at Digital Sky Survey, at http://www.sdss.org. For related information, see other background information, see preprint by M. Turner, http~/xxx.lanl.gov/abslastro-phl9811364(to be published in Proc. Astron. Soc. Pacific, Feb. IW), "Cosmology Solved? Quite Possibly!" [45] Weinberg, "The cosmological constant problem," Rev. Mod. Phys. 61 (I), pp. 1-23 (1988).

NEW DEVELOPMENTS IN ELECTROMAGNETIC FIELD THEORY BO LEHNERT Alfven Laboratory Royal Institute of Technology SE-100 4 4 Stockholm, Sweden

Abstract Conventional electromagnetic theory and quantum mechanics have been very successful in their applications to numerous problems in physics. Nevertheless there are questions leading to difficulties with Maxwell s equations which are not removed by and not associated with quantum mechanics. As a consequence, several new developments have been elaborated which both contribute to the understanding of so far unsolved problems, and predict new features of the electromagnetic field to exist. In this review some examples are first given on the problems which arise in conventional theory. This is followed by short descriptions of the main characteristics of modified field theories being based on the hypothesis of additional electric currents in vacuo,as well as on more generalized forms including magnetic monopoles and a unification of electromagnetism and gravitation. The new aspects which result from these extended theories are manifold, including both time-dependent states with wave phenomena and timeindependent states with electromagnetic equilibria. The former states are discussed with emphasis on plane waves, axisyrnmetric models of the individual photon with its integrated parameters and unified wave particle concepts, the possible existence of a nonzero photon rest mass. the thermodynamics of a photon gas. the transition of a beam of photon wave packets into a plane wave. and superluminous phenomena including the basis of tachyon theory. The latter states are discussed with emphasis on particle-shaped electrically neutral and charged equilibria providing possible models for neutrinos and charged leptons, features of the electron behaving nearly as a point charge, string-shaped equilibria, and instantaneous interaction at a distance including longitudinal field components. 1. Introduction Conventional electromagnetic field theory based on Maxwell's equations and quantum mechanics has been very successful in its application to numerous problems in physics. and has sometimes manifested itself in an extremely good agreement with experimental results. Nevertheless there are are as within which these joint theories do not seem to provide fully 125 RL Arnoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 125-146. C2002 Kluwer Academic Publishers. Printed in the Netherlands.

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adequate descriptions of physical reality. Thus there are unsolved problems leading to difficulties with Maxwell's equations which are not removed by and not directly associated with quantum mechanics [1-2]. Due to these circumstances a number of modified and new approaches have been elaborated during the last decades. Among the reviews and conference proceedings describing this development, those by Lakhtakia [3], Barrett and Grimes [4], Evans and Vigier [5]. Evans et al. [6-7], Hunter et al. [8] and Dvoeglazov [9] can be mentioned here. The purpose of these approaches can be considered two-fold: •

To contribute to the understanding of so far unsolved problems. To predict new features of the electromagnetic field.

The present review summarizes these new developments in electromagnetic field theory. After a description of some cases where standard theory fails to give fully adequate solutions, a number ofmodified field theories will be shortly reviewed, then followed by the aspects which result from these developments. Within this review it has not been possible to dojustice to all the relevant contributions, and the references have mainly been concentrated to some more recent investigations.

2. Unsolved Problems in Conventional Electromagnetic Theory There are a number of cases which illustrate the failure of standard electromagnetic theory based on Maxwell's equations: Light appears to be made of waves and simultaneously of particles. In conventional (i) theory the individual photon is on one hand conceived to be a massless particle, still having an angular momentum, and is on the other hand regarded as a wave having the frequency v and the energy hv, whereas the angular momentum is independent of the frequency. This dualism is not fully understandable in terms of conventional theory [5]. (ii) The photon can also sometimes be considered as a plane wave, but there are also experiments which indicate it can behave like a bullet. In investigations on interference patterns created by individual photons on a screen [10], the impinging photons produce dotlike marks on the latter, as being made by needle-shaped objects. (iii) In attempts to develop conventional electrodynamic models of the individual photon, there is a difficulty in finding axisymmetric solutions which both converge at the photon centre and vanish at infinity. This was realized by Thomson [11], and later by other investigators [12]. (iv) During the process of total reflection at a vacuum boundary, the reflected light beam has been observed to be subject to a parallel displacement with respect to the incident beam. For this so called "Goos-Hanchen effect" the displacement was further found to have a maximum for parallel polarization of the incident electric field, and a minimum for perpendicular polarization [13- 14]. At an arbitrary polarization angle the displacement does not acquire an intermediate value but splits into two values for parallel and perpendicular polarization. This behavior is not explained by conventional theory.

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The Fresnel laws of reflection and refraction of light in non-dissipative media have (v) been known for some 180 years. However. these laws do not apply to the total reflection of an incident wave at the boundary between a dissipative medium and a vacuum region [15]. (vi) In a rotating interferometer fringe shifts have been observed between light beams which propagate parallel and antiparallel with the direction of rotation [4]. This so called "Sagnaceffect" requires an unconventional explanation. (vii) Electromagnetic wave phenomena and the related photon concept are even in our days somewhat of an enigma in more than one respect. Thus, the latter concept should in principle apply to wave lengths ranging from about 10e-15 m of gamma radiation to about 10e5 of long radio waves. This leads to a so far not fully conceivable transition from a beam of individual photons to a nearly plane electromagnetic wave. (viii) As the only explicit time-dependent solution of Cauchy's problem, the LienardWiechert potentials are not adequate to describe the entire electromagnetic field [2]. With these potentials only, the part of the field is missing which is responsible for the interparticle long-range Coulomb interaction. (ix) There are a number of observations which appear to indicate that superluminal phenomena should exist [16]. Examples are given by fast mini-quasar expansion, photons tunneling through a barrier at speeds larger than c, and the propagation of so-called Xshaped waves. These phenomena cannot be explained in terms of purely transverse waves resulting from Maxwell's equations, and they require a longitudinal wave component to be present in the vacuum [17]. (x) It has been found that Planck's distribution law cannot be invariant to adiabatic changes of a photon gas, because such changes would then become adiabatic and isothermal at the same tmie [18]. To remove such a discrepancy, longitudinal modes have possibly to be present which do not exist in conventional theory [18- 19]. (xi) It is not possible for conventional electromagnetic models of theelectron to explain the observed property of a "point charge" with an excessively small radial dimension. Nor does divergence in self-energy of a point charge vanish in quantum field theory where the process of renormalization has been applied to solve the problem.

3. Main Characteristics of Modified Field Theories Before turning to details, we shall here describe the main features of some of the modified theories.

3.1. THEORIES BASED ON ADDITIONAL VACUUM CURRENTS The displacement current density in vacuo was introduced by Maxwell. In combination with the Lorentz condition, the basic equations of conventional electromagnetic field theory in vacuo can be cast into the fourdimensional form

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of the d' Alembert equation where Au, = ( A , i 0 / c ) is a four-vector with A as the magnetic vector potential in three-space. In a number of extended theories equation (1) has been modified into a Proca type

(2)

A, = P O J ,

with a four-current density J ,

. Part of these theories will be briefly outlined here.

3.1.1 Quanturiz Mechunical Theory of the Electron In the theory of the electron by Dirac[20] the relativistic wave function \3Y has four components in spin-space. With the Hermitian adjoint wave function mechanical form ofthe charge and current densities become [21-221

v, the quantum

where ai are the Dirac matrices of the three spatial directions (x ,y, z). There is more than one set of choices of these matrices [23]. This form could be interpreted as the result of an electronic charge being "smeared out" over the volume of the electron. 3.1.2. Theory of the Photon with a Rest Mass At an early stage Einstein [24] as well as Bass and Schrodinger [25] have considered the possibility for the photon to have a very small but nonzero rest mass mo. Later de Broglie and Vigier [26] and Evans and Vigier [5] derived a corresponding form of the four-current in the Proca type equation (2) as given by

As a consequence, the solutions of the field equations were also found to include longitudinal fields. Thereby Evans [27] has drawn attention to a longitudinal magnetic field part, 3.1.3.

d3) , of the photon in the direction of propagation.

Nonzero Electric Field Divergence Theoy

One approach is based on a nonzero electric field divergence and an associated "space-charge current", in combination with Lorentz invariance [28-321. The four-current then becomes

The divergence can be understood as the effect of a polarization ofthe vacuum state [28]. It introduces an extra degree of freedom, leading to new possibilities such as "free"


129

dynamic states of wave phenomena including non-transverse and longitudinal modes, and "bound" electromagnetic equilibrium states. The velocity vector C corresponds in an axisymmetric case to the two values fc , a vanishing divergence, and a nonzero curl. With a time part icF of the four-current (5) resulting from the nonzero electric field divergence, the Lorentz invariance thus requires the space par^ to adopt the form F C . There is a certain analogy between the currents (3) - (3, but in the theory of the latter current (5) the electric charge and the mass should f i s t come out of the integrated solutions. 3.1.4. Nonzero Electric Conductivity Theory Maxwell's equations in the vacuum were already proposed by Bartlett and Corle [33] to modified by assigning a small nonzero electric conductivity to the formalism. As pointed out by Harmuth [MI, there was never a satisfactory concept of propagation velocity of signals within the framework of Maxwell's theory. Thus, the equations of the latter fail for waves with non-negligible relative frequency bandwidth when propagating in a dissipative medium. To meet this problem, a nonzero electric conductivity o and a corresponding three-space current density in vacuo. J* = 0 E (6) was thus introduced. The concept of this electric conductivity was later reconsidered by Vigier [35] who showed that the introduction of the current density (6) is equivalent to adding a related nonzero photon rest mass to the system. The dissipative "tired light" mechanism underlying this conductivity can be related to a nonzero energy of the vacuum ground state, as being predicted by quantum physics [5, 361. That the current (6) is related to the form (4) of a four-current can also be understood from the conventional field equations for homogeneous conducting media [37]. The effects of the nonzero electric conductivity were further investigated by Roy et al.38-40' 32, They have shown that the introduction of a nonzero conductivity yields a dispersion relation which results in phase and group velocities depending on a corresponding nonzero photon rest mass, due to a "tired light" effect. 3.1.5.

Singk Charge Theop

A set of first-order field equations was proposed by Hertz [41-43] in which the partial time derivatives in Maxwell's equations were substituted by total time derivatives

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Here vd denotes a constant velocity parameter which was interpreted as the velocity of the ether. Hertz' theory was discarded and forgotten at his time, because it spoiled the spacetie symmetry of Maxwell's equations. Recently Chubykalo and Smimov-Rueda [Z, 441 have presented a renovated version of Hertz' theory, being in accordance with Einstein's relativity principle. For a single point-shaped charged particle moving at the velocity v, the displacement current in Maxwell's equations is then modified into a "convection displacement current"

The approach by Chubykalo and Smirnov-Ruedafurther includes longitudinal modes and Coulomb long-range electromagnetic fields which cannot be described by the LienardWiechert potentials [Z, 451. 3.1.6.

Problerils with the Approaches o f Additional Vacuuril Currents

Two objections may at a first glance be raised against the parts of the approaches which lead to a nonzero rest mass. The first concerns the problem of gauge invariance. An answer on this point has recently been given by an analysis [46] using covariant derivatives in connection with the electric field divergence theory of the form (5). This also becomes consistent with an earlier conclusion that gauge Invariance does not require the photon rest mass to be zero [47]. The second objection concerns the supposition that a nonzero rest mass would provide a photon gas with three degrees of freedom, i.e, two transverse and one longitudinal. This would alter Planck's radiation law by a factor of 312, in contradiction with experience [32]. There are, however, arguments which also resolve this problem, as shown later in this context. 3.2. GENERALIZED THEORIES There have also been made approaches of a more general character than those based on the introduction of additional vacuum currents. as demonstrated here by a few examples. 3.2.1.

Magnetic Monopoles

One may raise the question why only the divergence of the electric field should be permitted to be nonzero, and not also the divergence of the magnetic field. In any case, there are a number of investigators who have included the latter and corresponding magnetic monopoles in their theories [34,48,49], also from the quantum theoretical point of view [32]. According to Dirac [50], the magnetic monopole concept is an open question. This concept also leads to a quantization condition €or the electric charge, and a similar result is derived from the tlHooft-Polyakov monopole [511.

NEW DEVELOPMENTS IN EM THEORY 3.2.2.

Unification of Electromagnetism and Gravitation

In attempts to make further generalizations, there is a profound difference between geometrical and Maxwellian theories [7]. It is stated by Ryder [51] that in electrodynamics the field is only an "actor" on the space-time "stage", whereas in gravity the "actor" becomes the "space-time stage" itself. There are several reviews on theories aiming at a unification of electromagnetism and gravitation, of which those recently made by Evans [52] and described by Roy [32] can be mentioned here among others. From the quantum theoretical point of view the Dirac magnetic monopole has been discussed in connection with the massive photon [53-551. An attempt has been made by Israelit [55] to generalize the Weyl-Dirac geometry by constructing a framework which includes both a massive photon and a Dirac monopole, thereby offering a basis for deriving electromagnetism and gravitation from geometry. The most general equations of a unified field theory of electromagnetism and gravitation have been elaborated in a series of investigations by Sachs [56]. A proposal for unification is also due to Evans [52] who considers electromagnetism and gravitation as different interlinked parts of the %emann tensor. Then electromagnetism can be regarded as a "twisting" and gravitation as a "warping" of space-time. The electromagnetic field tensor

Gpv = (h / e)RP ( A ) is

equivalent to a space-time curvature expressed by an

then made

antisymmetric tensor

RP(lc) obtained From contraction of the Riemann tensor. This approach further connects the longitudinal magnetic field [27] of Section 3.1.2. with the Riemannian space-time [52, 571. The corresponding geodesic equations of a plane wave in vacuo then have a space part which is a helix [52, 121. Analogous field equations of the Proca type have also been proposed to form a model of the universe as a collection of dust particles and nonzero rest mass gravitons which possess a collective mode behavior [58]. Such an approach could also connect the graviton mass with the problem of dark matter in cosmological models [58, 591.

4. New Aspects Due to the Extended Theories The extended theories of this review lead to results which both clarify some unsolved problems and end up with new concepts, thus having impacts on the following areas: Time-Dependent Stares with Wave Phenomena These include plane waves, individual photon models, the transition to a light beam, and superluminosity. Time-independent States with Electromagnetic Equilibria These include particle-shaped states, string-shaped states, and nonlocality effects.

B. LEHNERT 4.1. PLANE WAVES

The plane waves in vacuo become modified both by a nonzero electric conductivity and a nonzero electric field divergence.

4.1.1.

Nonzero Electric ConductiviQ

Staiting from the field equations with an additional small current (6), a plane wave of the form exp[i w t + kz)] yields a dispersion relation with phase and group velocities which differ slightly from C due to a resulting photon rest mass, thereby leading to a dispersion effect [35, 38-40]. These properties provide an explanation of the cosmic redshift [35, 38-40] being an alternative to the conventional geometrical Hubble expansion, and giving rise to a frictional "tired light" effect. If the latter would dominate, it should be related to the

= E O X , ~ / z f i w h e r e X, is the relative dielectric constant. Since there are many strong commonly accepted arguments in favour of the Big Bang model, the relative importance of this tired light effect remains an open question.

Hubble parameter H by

4.1 -2.

0

Nonzero Electric Field Divergence

The field equations with the four-current (5) result in three wave types: When div E = 0 and curl 33 f 0, the result is a conventional transverse electromagnetic wave. henceforth denoted as an "EM wave". When div E f 0 and curl E = 0 a purely longitudinal electric space-charge wave arises, here denoted as an "S wave". When both div E f 0 and curl E f 0, a hybrid non trunsverse electromagnetic space-charge wave appears, here denoted as an "EMS wave". For plane elementary modes of the form exp[i(- w )t + k r )] we can introduce the notation k E = kE(cos 2 ) where the angle x represents the new degree of freedom. The set of plane wave modes then ranges from the EM mode for

x = n / 2 . via the

a / 2 > x > 0> 0, to the S mode for 2 =O. The group velocities are v, = ck / k for the EM and S modes. and v, = C for the EMS mode. Poynting's

EMS modes for

theorem for the energy flow of plane waves in vacm applies to the EM and EMS modes, but not to the S mode. The problem (v) of Section 2 concerning total reflection of an incident conventional EM wave at the vacuum boundary of a dissipative medium can now be


133

tackled. Thus the matching of the field components. which is impossible for a conventional EM wave in the vacuum region 15. 29-32. can now be performed by EMS waves in the same region. 4.2. INDIVIDUAL PHOTON MODELS We now turn to individual photon models being based on axisymmetric solutions of a Proca type equation. Thereby there are connections between the photon rest mass theory and the nonzero electric field divergence theory outlined in Sections 3.1.2 and 3.1.3. 4.2.1. Basic Features of Axisymetric Model

With a cylindric frame @, p , z) and considering axisymmetric modes of the form

j?B)exp[i(- w t + kz)] propagating in the axial z direction, the basic equations of the system (2) and (5) give the following general results: For the conventional EM mode with div E = 0, the general solution either diverges at the origin r = 0 or at infinite values of r. Such a behaviour was already realized by Thomson [ll], and further by Heitler [60] as well as by Hunter and Wadlinger [12]. This mode is henceforth excluded fkom the axisymmetric analysis. For the EMS mode with div E # 0, the general solution can be shown to become derivable from a generating function which yields convergent solutions within entire space [29-321. For this axisymmetric normal EMS mode the velocity vector of equation (5) is now written as C = c (0, cos a , sin LZ ) (9) where O > 1. The results of the integration. which can as well be performed in the rest frame K', are as follows [30-321. The total electric charge vanishes. For electric neutrality it is then not necessary to assert that the photon is its own antiphoton. The total magnetic moment vanishes. but the local magnetic field is still nonzero. •

The energy density can be expressed in the two alternative forms

NEW DEVELOPMENTS IN EM THEORY of a field energy density wf and a source energy density same integrated total energy W, but the moments of W,

135

wS. They give the and

wbwith

an

arbitrary function become different. The present theory thus results in a total mass W / c2 and a photon rest mass

mo being related by

where v is the frequency associated with the mean wave number ko of a wave packet of narrow line width. A Lorentz transforiination to the rest frame yields the mass

m' = m (COS a

) which confirms the relations (14).

With the form (14) the axial symmetry and the symmetry with respect to z-vt make the integrals of the forces F E and j x B = FC x B vanish as well as those of E j = F E C . It is then possible to use the conventional [61] form

s=rxS/c S= ExB/p,

(15) for the density of the angular momentum, with S as the Poynting vector and r standing for the radius vector from the origin. Here the integral ofS is equal to the total angular momentum S = h / 2n for the photon as a boson. From

expression (14) with associated field quantities we obtain 2n? = c l v = A , /(sina)=AO (1 6) where ? is a sharply defined radius for y )) 1 in equation (12). This result agrees within an error of 0.5 percent with microwave transmission experiments in presence of an aperture, as being conducted by Hunter and Wadlinger [12]. With the angular momentum h / 4 n of a fermion, there would arise a discrepancy by a factor of two. In earlier interference experiments with individual photo [lo], dot-shaped marks were father observed at a screen. For impinging individual photons these marks seem to be consistent with the radial extension of the packet given by equation (16). The axial magnetic field B, of the wave packet is thus associated with the nonzero rest mass moand the total angular momentum (spin) s. The latter comes out the same in the Erames K and K: The three axisymmetric field components of E and B form a helical structure. These properties are analogous to the helical spin field configuration of B 4.2.3.

( ~and ) its cyclic field relations [5-7, 27, 571.

The Physical Picture of the Individual Photon Model

136

B. LEHNERT

An essential feature of de Brogue's picture of the wave-particle duality consists of regarding the particle and the associated wave as simultaneously existing physically real entities. The present wave packet model is consistent with this picture. The total energy is

hv = mc2 in the laboratory frame K. Thereby the fraction ( m- mo )c2 can be regarded as the energy ofa "free" pilot wave ofradiation, and the fmction m o c 2 as the energy of a

"bound" particle state of "self-confined" radiation. The rest mass mo thus represents an integrating part of the total field energy. The "bound" radiation moves around the axis of according ) to equation (9), and the pilot wave symmetry at the velocity C, = ~ ( C O S O propagates at the velocity CZ= c(sin a ) . With the sharply defined radius

? of equation

(16) the "bound" part of the radiation field becomes associated with a frequency

of revolution around the z axis in the laboratory frame K. In combination with relations (16) and (14) this yields

vo = v(c0sa ) = v(mo / m)

(18)

as being supported by the idea that all parts of the field energy should be included in the same way in the total energy hv. The relation

by de Broglie then comes out of equation (18). Its connection with equations (9) - (18) has been confirmed in terms of gauge theory [46]. In the rest frame K' the velocity vector (11) has a component C; = c only, and the photon radius in the perpendicular r direction remains the same as in equation (16) with v as the f ~ q u e n c yin K. With a Lorentz mnsfoimation of time from K to K' the frequency (inverted time) should further change from v to v= v(cos a ), thus making v' = V,

of equation (18). In K' the frequency of revolution is expected to become

v' = c / n? for the "bound" radiation. Combining this frequency with relations (16) and (IS), the angular momentum in K' becomes so = mo;c = h / 2~ which is the same as in the laboratory frame. This supports the analysis and its physical interpretation. Provided that mo / rn is independent of the frequency v, the results (17) - (19) and (10) thus permit the angle a also to be independent of v. There are two important consequences of this analysis:


137

The two-frequency paradox by de Broglie[6] is resolved, in the sense that the frequency vo = v ' becomes coupled to the frequency v by relation (18). The phase and group velocities become independent of frequency for a constant angle a . thereby resulting in vanishing dispersion effects. If there would arise a frequency dependence of a , the extremely small rest mass would only give rise to small dispersion effects, even at large cosmical distances. The present physical picture thus increases the understanding of the points (i) - (iii) of Section 2. 4.2.4.

The Then~zodynarlzicsqf a Photon Gas

With a nonzero rest mass one would at First glance expect a photon gas to have three degrees of Ereedom, two transverse and one longitudinal. This alters Planck's radiation law by a factor of 312, in contradiction with experience [32]. A detailed analysis shows,

) field cannot be involved in a process of light absorption [5]. however, that the B ' ~ spin This agrees with the present analysis where the spin field is "carried away" by the pilot field, and Planck's law should be recovered in all practical cases [32], Moreover, transverse photons cannot penetrate the walls of a cavity, whereas this is the case for longitudinal photons which would not contribute to the thermal equilibrium [25]. The equations of state of a photon gas have been considered by Mezaros [18] and Molnar et aL [19]. They find that Planck's distribution cannot be invariant to an adiabatic change occurring in an ensemble of photons. This dilemma is due to the fact that the changes cannot be adiabatic and isothermal at the same time. Probably this contradiction is due to the lack of longitudinal magnetic flux density. Thus, in an adiabatically deformed photon gas the intensity will change in time, and so will a field like B ( ~ )This . result can contribute to the clarification of point (x) in Section 2, but further analysis may become necessary. 4.2.5.

The Photon Rest Mass

The possible existence of a nonzero photon rest mass was already proposed by Einstein [24], Bass and Schrodinger [25], de Broglie and Vigier [26] and further by Evans and Vigier [5] among others. It includes crucial points such as the relation to the Michelson-Morley experiment, and the so far undetermined magnitude ofthe mass and its experimental determination. The velocity w of the earth around the sun is (wlE lo-' c . Ifthis would be the velocity with respect to a stationary ether, and if massive photons would move at the velocity v = c(sin a ) in the same ether, then the velocity u of photons recorded at the earth's surface would become

B. LEHNERT

For the magnitude of the rest mass and the velocity according to equation (14), the following conditions will apply: For cos a 5 I o - ,~corresponding to a rest mass

m, c into it [71,16]. This leads to an imaginary rest mass, with difficulties of interpretation [711. These central concepts also come out of the approach of Section 3.1.3. To satisfy the Lorentz condition one can thus replace the form (9) of the velocity vector by

C=c(O,i sin ha,cosha)=c(O,C,,C,)

NEW DEVELOPMENTS I N EM THEORY For propagating axisymmetric normal modes of the form exp[i(- w t tachyon mode yields the dispersion relation

w

=kc,=kv

141

+ h)l an EMS-lie

v = c(cos ha) > c

(24)

which replaces equations (10). If we further assume relations (14) to be adopted and modified to apply to the tachyon case, the result would become

hv=mc2 = m o c 2 [ l - ( v l c2)]-112 =m o c 2 /i(sin h a )

(25)

or

Consequently, this leads to what could be conceived as an imaginary rest mass, as is also obtained in current tachyon theory. 4.5. STEADY ELECTROMAGNETIC EQUlLlBl2LA With equations (2) and (5) steady equilibrium states are determined by c2

CUP!curl A = - c ( v ~ ~ ) = c F I E ~

(27)

We limit ourselves to axisymmetric states oftwo kinds: "Particle-shaped" equilibria where the configuration becomes bounded both in the radial and axial directions. "String-shaped" equilibria where the configuration is uniform in the axial direction. For both these kinds the general solution can be expressed in terms of a generating function F = CA - 0 where the vector potential A only has a component in the y, direction of a spherical frame ( r ,8 , y, ). Separable functions are considered where and

F = GoG= G,R(p) T(8)with Go as a characteristic amplitude, p = r / P,

uo as a characteristic radius.

4.5.1.

Particle-Shaped States

By integration of the obtained field quantities, expressions for a net electric charge qo, magnetic moment Mo, mass mo, and angular momentum (spin) so are obtained. There are two classes of solutions being ofpossible interest to leptons [28-321;

142

•

B. LEHNERT For a convergent radial part R there is a vanishing charge

qo and magnetic

Mo. A nonzero and very small mass m, can be obtained, and the spin can be made equal to S, = h / 452 of a fermion. These solutions could become moment

models for the neutrino. The latter moves nearly at the speed of light, has weak interaction with its surroundings, and has no color charge. For a divergent radial part R at the origin

P = 0 solutions can be obtained for

qo, Mo,mo, and so = h / 4n, by permitting the characteristic radius roto approach the very small range of a "point charge". nonzero and finite

Thereby the parameter ranges of F can be chosen to include

lqol

= e of the

elementary charge. as well as to satisfy the value of the magnetic moment by Dirac and Feynman [72].On pure physical grounds it appears to be unacceptable to have a radius being exactly equal to zero. By applying a small general relativistic correction, a nonzero radid of the order of ro = l ~ - ' ~ canm be obtained for the electron. Here we turn to the question of Section 2(xi) on the infinite self-energy of a point charge in classical and quantum theory. The latter uses a renormalization procedure, by subtracting two "infinities" to end up with a finite result, Despite of the success of such a procedure, there ought to be searched for a more satisfactory way from the physical point of view [51]. The present theory may possibly provide such an alternative, where instead a product of an "infinity" with a "zero" yields a finite result. 4.5 -2.

String-Shaped States

The string-shaped equilibria which result from equations (27) can serve as an analogous model which reproduces several of the desirable features of the earlier proposed string configuration of the hadron color field structure. These equilibria have a constant longitudinal stress which tends to pull the ends of the configuration towards each other. The magnetic field is thereby located to a narrow channel, and the system has no net electric charge. Since the divergence of the magnetic field is zero, no model based on magnetic poles is needed. 4.6. NONLOCALITY EFFECTS An increasing interest has recently been taken in instantaneous long-range interaction. The investigations within this field are yet at a preliminary stage, also including difficult concepts and interpretations.

NEW DEVELOPMENTS IN EM THEORY 4.6.1.

General Questions

It was already pointed out by Dirac 501 that, as long as dealing only with transverse waves, we cannot bring in the Coulomb interactions. There must then also arise longitudinal interactions between pairs. In fact, as already argued by Faraday and Newton and further stressed by Chubykalo and Smirnov-Rueda [2] among others, instantaneous long-range Lnteraction takes place not instead of but along with the short-range interaction in classical field theory. This point of view has also been expressed by Pope [73] in stating that instantaneous action-at-adistance and the finite speed of light are generally considered as antithetical, but it is well known that in relativistic physics light has both finite and infinite speed. Thereby c is not a velocity but is a space-time constant having the dimension of velocity. In this connection Argyris and Ciubotariu [58] point out that the unquantized longitudinal-scalar part ofthe field yields the Coulomb potential, and that transverse photons transport energy whereas longitudinal (virtual) photons do not cany energy away. Thus, there is direct interaction between a transverse photon and the gravitation field of a black hole, but not with a longitudinal photon. The Coulomb field is therefore able to cross the event horizon of a black hole. 4.6.2.

Instantaneous Long-Range Interuction

h considerations on a s i i e moving charged particle, Chubykalo and Smirnov Rueda [2,44-45] have shown the Lienard-Wiechert potentials mentioned in Section 2(viii) to be incomplete. by not being able to describe long-range instantaneous Coulomb interaction. tn a modified version of the earlier theory by Hertz [41-431, such an interaction is included by Chubykalo and Smirnov-Rueda. The same authors have also made an analysis [2,44-45] in which a Proca type equation is divided into two pairs. The first manifests the instantaneous and longitudinal aspects of electromagnetic nature, as represented by functions f [R(t)3 of an implicit

time dependence. For a single charge system this would then lead to the form R(t) = r - )b ( t )where r is a fixed vector from the point of observation to the origin, and

rq(f)

is the position of the moving charge. The implicit time dependence then implies

that all explicit time derivatives disappear from the basic equations of the first part of the divided pairs. The second part is responsible for transverse wave phenomena. as represented by functions g(r, t) of an explicit time dependence. There are a number of arguments which could support the existence of long-range instantaneous interaction, Thus, superluminal phenomena cannot, be explained from Maxwell's equations without longitudinal wave concepts [17]. The energy of "longitudinal modes" cannot be stored locally in space but can be spread by an arbitrary velocity [57]. Moreover, non-locality behavior is supported by observations as a Eundamental property of the universe [74]. There are also several quantum mechanical arguments in favor of long-range interaction, such as that of the Aharonov-Bohm effect [75] and those related to the discussion on the Einstein-Podolsky-Rosen paradox [76-771

144

B. LEHNERT

Instantaneous action at a distance as represented by longitudinal components can thus be interpreted as a classical equivalent ofnonlocal quantum interactions [2]. With these arguments, and with the similarity between electromagnetic and gravitational long-range interaction, it should be justified to represent the former with the implicit time dependence part in the basic equations. There are similarities between electromagnetism and gravitation which could be Important to the investigations on long-range interaction. Thus, there is a resemblance between the Coulomb and Newton potentials [58]. A holistic view of electromagnetism and gravitation would imply that action at a &stance occurs in a similar way in gravitation [17] and vice versa. This is supported by the general principle that there exists no screen against gravitational forces acting between distant massive sources [35, 621. In other words, the position, velocity and acceleration of a source of gravity would then be felt by the target body in much less than the light-time between them.

5. Conclusion The new developments in electromagnetic field theory have here been seen to provide a number of contubutions to the improved understanding ofproblems which have so far not been clarified in terms of conventional analysis. Examples have thereby been given in connection with the points (i) - (xi) of Section 2. This development has also resulted in a number ofpredicted features of the electromagnetic field. in the form of new models of the photon and of leptons. The areas of new development do not seem to be exhausted by far, and there are several important questions which require further investigations. Examples are here given by the magnitude ofthe photon rest mass, the tired light hypothesis versus the Big Bang concept, superluminosity, the Sagnac effect, the light beam transition, the absolute values of the elementary electronic charge and of the masses of leptons, long-range interaction and the general concepts of causality and locality.

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COMPARISON OF NEAR AND FAR FIELD DOUBLE-SLIT INTERFEROMETRY FOR DISPERSION OF THE PHOTON WAVEPACKET R. L. AMOROSO Noetic Advanced Studies Institute I20 Village Square, MS49 Orinda, CA 94563-2502 USA

J-P VlGlER Pierre et Marie Curie Universitb Gravitation et Cosmologie Relativistes Tour 22-Boite I42,4 place Jussieu, 75005 Paris, France M. KAFATOS Center for Earth Observing and Space Research George Mason University Faiijfax,VA 22030-4444 USA G. HUNTER Department of Chemistry York University, 4700 Keele St. Toronto, Canada M3JIP3

Abstract. Extending EM and Quantum Theory suggests the possibility of photon mass, additional terms for Maxwell's equations, reality of de Broglie-Bohm causality and the Vigier model of extended charged particles. Experimental tests indicative of these hypotheses can be performed with double-slit interferometry of single visible wavelengths comparing near and far field sources over laboratory and various cosmological distances to observe the possibility of spreading of the photon wavepacket during propagation. These observations could determine whether nonlinearities causing non-dispersivity are associated with Maxwell's equations. If so, this may be an indirect determination of nonzero restmass photon anisotropy, de Broglie photon piloting and vacuum permittivity reincarnating the Michelson-Morley experiment in terms of a Dirac covariant ether.

1. Introduction

Traditional thinking suggests that EM radiation coming from a point source is subject to spreading of the photon wave-packet over x = -ict. Recently however, families of 147 R.L. Antoroso et a1 (eds.).Gravitation and Cosmology: From the Hubble Radius to the Pllnck Scale, 147-156. O 2002 Kluwer Academic Publishers. Printed in the Netherlur~ds.

148

R. AMOROSO, J-P VIGIER, M. KAFATOS, G. HUNTER

nondispersive waves have been found for Maxwell, Klein-Gordon, Dirac, Weyl and Schriidinger field equations [I-7, 31-33]. Courant and Hilbert [8] were probably among the first to make this distinction. In this paper we briefly review classical and extended theoretical pre ctions of photon propagation, present an experimental design for possible empirical tests and relevant discussion of the physical consequences for the de Broglie, Bohrn and Vigier formalisms [9] of extended electromagnetic theory that suggest the photon wavepacket is piloted and does not spread over cosmological distances as suggested by current classically oriented interpretations of Maxwell's field equations. The experiment is accomplished by comparison of the double-slit Fringes of monochromatic light from near and far field monochromatic emission sources. Recent EM-Theory discussions on the possible existence of photon mass di (my) and U(l) group invariance of EM-Theory in view of the putative

(my) implies the

introduction of.new terms for Maxwell-Lorentz equations. This cannot be considered as purely theoretical as in the past because of recent new experimental evidence [lo]. The aim of the present text is to discuss the two possible interpretations of Maxwell's equations: 1. Photon propagation without trajectory - random probability distributions [classical]. 2. Photon propagation with trajectory - piloted with no or minute spreading of the wavepacket [extended]. The former is in accord with the standard model of classically oriented Copenhagen approaches to the EM and quantum formalism; and the latter the Vigier-Bohm-de Broglie extended charge particle causal approaches.

1.1 STAGES IN DEVELOPMENT OF THE THEORY OF LIGHT As generally known the present status of the theory of light is the result of three stages of development. a) The Maxwell - Heaviside - Hertz stage which developed (in the middle of the last century) the Maxwellian linear equations of light (and Hertzian waves) with transverse continuous waves (with zero-mass and U(l) invariance) of velocity with the separate existence of instantaneous Coulomb interaction. Technically this led to the huge development of electricaVphotopaphic devices in the modern world. b) The Einstein stage (1905) with 1) The discovery of the photon (( Lichtquanten )) carrying an energy E=hv Ill]. 2) The idea (1916) that photons carry the observed electromagnetic energy in the form of unidirectional (( Nadelstrahlung )), of oriented beams and not in the form of spherical emission (( Kugelstrahlung )) which did not exist 121. 3) The observed quantized electromagnetic potential for inertial massive charges only emit energy when accelerated; except in the case of Bohr orbits, where it

NEAR AND FAR FIELD DOUBLE-SLIT INTERFEROMETRY

was postulated if one introduced Poincari forces to keep the sub-charges together and that the inertia of the gravitational and EM charged part in any particle was separately covariant [lo]. c) The quantum stage (1920-2000) where the Copenhagen proponents dropped the stepby-step description of the propagation of light only discussing the statistical properties. Later photon emission and absorption was shown experimentally by Aspect [13] to correspond to (individual photon emission followed by absorption) linear recoils of the corresponding sources. Two theories developed in parallel :

c l ) The Copenhagen interpretation. The QED version suggests probability waves or quantized photons ; never the two simultaneously. cz) The Einstein - de Broglie interpretation (dropped between 1925 and 1950) allowing simultaneous real waves and piloted photon particles where both carry energy momentum distributions mostly concentrated on the latter, Both interpretations recover FAPP the same experimental facts. d) The recent period in which new properties of EM waves have been discovered that are generating renewed discussion of differences between the cI & c2 interpretations. We only mention here

-

the simultaneous existence of two different electron radii (the Compton and charge radii) in scattering experiments [14]. observed anomalies in e*e- scattering [lo]. Recent observations in EM theory [15] and excess energy in ~ scattering [16].

i p~ + +e

2. General Nature Of The Photon Wave-Packet A wave-packet is a balanced superposition of waves of one predominant wave number k with phase amplitudes that interfere constructively over the small region ct or AX outside of which the amplitude reduces to zero quickly by destructive interference. This is not true for ideal monochromatic wavepackets; Figure 1A shows a wave function or ' ' that depends on space coordinates x,y,z and time t simplified to x,t with amplitude $ average wavelength A. over the limit defined by a light pulse passing through a shutter, since an ordinary plane wave would be spread over all space [17,18].

150


Figure 1A is plotted in Figure 1B according to equation I [17] as a function of x - xo reaching a maximum at x = xoto zero where x - xo = z / Ak : thus obtaining a wave function concentrated in a packet where

A = 2~ / ko.

The Fourier transform of

which, as shown in Figure IC, is the wavepacket for a single photon. Equation 2 is a general type of wavepacket for any function thus defined [17, 181.

2.1 REVIEW OF THE CLASSICAL DOUBLE-SLIT EXPERIMENTAL FORMALISM A description of Young's 1801 classic fringe experiment for two slits [19] of width a, at a distance A apart as shown in Figure 2 is described by [20]:

f ( x ) =s(x)x[S(x+ A12)+S(x- A1211

(3)

where s(x) is the transmission function for one slit of width a. For very narrow slits a goes to zero and the incident amplitude is proportional to lla so that s(x) becomes S(x).

The intensity of the diffraction pattern (Figure 2B) [20] is cos2 fringes of period 112A modulated by a (sin x) 1x function going to zero for p = a-I . This is Young=s well known fringe experiment [19,20],

NEAR AND FAR FIELD DOUBLE-SLIT INTERFEROMETRY

Figure 2. Diagram of a double-slit set up distance A apart and shaving the observed intensity of the diffraction pattern from equation 4 1201.

For a coherent wavepacket passing through a double-slit interference creates phase difference A@ such that a maximum occurs at d sin B = rnR where rn = 0,1,2 ; and a minima at d sin B = rn1/ 2 R where m = 12.3 represents A 4 of 180". d is the distance between slits (A in fig. 2) and B is the angle made by an arbitrary point P on the screen, the central point between the slits q(x) and the corresponding center of the screen. The distance between the two screens D must be much larger than d so that the distances r, & r2 from the slits to P can be considered parallel [211.

2.2 CURRENT CLASSICAL INTERPRETATION OF DOUBLE-SLIT PHENOMENA Historically Einstein first proposed in 1905 that radiant electromagnetic energy should appear as cc Lichtquarzta )> About 10 years later he assumed that these quanta should also be spatially quantized with a unique orientation. The essential conclusion of his research was that Maxwell's cc Kugelstruhimg )) - spherical radiation around a source - does not exist but that elementary light energy h v always appears in a unique direction creating a recoil upon emission hv / c which he called ) or 'needle radiation', giving deference to Einstein's original term [12]. This ce nadelstruhiozng )) is also responsible for radiation pressure. According to classical interpretations of Maxwell's theory dispersion of the wave-packet over distance is expected [17,18]. This is partly due to the fact that in the Copenhagen view emission is point-lie so any packet should expand. The uncertainty principle can be derived fmm propagation of the wavepacket which in classical terms propagates accor ng to the Schtklinger equation. This is described by Bohm [17] as :

R. AMOROSO, J-P VIGIER, M. KAFATOS. G. HUNTER

152

where a wavepacket with an initial diameter of AXO will spread to Ax E At / Ax, as t becomes limitless. The narrower the wavepacket originally the more rapid the spreading. (see Figure 3) According to Bohm [17] the reason for the spread is in terms of the uncertainty principle. The region AxO confining the packet has a number of wavelengths near

AXo

SO

that even though the average velocity of the wavepacket is equal to the

group velocity, the actual velocity will fluctuate, and the distance propagated by the packet isn't fully determined. It can fluctuate by

According to Bohm [17] photons have momentum as evidenced in the radiation pressure during absorption , such that the energy and momentum of light quanta is the same as a zero mass particle in 3D space. It is the wave properties of a wavepacket that produces the AxAk 2 1 that allows spreading because a particle will never spread ;but a collection of particles because of uncertainty in velocity gradually spread with At. (x) Thus although equations (5) & (6) used for illustration of classical wavepacket spreading are Schriidinger type equations, Maxwellian equations give a similar result for photons in the classical limit - i.e. - spreading. In our extended theoretical approach that includes photon mass [10,22]; piloting effects prevent spreading as At +a. There may be an infinitesimal spreading over cosmological distance. This is the current empirical knowledge limit; where further understanding can only be achieved by experimentation like the suggestion in section 3.

Figure 3. Dispersion and change of shape of a wavepacket during propagation. Broader packets with many

wavelenghts near k

(koL2 1) distort relatively little; a narrow k packet (koL5 1) rapidly broadens [23].

2.3 EXTENDED THEORETICAL APPROACH TO PHOTON PROPAGATION

De Broglie wave mechanics creates a relationship between wave numbers and momentum not considered in classical mechanics. In the de Broglie mode a classical wave of a wave number k can be of arbitrary amplitude and momentum; and whenever position or

NEAR AND FAR FlELD DOUBLE-SLIT INTERFEROMETRY

153

momentum is measured a definite number results. The de Broglie relation p = Rk implies a definite wave number k for a definite momentum. This is contrary to a classical description of a wavepacket, which suggests a range of wave numbers and positions [17]. Concurrently defined position and momentum values is considered equivalent to the assumption of t( hidden variables )) that constantly determine these values. This is inconsistent with the standard Copenhagen interpretation of quantum theory which is statistical and not causal.

2.4 RECENT WORK ON NON-DISPERSIVE PROPAGATION MODES Rather than the classically oriented Schriidinger equation which suggests spreading of the wavepacket (in a variation of Figure 3). nondispersive wave modes would be expected to propagate according to the de Broglie relativistic Klein-Gordon type equation [1.24]. Recent work on non-dispersive modes ofthe wavepacket by numerous authors [l-81 has demonstrated mathematically the possibility for the existence of 'real' non-dispersive modes of the photon wavepacket satisfying linear Maxwell equations but considered contrary to the prevailing opinion. This has urged empirical testing of the issues at the heart ofthe matter and is our main inspiration for writing this paper. We give here only the very briefest review of this recent theoretical work and refer parties interested in deeper analysis to the main references [I-81. According to established wave mechanics a de Broglie wave with infinite wavelength is said to be associated with all particles and have a wave function uniform throughout all space. A particle's internal vibration and infinite de Broglie wave stays in phase at the particle's location. This suggests how the de Broglie wave pilots a particle's motion with no spreading; whereas a Schrodmger wavepacket spreads because of uncertainty in momentum [I]. Because the de Broglie relation is relativistic a Lorentz transformation might be involved between the particle's point of origin and present position during propagation, canceling insertion of any would be classical uncertainty effects and maintaining phase coherence between the particle's internal motion and the wave function in the de Broglie relativistic-piloted regime. This might be considered reminiscent of error correction modes discussed in terms of quantum computing: If w(k,,- w(ko) = p(kl - k, ) then dw(k,) l dkl = ,u for all kl and therefore

F(x)

d 'a)(k,)dk,= 0. With these conditions de Broglie theory yields [I]

(exp $m(ko )t - koxJ)[sin Ak(x -

a/ Ak(x - ert) or also in the form

F (x,t ) = G ( x , t)H(x, t )

(7)

Using a different technique, Hillion [2] uses electromagnetic theory to derive nonhomo~eniousnondispersive waves from Maxwell's equations. With the variables

-

C=x+iy,g=x-iy,e=r-x becomes

0

,q=z+xo,i=fi,

acacy + a4any = 0.

the wave e q u a t i o n ~ y - i 3 ~ y = O

This was fmt shown by Courant and Hilbert [S] and has

154


nondispersive solutions of the type cy = g(x,x0)F(u), x =(x, y,z,) where the phase u is a solution of the characteristic equation d ud -u + dpi7,u = 0,where F is arbitrary with 4'4'

continuous partial derivatives and g is an attenuation factor [2]. Brittingham [7Jderives on the other hand homogenous nondispersive solutions to Maxwell's equations in the soliton regime with both linear and nonlinear parameters. Beil [31-331 has applied the Brittingham solutions to modeling ofthe photon as a specific realization of Nadelstralung, which Einstein conjectured was the only k i d of radiation that is consistent with relativistic dynamics. Finally Shaarawi [6] derives Brittingham l i e nondispersive solutions for the wavepacket applicable to Klein-Gordon equations which can be used as local de Broglie scalar wave particles.

3. Proposed Experimental Design A. Stellar objects with emission spectra compatible with the telescopes instrumentation are chosen for observation. As a baseline fixed sources in our galaxy of about 100 light years distance are selected for each of 3 wavelengths (red, yellowlgreen, blue). Narrow pass filters of single wavelengths are used with 3 far-field stars of about 2 million light years from the local group in the Coma cluster and beyond for comparison with the control stars from our galaxy for each of the 3 wavelengths. B. The comparisons with near and far-field stars are made for possible spreading during propagation of the photon wavepacket over cosmological distances in the fringe patterns of standard double-slit interferometry. It is suggested that 3 control stars be compared with 3 far-field stars for a 12 star database. To ensure uniformity of stellar types, we suggest bright Cepheid's or M-giants and 0-supergiants in the local Or sources of greater group of galaxies, and the Virgo and Coma clusters. observational ease such as emission nebulae with Balmer and HI1 and Ha lines 1251C. Because of the foregoing discussions on the nature of the photon wavepacket during propagation the experiment might optimally be performed with additional filters allowing passage of only single photons. However in case any physical parameters might be missing from current theoretical predictions it would be useful ifpractical to also perform the experiment with continuous wave trains of multi photon wavepackets for experimental diversity and exploration of group dynamics. As in figure 2 and accompanying discussion it is deemed important to perform the experiment with double-slit designs with 3 A dimensions or spacing; 1. Optimally maximum, 2. Median and 3; Optimal minimum. Account of work on measurements of photon radius [26] should probably also be taken into account for optimal a distance as also shown in figure 2A. D. To simplify the experiment for preliminary results the model can be done initially with the near-field control group performed in an Teran laboratory setting. The far field sources could then be Haemission lines in the solar chromosphere. E. Anticipated results. The current model formalism suggests significant spreading of the wavepacket over cosmological distance because of uncertainties in momentum.

NEAR AND FAR FIELD DOUBLE-SLST INTERFEROMETRY

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However our view according to extended theoretical models of Vigier, de Broglie, and Bohm that there might be infinitesimal spreading of the wavepacket because of de Broglie-Bohm piloting. Until we have preliinary tests we are unsure of sufficient limits in discerning the degree of spreading within the current instrumentation Limits of CCD cameras and computer analysis of the data. The mathematical predictions for spreading will be included in the proposals for telescope time. By the time of publication we anticipate having at least preliinary data. F. Comparison of dispersion for dust-free and clear spacetime regions as test of gravity effects and redshift from TIFFT.

48#e I

A

B

C

D

Figure 4. Experimental set up. A. source, B. monochromatic filter, C. intensity filter, D. double slits, E. CCD array, F. analyzer.

4. Conclusion And Summary Superficially the nature of a wavepacket and its spreading during propagation seems straight forward; but the subtleties involved are at the heart of wave mechanics and quantum theory which is by no means complete and entails persistent discussion on the merits of Copenhagen vs. extended forms of quantum theory. A definitive delineation iq not possible in terms of any type of current theoretical discussion alone. Therefore if technically feasible experiments on the nature of the wavepacket and its propagation like those proposed here might advance our understanding of Quantum Theory. Classical approaches predict wavepacket spreading because of uncertainty relationships. The de Broglie-Bohm approaches predict coherence over all space and time in view of putative causal action of the pilot wave or quantum potential. This is the well known assumption of hidden variables deemed inconsistent with the classically oriented Copenhagen model. A final understanding of the photon and its propagation is far from being understood [27]. To understand the anticipated experimental results photon propagation may not only have to be perceived in terms of internal de Broglie Lorentz transformations [I] but also deeper aspects of nonlocality [28] which might only be clarified in terms of a post big bang cosmology and the attendant understanding of spacetime hyperstructure [29] which non-zero photon restmass seems to demand. A Newtonian ether was disallowed by Einstein's relativistic dynamics and the Michelson-Morley experiment. Einstein himself said that relativity did not preclude an ether. We revisit this issue in terms of a Dirac covariant subquantum stochastic ether with correspondence to relativity and inclusive of de Broglie Bohm Vigier charged particle models [30].


References 1. Mackinnon, L. 1978, A nondispersive de Broglie wave packet, Found. Phys. 8:3/4, 157-176. 2. Hillion, P. 1991, Nonhomogenious nondispersive electromagnetic waves, Phys. Rev. A, 454, pp.26222627. 3. Peshkin, M. 1999, Force-free interactions and nondispersive phase shifts in interferometry, Found. Phys. 29:3, pp. 481-489. 4. Rodrigues. W.A., & Lu. J-Y., 1997. On the existence ofundistorted progressive waves. Found. Phys. 27:3. pp. 435-508. 5. Ignatovich. V.K.. 1978. Nonspreading wave packets in quantum mechanics. Found. Phys. 8:7/8. pp. 565571. 6. Shaarawi, A.M. & Ziolkowski, R.W., 1990, A Novel approach to the synthesis of nondispemive wave packet solutions to the Klein-Gordon and Dirac equations, J. Math. Physics, 3:10, pp. 2511-2519. 7. Brittingham, J.N. 1983, Focus waves modes in inhomogenious Maxwell=s equations: Transverse electric mode, J Appl. Phys. 543, 1179-1189. 8. Courant, R. & Hilbert, D., 1962. Methods of Mathematical Physics, Vol. 2, Interscience :New York 9. Gueret, Ph. and Vigia, J-P, 1982, de Broglie's wave particle duality and the stochastic interpretation of quantum mechanics. Found. Phys. 12:12. pp. 1057-1083. 10. Vigier, J-P, 2000, Photon mass and Heaviside force, Phys. Let. A, 270,221-23 1. 11. Einstein. A. 1905. Ann. Phys. 7.132. 12. Cormier-Delanoue. C., 1988, Sur remission d'une radiation electromagneticiquepar une charge electrique en movement rectiligne accelere, Annales de laFondation Louis de Broglie, 13 :I, pp. 43-63. (Translation by E P Vigier, edited by R.L. Amoroso) 13. Aspect, A., 1978, Phys Rev D, 14,1944; Phys Rev Let, 1980,47,480. 14. Anderson, J.D., Lain, P.H., Lau, E.L., Liu, AS., Nieto, M.M., & Turyshev, S.G. 1988, Phys. Rev. Let. 81,2858. 15. Vigier, J-P, 1997. Phys Let A, 234.75. 16. Vigier, J-P, 1993, New Hydrogen (Deuterium) Bohr orbits in quantum chemistry and cold fusion processes. Proc. ICC 54 Hawaii. 17. Bohm, D. 1963, Quantum Theory, Prentice-Hall: Englewood Cliffs. 18. Schiff. L.I.. 1987. Quantum Mechanics. London : McGraw-Hill. 19. Young, T., 1803, Philosophical Transactions. 20. Cowley, J.M., 1975, Diffraction Physics, Amsterdam : North Holland. 21. Halliday, D., & Resnick, R., 1963, Physics tor Students of Science & Engineering, New York: Wiley & Sons. 22. Arnoroso, R.L., Kafatos, M. & Ecimovic, P., The origin of'cosmologicalredshift in spin exchange vacuum compactificationand nonzero rest mass photon anisotropy, in G.Hunter, S. Jeffers & J-P Vigier (eds.) Causality and Locality in Modem Physics, Dordrecht: Kluwer. 23. Jackson, J.D., 1999, Classical Electrodynamics, New York : Wiley & Sons. 24. De Broglie. L. 1925, Ann. Phys. 322. 25. Kautinann, W.J., 1988, Universe, New York : W.H. Freeman. 26. Hunter, G, & Wadlinger, R.L.P., 1989, Physics Essays, Vol. 2. 27. Whitney, C.K., 1998, The mass-connected photon, in G-Hunter, S. Jeffers & J-P Vigier (eds.) Causality and Locality in Modem Physics, Dordrecht: Kluwer. 28. Nadeau, R. & Kafatos, M., 1999, The Non-Local Universe, London: Oxford 29. Amoroso, R.L. 2001, The continuous state universe, in R.L. Amoroso, G. Hunter, M. Kafatos & J-P Vigier (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale Kluwer Academic Publishers. 30. Vigia, J-P, 1983, Dirac's ether in relativistic quantum mechanics, Foundations of Phys. 13:2,253-285. 31. Beil, R.G. 1993, Found Phys. 23, 1587. 32. Beil, RG. 1995, Found. Phys. 25,717. 33. Beil, R.G. 1997, pp. 9-16, in The Present Status ofthe Quantum Theory oflight, S. Jeffers et al, (eds.) Dordrecht: Kluwer Academic.

PHOTON DLAMETER MEASUREMENTS

GEOFFREY HUNTER*, MARIAN KOWALSKI REZA MANI, ROBERT L.P.WADLINGER** o Centre for Research in Earth and Space Science York University, Toronto, Canada M3J 1 P3. FRlTZ ENGLER Itibar Industries (Division of Radareseareb Inc.) 3650 Weston Road. Weston. Ontario. Canada MDL 1 W2.

TIM RICHARDSON Bio-Microtech Inc., 670 Hardwick Road, Unit 4 P.O.Box 23, Bolton, Ontario, Canada L7E 5T1.

Abstract. The 1985-89 Hunter-Wadlinger electromagnetic theory of the photon predicted that the photon is a soliton-wave with the shape and size of a circular ellipsoid of length

/Z

the wavelength).

/Z / . This prediction is being tested by three diameter measurements: 1) those canied out in 1985-86 with microwaves, 2) in progress measurementson 10 micron photons from a CO,

with a diameter of

laser, and 3) an imminent experiment with monochromatic visible light (400-800 nm).

1. Introduction In 1985-86 we measured the transmission of X & K-band microwaves through circular and rectangular apertures. The transmitted intensity plotted as a function of window area (slit width) was linear with the slit-width intercept at zero transmission as the photon diameter. The measurements confirmed the theoretically predicted photon diameter of /Z / w with an In a 1999-2001 experiment with 10.5 pm photons the experimental error of 0.5%.

"Corresponding author, email: gbunter-yorku.ca **Deceased 1991 3 Note that

/ fl differsfrom

a / 3 by 5% - 10times the experimentalerror. 157

R.L. Anzoroso et a1 (eds.), Gravitation and Cosntology: F r m the Hubble Ruclius to tlze Planck Scale, 157-166. O 2002 Kluwer Academic Publislzers. Printed in the Netlzerlands

158

G. HUNTER

radiation impinges upon gratings of slits (from lpm to 10 pm ) with slit-spacing proportional to slit width so that all gratings have the same window area. Thus transmitted intensity is expected to be constant with a sharp cutaff for slit-widths less than A / w . In July 2000 T. Richardson realized the possibility of photon diameter measurements with visible light by illuminating a Bio-Microtech microscope "test slide" with variable wavelength light from a monochromatorto observe the wavelength at which slits on the slide no longer transmit light. The slide has several gratings with slit-widths from 100 nm to 500 nm. This yield s a direct measurement of the diameter of the photons that will just pass through the slits of each grating.

2. Motivation for the Experiments Our theory of a freely propagating photon [I, 21 yielded a soliton (limited by the principle of causality) with the photon's electromagnetic field contained within a circular ellipsoid of length and cross-sectional circumference both equal to the wavelength ( A ), the ellipsoid's long axis being the axis ofpropagation. That the photon is about one wavelength long is supported by laser pulses less than two wavelengths long [31. The soliton model predicts that a beam of monochromatic, circularly polarized photons will readily pass through apertures whose smallest linear dimension is greater than the soliton's diameter of /Z / w . For smaller apertures one would expect strong attenuationldiffraction based upon the simple mechanical notion that the soliton cannot pass undeflected through an aperture that is smaller than its own diameter of /Z / w For larger, uniformly illuminated apertures one would expect the transmitted power to be proportional to the difference between the area ofthe aperture and the soliton's cross-sectional area. In 1985 our literature search covering electromagnetic radiation fmm 7 -rays to radio waves failed to reveal definitive measurements to confirm or deny the above hypothesis. However, microwaves having wavelengths of 1-10 cm seemed to be most amenable to laboratory scale experiments, and several reported measurements with microwaves broadly supported the hypothesis. Most of these measurements were ofelectric field intensity measurements in the vicinity of apertures, designed to investigate diffraction patterns. The post-war work of Andrews is extensive [4]. Diffiction measurements were made on slits by Hadlock [S], on circular holes by Robinson N] and on a variety of other shapes by Buchsbaum et al. [7]. Theoretical work by Meixner and Andrejewski [8] based upon plane waves (reviewed in English by King and Wu [9], also predicted strong attenuation of the transmissioncoefficient for slits and circular aperturessmaller than R / w . Despite this supporting evidence, no d i i t measurements of the transmission coefficient could be found, and so we decided to do them ourselves.

.

PHOTON DIAMETER

3. Summary of the Experiments In all of the experiments a beam of monochromatic photons impinges upon one or more holes or slits in an otherwise opaque screen; the measurements consist of observing those photons which are transmitted through the screen undeflected. Thus the observed photons are those that have not interacted with the screen, which means that we are truly measuring a property of freely propagating photons. The principle of the original experiments was to allow a beam of circularly polarized microwaves to impinge upon a circular or rectangular aperture in a metal screen, and then measure the transmitted power on the far side of the screen. The microwave generator, the centre of the aperture, and the receiver were coaxial, and the screen was perpendicular to this axis. For a uniform flux of infinitesimalparticles, one would expect the transmitted power to be proportional to the area ofthe aperture. For finite-sized particles the effective width of the aperture is the difference between its actual width and the particle diameter; extrapolation of measurements of the transmitted power to zero as a function of the aperture width will yield the particle diameter. There were, however, a number of experimental exigencies that had to be taken into account; the requisite technique for an accurate measurement evolved through a series of experiments carried out over a 6-month period. In the light of our experience with these experiments. the relevant experimental exigencies are as follows: The transmitted power decreases towards zero as the aperture size approaches the critical size 2 l w ; discriminating low power transmitted radiation from instrumental noise is error prone. rn

The fraction oftransmittedradiation that is diffracted (i.e. bent through an angle as it passes through the aperture) increases as the aperture size is reduced towards the critical size of A l w . In terms of the theory diffraction occurs when a soliton impinges upon the wall ofthe aperture, the angle of bending being a function of the impact parameter of the collision. The proportion of incident photons that collide with the aperture walls will increase towards 100% as the aperture size approaches the critical size, 2 / n . For the purpose ofthe experiment we want to detect only the non- diffracted light; i.e. the solitons that pass through the aperture without colliding with its walls. Separating diffracted from non- diffracted light requires an appropriate experimental arrangement.

rn

Currents induced in the nominally opaque screen may cause some radiation to appear on the far side ofthe screen. One must be careful to separate this re-radiation through the screen from the radiation that passes through the aperture. While the relative intensity of re-ra ation tends to be small, it can be significant if the detector is close to the aperture.

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G. HUNTER

The extrapolation to zero transmitted power assumes monochromatic radiation. Any higher frequency radiation in the incident beam (harmonics in the microwave experiments) will dominate the transmitted power for aperture sizes close to the critical size, /Z / w This must be considered in analyzing the measurements.

.

4. 1985-86 Experimental Results The experiments were carried out in the research laboratory of Tribar Industries over a 6-month period from October 1985 to April 1986~.As in most experimental studies, experience gained in the early experiments led to an improved arrangement designed to improve the accuracy of the measurements in the later experiments. The microwave generators were standard production models based upon a turnstile junction and a horn antenna; they produced a beam of circularly polarized microwaves. The transmitted power was measured initially with a receiving antenna coupled to an amplifier and millivolt meter, and later with a Hewlett-Packard power meter. Hole Diameter D Imm)

Receiver Signal V (Millivolts)

92

28.0

60

14.0 1.4 4.4 0.7

30 15 12

Table I. The first experiment1 October 15, 1985: X-band radiation, circular Roles in aluminum foil. A graph of V against D shows that the lower 3 points lie on a straight line, but the upper 2 points (D760,92) deviate significantly from the line. The unmodified raw data yields a least-squares straight line through the 3 lower points @=12,15,30) having a D-axis intercept Do = 8.4 f 0.3 mm.The theoretically expected result is 9.07 mm. The first set of measurements were made with circular apertures cut in a thin aluminum screen with compasses; we aimed to control the diameter to within 0.1 mm. The measurements with X-band radiation (10.525 GHz., /Z = 28.48mm, / n = 9.07mm)are shown in Table 1. The non-linearity for the large holes may be due to non-uniformity of the beam intensity over the area of the hole; this was also observed in some of the later experiments. The useful data (3 points) is minimal for calculating statistics. This first experiment is recorded here to show that even a very simple set-up yields the result that there is practically no transmission through holes smaller than /Z / w .

2 Previous attempts (since 1986 to publish these experimental results were thwarted by referee resistance to the intepretation of the experimental measurements.

In experiments recorded in Tables 2 and 3 many more holes were used to improve the statistics, and the diameters of the drilled holes were accurately measured to the

Hole Diameter thou.

1243 1005 754

Receiver Signal P/mwatts

D/m

31.6 25.5 19.2

629

16.0

500 451 420 373 356 328

12.7 11.5 10.7 9.5 9.0 8.3

,

0.1050 0.0640 0.0350 0.0225 0.0090 0.0067 0.0035 0.001 5 0.0017 0.0003

Square Root of Signal

Computed 1st Harmonic

V

H

0.324

0.1820 0.1410 0.0986 0.0771 0.0549 0.0468 0.0414 0.0334 0.0300 0.0253

0.253 0.187 0.150 0.095

0.082 0.059 0.039 0.04 1 0.0 17

Computed Funclamntal I;-H

0.1420 0.1120 0.0884

0.0729 0.0401

0.0352 0.0176 0.0056

-

-

Table 2. Experiment) November 9, 1985: X-band radiation, circular holes drilled in thin (0.5 mm) aluminum plates and calibrated with an engineer's micrometer measuring thousandths of an inch (thou.). The Computed 1st Hannmic H 0.03 x (D -4.54) / (0.00 - 4.54) was based upon the mean value ofbe signals at D = 8.3 and D = 9.0. Visual inspection shows that the upper 3 points (D = 19.2, 25.5,3 1.6) deviate from a straight line. The least-squares straight lines of V-H vs D through 3,4, or 5 points (D = 9.5 16.0) all yield Do= 9.0 k 0.4 mm (5% error). The theoretically expected result is 9.07 mm.

the nearest thousandth of an inch. A disadvantage of this technique is that the drilling necessitated using thicker aluminum than in the first experiment; diffraction (scattering) from the internal walls of the hole shouW increase with screen thickness. IdeaHy one would use an infinitesimallythin screen. The calibrated Hewlett- Packard power meter was the dectector in these experiments. Table 2 records the results of the X-band radiation experiment, arid Table 3 the similar experiment with K-band radiation(24.15 OHz, 2 = 12.41 mm, 2 / a = 3.95 mm). Although the results were satisfactory the errors (computed from the least squares straight line) were quite large (5% and 8% respectively), and hence it was felt that the accuracy could be improved by working with a slit, because the aperture area is proportional to the slit width, whereas for a circular aperture it is proportional to the square of the diameter. The last experiment (Table 4) employed a slit instead of a circular hole; the slit width was varied by moving the aluminum plates that formed its edges. After experience with previous slit experiments the exit side of the slit was shielded with absorbing foam to prevent diffracted radiation from entering the receiving antenna. After correcting for harmonic content of the beam a satisfactory result was

G. HUNTER

162

obtained; the theoretical photon diameter of A / w was confirmed within t experimental error of 0.5%. It is note-worthy that this result discriminates between effective diameters of R / x and A / 3 ,for A / x differs by 5% from A / 3 - ten times the experimental error of 0.5%.

Hole Diameter

Receiver

thou.

Dimm

Signal P/mwarts

500

12.7

0.0145

Square Root of Signal

V 0.120

Computed l9 Harmonic H 0.0486

Computed Fundarn1 V-H 0.0714

Table 3. Experiment, November 9, 1985: K-band radiation, circular holes drilled in thin (0.5 mm) aluminum plates and calibrated with an engineer's micrometer measuring thousandths of an inch (thou.). Computed 1st Harmonic H= 0.009 x (D 1.9757)/(3.96 1.9757) based upon the signal at D = 3.96 mm. Visual inspection of a graph of V (or V-If)vs D shows that all the points lie close to a straight line, The least-squares straight line through all 11 points (D = 4.17-12.7) yields Do= 3.3 If:

-

-

0.3 mm (11% error) Erom the Computed Fundamental (V-H vs D). The theoretically expected result is 3.95 mm.

The experiments confirm the theoretical prediction [l, 21 that in transmission of light (experimentally microwaves) through apertures in a coil- duct mg screen, the light behaves as would a stream of particles (photonsolitons) of diameter A / i7c . This is experimental confirmation of our photon-soliton model [l, 21, in which the photon's electromagnetic field is contained within a circular ellipsoid whose length and cross-sectional circumference are both equal to the wavelength /Z The precision ofthese experiments (0.5%) is not sufficient to discriminate between our photon model and the classical, continuous field theory of Meixner and Andrejewski [8]. However, the classical prediction of strong attenuation below about 2 / w results from an elaborate algebraic and numerical analysis; the classical theory fails to provide a simple explanation for the phenomenon. In contrast, the finite-photon-soliton model provides a very

.

PHOTON DIAMETER 2

Receiver Signal P /rm~atts

Computed 1st

Computed

Harmonic

Fundamental

H

P-H

22.6 19.5

58

12.68 10.51

39.49

16.8 15.7 13.7

48 43

30

11.3

15

10.3 8.1

11.5

Slit Width

D imm

6.0

50

2.5 0.5

8.61 7.84 6.43 4.75 4.05 2.5 1.03

45.32 39.39 35.16

23.57 10.25 7.45

-

A

Table 4. Experiment, March 29, 1986: X-band radiation, rectangular slit (constant: length) in thin (0.5 mm) aluminum plates. The Computed 1st Harmonic H = 2.5 x (D-4.54)/(8.1-4.54) was based upon the signal at D = 8.1. Visual inspection shows that the upper 2 points (D 19.5,22.6) deviate h m astraight line. The least-squares straight line of P-H vs D through 3-5 points

(-10.3-16.8)

yields DO = 9.0710.04 rnm (0.5% error). The theoreticallyexpected result is 9.07 mm.

simple explanation for a phenomenon that is now commonly witnessed whenever a person looks through a metal screen into a microwave oven to see food cooking; the visible photons ( A / ; r ~m 0.0002 mm) readily pass through the 2 mm holes in the screen to allow the observer to see inside, while the microwave photons ( /Z / R 40 mm)do not.

5. The Photon's Intrinsic Intensity The theory [l, 21 predicts that the intrinsic intensity of the photon,

I,, is given by:

and at beam intensities substantially (say by an order ofmagnitude) lower than

I , ,the beam

is believed to consist of separately propagating spheroidal solitons; this accords with and quantifies Einstein's original concept of a beam of light as consisting of spatially separated photons [I I]. At beam intensities higher than Zp the spheroidal solitons necessarily overlap; this occurs in focused laser beams and is the essential physical condition required for multiphoton absorption to take place.

G. HUNTER The overlapping also explains (in view ofthe R

-'dependence of l pin (1)) why

long wavelength (radio) radiation behaves as a classical electromagnetic field rather than a beam of separately propagating photons; e-g. for v = 1 o6Hertz (

I , = 9.26 x

wattdcrn

, which

=300 meters),

is considerably smaller than typical operating

intensities ofradio waves. For X-band ( L = 28.48mm) and K-band ( 1 = 12.41 mrn) photons

I, has the

o-'

values 1.14 x 1 wattslcm and 3.16 x 1W4wattslcm respectively. These intrinsic photon intensities are just about the same as the working beam intensity of about 1

x 1 o - w~ d c m employed in the experiments recorded in Tables 2 and 3, the implication being that the beams consisted of closely bunched photon-solitons with some overlapping of them.

6. The /Z = 10.5 p Experiment In view ofthe above analysis it is desirable to repeat the experiments with shorter wavelength radiation. However, the wavelength must be long enough for fabrication ofholes or slits as small as / x (somewhat smaller in practice). These considerations led to the prospect of repeating the photon diameter measurements with the 10.5 ,urn radiation from a carbon dioxide laser. In continuous wave

(CW) mode this laser is highly monochromatic, and beams of lower intensity than 1, = 6

watu'cm are readily produced and measured. Furthermore the fabrication techniques employed in the microelectronicsindustry can fabricate holes/slits in a layer of metal at about the 1 micrometer level, thus satisfying the theoretical expectation of needing a range of slits to 2about 10 times this value. Thus the from somewhat smaller than 10.5 / a = 3 . 3 ~ ~

,

availability of the CO , CW laser and VLSI fabrication technology mikes the experiment with , I= 10.5 jim radiation feasible at a working beam intensity less than the intrinsic photon intensity of 6 watt/cm . An improvement to having a single hole or a single slit was suggested by Dr. Tyler Ivanco [12]. His idea was to construct aregular array ofholes or slits in a screen that is larger than the beam diameter. The window (non-metal) area of the array would be say 10-5Wo of its overall area. The array would be drawn using VLSI design software, and a set of screens with differently sized holedslits would be produced by (photographic) reduction of the original array by different scale factors. Thus all ofthe different screens would have the same ratio of window area to beam area and hence if photo were infinitesimal one would expect all the screens to transmit the same radiation intensity. The different screens constructed in this way would all have the same effective

PHOTON DIAMETER

165

window area (10-50% ofthe cross-sectional area ofthe beam); they would only differ in the hole diameter (or slit width). Thus all the screens having holes substantially larger in diameter than 2 / a should transmit the same beam intensity. At smaller hole diameters one would expect a sharp attenuation in transmission as the hole diameter drops below /Z / a even though the window area is unchanged. Forthe above reasons we anticipatethatthis experiment should yield ameasurement ofthe effective diameter ofphotons that is more accurate than the 0.5% ofour March, 1986 experiment. It may even discriminate between the finite photon-soliton model [l, 21 and the classical, algebraic theory [8]. Four gratings (with slit widths of 2.5, 3.2, 3.8 & 5 ,urn and slitspacing =10 x slit-width) were manufactured early in the year 2000 by the Canadian Microelectronics corporation3 as 3mm x 3mm silicon chips with the required pattern of slits in one of the metal layers of a VLSI circuit. Attempts to observe the transmission of CO laser radiation through these gratings were carried out in the laboratory of Reza Mani during the summer of 2000, but after much experimentation we realized that a layer of titanium oxide covering the whole area ofthe chip was preventing any transmission. Negotiations with CMC for the manufacture of chips having only the designed slotted metal layer on the silicon wafer base broke down because it would entail non-standard processing in Mitel's silicon foundry, and because CMC's mandate is to facilitate the manufacture of VLSI circuits for electrical engineering researchers at Canadian universities.

7. Visible Photon Measurements These measurements were conceived by T. Richardson in July 2000; the Bio-Microtech microscope, test slide, and the monochromator and adapters required for the experiment were assembled in August 2000 with the intent ion ofdoing the measurements on August 28, 2000. However, due to an email miscommunication this plan was not realized and the experiment is pending - dependent upon an available time-slot in the commercially active laboratories of Bio Microtech; our hope is that this will occur in December 2000 or January 2001. The experiment will be conducted by T. Richardson and G. Hunter. White light will be input to a monochromator having a wavelengthcalibrated output slit. The monochromatic light will be directed into the light-source condenser of the high-performance microscope, where it will illuminate the test-slide containing a pattern of 5 gratings (metal lines on silica). The gratings ranged from 100 nrn wide lines on 200 nrn centres, to 500 itnr wide lines on 1000 nrt1 centres. The wavelength ofthe monochromatic light will be increased to determine the wavelength at which a specific grating no longer transmits any light; i.e. appears to be black-- the lines disappear; aphoto-electric camerawill be used to allow measurements in the ultra-violet arid infra-red as well as at visible

3 CMC - an agency of the Canadian Natural Sciences and Engineering Research Council located at Queens' University, Kingston, Ontario, Canada

166

G. HUNTER

wavelengths. The effective diameter of the photons of known wavelength will be equated with the slit-width of the dark grating. The expectation is these measurements will also yield an effective photon-diameter of A / n the theoretical prediction for circularly polarized radiation. The experimental measurements will also investigate whether the effective diameter depends upon the polarization (linear or circular) of the radiation.

References [l] G. Hunter and R.L.P. Wadlinger, Physical Photons: Theory, Experiment, Interpretation, in: Quantum Uncertainties: Recent and Future Experiments and Interpretations: Proceedings of tile NATO Workshop, University of Bridgeport, Connecticut, USA 1986, NATO AS1 Series B, Vo1.162 (Plenum Press, 1987). [2] G. Hunter and R.L.R Wadlinger, Physics EssQvs, Vol. 2(1989) 158-172. [3]C. Spielman, C. Ran, N.H, Burnett, T. Brabec, NI. Geissler, A. Scrinzi, M Scinjirer, and F. Krausz, IEEE J.Selected Topics in Quantum Electronics, Vo1.4 (1998) 249-264. r4lC.L. Andrews, Optics of the Electromagnetic Spectrum (Prentice-Hall, Englewmd Cliffs, NJ., 1960) p.328. [5] R.K. Hadlock. J.APP1.PIkvs. Vo1.629 (1958) 918. [6] H.L. Robinson, J-Appl-Phys. Vo1.624 (1953) 35. [7]SJ. Buchsbauni, A.R. Mime. D.C. Hogg, 0. Bekefi, and C.A. \Voonton. J.Appl.Phys. Vo1.626 (1955) 706. [8] J. Meixner and W. Andrejewski, Ann. Physik, 7,157 (1950). [9]R.W.P. King and T.T. Wu. The Scattering and Diffraction of Waves (Haward University Press, Cambridge, Massachusetts, U.S.A. 1959). [lo] J.D. Kraus, Antennas (McGraw-jim, New York, 1950) p.178. [l 11S. Dmer, D. Irague, 0. Lochalt and F. Selleri (Editors), The Wave-Particle Dualism (D.Reide1 Company, Dordrecht, Holland, 1984). [12] T.A. rvanco, PhD. Dissertation, York University, Toronto, Canada, 1987.

WHAT IS THE EVANS-VIGIER FIELD?

VALERJ V. DVOEGLAZOV Escuelu de Fisica, Universidad Autbnomu de Zncatecus Apartado Postal C-580, Zacatecas 98068, Zac., Mejico E-mail: [email protected] e k r n x URL: http://ahobon. reduaz.mnd- vale ri/valen hbn

Abstract. We explain connections of the Evans-Vigier model with theories proposed previously. The Comay's criticism is proved to be irrelevant.

The content of the present talk is the following:

- Evans-Vigier definitions of the B ( ~field ) [I]; - Lorentz transformation properties of the

field and

the B-Cyclic

Theorem[2, 31; - Clarifications of the Ogievetskii-Polubarinov, Hayashi and Kalb-Ramond papers [4, 5, 61; - Connections between various formulations of massive/massless J= 1 field; - Conclusions of relativistic covariance and relevance of the Evans-Vigier postulates. In 1994-2000 I presented a set of papers [7] devoted to clarifications of the Weinberg (and Weinberg-like [8,9]) theories and the conceptof OgievetskE-Polubarinov notoph. In 1995-96 I received numerous e-mail communications from Dr. M. Evans, who promoted a new concept of the longitudinal ) (which is phaseless magnetic field associated with plane waves, the B ( ~field later obtained the name of M. Evans and J.-P. Vigier). Reasons for continuing the discussion during 2-3 years were: 1) the problem of massless limits of all relativistic equations does indeed exist; 2) the dynamical Maxwell equations have indeed additional solutions with energy E = 0 (apart of those 167 R.L knoroso et a1 (eds.),Gravitation and Cosmology: From the Hubbk Radius to the Planck Scale, 167-182. O 2002 Kluwer Academic Publishers. Printed in the Netherlan~i~~.

VALERI V. DVOEGLAZOV

168

with E = &lrcl, see [lo, 11, 12, 13, 141;' 3) the B ( ~concept ) met strong non-positive criticism (e. g., ref [15, 16, 171) and the situation became even more controversial in the last years (partially, due to the Evans' illness). ) and What are misunderstandings of both the authors of the B ( ~model their critics? In Enigmatic Photon (1994), ref. [l], the following definitions of the longitudinal Evans-Vigier field have been given:2 Definition I . [p.3,formula (4a)l

B(') x B ( ~=) ~B(')B(~) * , et cyclic. Definition 2. [p.6.fonnula ( l2)]

and Definition 3. [p. 16,fonnula(4 I)]

The following notation was used: K is the wave number; 4 = wt - rc. r is the phase; B(*)and B ( ~are ) usual transverse modes of the magnetic field; are usual transverse modes of the vector potential. A(') and The main experimental prediction of Evans [la,b] that the magnetization induced during light-matter interaction (for instance, in the IFE)

M

+ +

= ~ 1 "01~ 713/2 where

1 1= 2

-EOC~i, &= C

~

I~

, (6)

has not been confirmed by the North Caroline group [18]. As one can see from Figure 4 of [18] "the behaviour of the experimental curve does not match with Evans calculations". Nevertheless, let us try to deepen understanding of the theoretical content of the Evans-Vigier model. In their papers and books [I] Evans and '1f we put energy to be equal to zero in the dynarnical Maxwell equations

we come to V x E = 0 and V x B = 0,i. e. to the conditions of longitudinality. The method of deriving this conclusion has been given in [19]. 2 I apologize for not citing all numerous papers of Evans et a1 and papers of their critics due to page restrictions on the papers of this volume.

THE EVANS V IGIER FIELD

169

Vigier used the following definition for the transverse antisymmetric tensor field:

= E(O) this formula describes the right-polarized radiation. Of If course, a similar formula can be written for the left-polarized radiation. These transverse solutions can been re-written to the real fields. For instance, Comay presented them in the following way [16c] in the reference frame C:

and analized the addition of BII= f i k to (9). Making boost to other frame 1 ' not parallel to the Poynting of reference X' he claimed that a) ~ ( ~ is 1 ' a real part; c) transverse vector; b) with the Evans postulates ~ ( ~ has fields change, whereas B ( ~is) left unchanged when the boost is done to the frame moving in the x direction. Comay concludes that these observations disprove the Evans claims on these particular questions. Furthermore, he ) is inconsistenl with the Relativity Theory. claimed that the B ( ~model According to [20, Eq.(11.149)] the Lorentz transformation rules for electric and magnetic fields are the following:

where

p=

v/c, P =

IPI =

1 tanh6, y = -

- = coshd, with 6 being the parameter of the Lorentz boost. We shall further use the natural . . unit system c = li = 1. After introducing the spin matrices (si)jk = -icz~".'knd deriving relevant relations:

170

VALERl V. DVOEGLAZOV

one can rewrite Eqs. (10,ll) to the form

Pure Lorentz transformations (without inversions) do not change signs of the phase of the field functions, so we should consider separately properties , can be regarded as the negative-energy of the set of B(') and E ( ~ )which ) ~ ( ~ the 1 , positive-energy solutions in QFI' and of another set of B ( ~ and solutions. Thus, in this framework one can deduce from Eqs. (12,13)

and

(when the definitions (7) are used). To find the transformed 3-vector ~ ( ~ 1 ' is just an algebraic exercise. Here it is

THE EVANS VIGIER FIELD

171

We know that the longitudinal mode in the Evans-Vigier theory can be considered as obtained from Definition 3. Thus, considering that transforms as zero-component of a four-vector and B ( ~as ) space components of a four-vector: [20, Eq.(11.19)]

we find from (22) that the relation between transverse and longitudinal modes preserves its form:

that may be considered as a proof of the relativistic covariance of the B'~' model. Moreover, we used that the phase factors in the formula (7) are fixed between the vector and axial-vector parts of the antisymemtric tensor field for both positive- and negative- frequency solutions if one wants to have pure real fields. Namely, B(') = +iE(l) and B ( ~=) - i ~ ( ~A)S ,we have just seen the B ( ~field ) in this case may be regarded as a part of a Cvector with respect to the pure Lorenrz transformations. We are now going to take off the abovementioned requirement and to consider the general case:

Our formula (26) can be re-written to the formulas generalizing (6a) and (6b) of ref. [2] (see also above (18,19)):

One can then repeat the procedure of ref. [2] (see the short presentation above) and find out that the B ( ~field ) may have va,i,us transformation laws when the transverse fields transform with the matrix A which can be extracted from (12,13). Since the Evans-Vigier field is defined by the

VALERl V. DVOEGLAZOV

172

formula (3) we again search the transformation law for the cross product of I the transverse modes B(') x B ( ~ ) =? ] with taking into account (27,28).

[

[B(l) x

(29)

B(2)]ir = ,ia

+. i ~ (ip k)] (I

+ r (P2y-+(Sl . pI2)). 2

. ~(3)+ j

23

We used again the Definition 3 that B ( ~=) ~ ( ' ) k . One can see that we recover the formula (8) of ref. [2] (see (22) above) when the phase factors are equal to a = -7r/2, ,8 = -7r/2. In the case 0=+ ~ / 2 and fl = +7r/2, the sign of p is changed to the opposite one.' But, we are able to obtain the transformation law as for antisymmetric tensor field, for instance when a = -n/2, /3 = + ~ / 2 Namely, .~ since under this choice of the phases

the formula (30) and the formula for opposite choice of phases lead precisely to the transformation laws of the antisymmetric tensor fields:

B") is a true scalar in such a case. What are reasons that we introduced additional phase factors in Helmoltz bivectors? In 1211 a similar problem has been considered in the (112, 0) @ (0, 112) (cf. also 17, 221). Ahluwalia identified additional phase factor(s) with Higgs-like fields and proposed some relations with a gravitational potential. However, the E field under definitions (a = -7r/2, p = +n/2) becomes to be pure imaginary. One can also propose a model with the corresponding introduction of phase factors in such a way that 3

By the way, in all his papers Evans used the choice of phase factors incompatible with the B-Cyclic Theorem in the sense that not all the components are entries of antisymmetric tensor fields therein. This is the main one but not the sole error of the Evans papers and books. 4 In the case a = +n/2 and = -n/2, the sign in the third term in parentheses (formula (30) is changed to the opposite one.


173

BL to be pure imaginary. Can these transverse fields be observable? Can the phase factors be observable? A question of experimental possibility of detection of this class of antisymmetric tensor fields (in fact, of the antihermitian modes on using the terminology of quantum optics) is still open. One should still note that several authors discussed recently unusual configurations of electromagnetic fields [23, 241. Let us now look for relations with old formalisms. The equations (10) of [4] is read

for antisymmetric tensor f,,expressed through cross product of polarization vectors in the momentum space. This is a generalized case comparing with the Evans-Vigier Definition 2 which is obtained if one restricts oneself by space indices. The dynamical equations in the Ogievetski'i-Polubarinov approach are

and the new Kalb-Ramond gauge invariance is defined with respect to transformations

It was proven that the OgievetskiY-Polubarinov equations are related to the Weinberg 2(2j 1) formalism [25, 261 and [7b-f,i]. Furthermore, they [4] also claimed "In the limit m -+ 0 (or v --+ c) the helicity becomes a relativistic invariant, and the concept of spin loses its meaning. The system of 2s + 1 states is no longer irreducible: it decomposes and describes a set of different particles with zero mass and helicities f3,&(3 - I),. . . f1,0 (for integer spin and if parity is conserved; the situation is analogous for half-integer spins)5." In fact, this hints that actually the Proca-Duffin-Kemmer j = 1 theory has two massless limit, a) the wellknown Maxwell theory and b) the notoph theory (h = 0). The notoph theory has been further developed by Hayashi [5] in the context of dilaton gravity, by Kalb and Ramond [6] in the string context. Hundreds (if not thousands) papers exist on the so-called Kalb-Ramond field (which is actually the notoph), including some speculations on its connection with Yang-Mills fields.

+

' ~ f .with [27]. 1 am grateful to an anonymous referee of Physics Essays who suggested to look for possible connections. However, the work [27] does not cite the previous OgievetskiY-Polubarinovstatement.


174

In [28] I tried to use the Ogievetskiy-Polubarinovdefinitions of fp, (see (32)) to construct the "potentials" f,,. We can obtained for a massive field

0

-P2

P1

0

\

This tensor coincides with the longitudinal components of the antisymmetric tensor obtained in refs. [9a,Eqs.(2.14,2.17)] (see also below and [7i, Eqs.(lGb,l7b)]) within normalizations and different forms of the spin basis. The longitudinal states reduce to zero in the massless case under appropriate choice of the normalization and only if a j = 1particle moves along with the third axis O.Z."inally, it is also useful to compare Eq. (35) with the formula (B2) in ref. [29] in order to realize the correct procedure for taking the massless limit. Thus, the results (at least in a mathematical sense) surprisingly depend on a) the normalization; b) the choice of the fiame of reference. In the Lagrangian approach we have

and

where (if one applies the duality relations). Thus, we observe that a) it is important to consider the parity matters (the dual tensor has different parity properties); b) we may look for connections with the dual electrodynamics [30]. The above surprising conclusions induced me to start form the basic group-theoretical postulates in order to understand the origins of the OgievetskiY-Polubarinov-Evans-Vigierresults. The set of Bargmann-Wigner equations, ref [31] for j = 1 is written, e.g., ref. [32]

%here is also another way of thinking: namely, to consider "unappropriate" normalization N = 1 and to remove divergent part (in m,-+ 0)by a new gauge transformation.

THE EVANS VIGIER FlELD

where one usually uses

In order to facilitate an analysis of parity properties of the corresponding fields one should introduce also the term -- ( - i 5 0 p u ~ ) a 0 f l , . In order to understand normalization matters one should put arbitrary (dimensional. in general) coefficients in this expansion or in definitions of the fields and 4-potentials [28]. The R matrix is

Matrices yP are chosen in the Weyl representation, i.e., y5 is assumed to be diagonal. The reflection operator R has the properties

They are necessary for the expansion (41) to be possible in such a form, i.e., in order the yPR, uPVRand (if considered) y 5 c r ~ uto~ be syinm-etrical matrices. I used the expansion which is similar to (41)

and obtained

If one renormalizes FP -+ 2mAP or Fp, -+ &F,, one obtains "textbooks" Proca equations. But, physical contents of the massless limits of these eyuations may b4 diflerent. Let us track origins of this conclusion in detail. If one advocates the following definitions [33, p.2091

176

and ($

VALERI V. DVOEGLAZOV = P"

/l p 1,

7= Ep/m),ref. [33, p.681 or ref. [34, p. 1081,

for the field operator of the 4-vector potential, ref. [34, p.1091 or ref. [35, p. 129]'v8

(54)

the normalization of the wave functions in the momentum representation is thus chosen to the unit, rt(p, h ) d ( ph) = -l.g We observe that in the massless limit all defined polarization vectors of the momentum space do not have good behaviour; the functions describing spin-1 particles tend to infinity. This is not satisfactory, in my opinion, even though one can still claim thal singularities may be removed by rotation and/or choice of a gauge parameter. After renormalizing the potentials, e. g., EP -+ u p m.& we come to the field functions in the momentum representation:

7~emember that the invariant integral measure over the Minkowski space for physical particles is

Therefore, we use the field operator as in (54). The coefficient (2n)3 can be considered at this stage as chosen for convenience. In ref. [33] the factor 1/(2Ep) was absorbed in creationlannihilation operators and instead of the field operator (54) the operator was used in which the tC'(p,h) functions for a massive spin-1 particle were substituted by up(p, h) = ( 2 ~ ~tp(p, ) h,), ~ ~ which ' ~may lead to confusions in searching massless limits m --+ 0 for classical polarization vectors. 6in general, it may be useful to consider front-form helicities (or "time-like" polarizations) too. But. we leave a presentation of a rigorous theory of this type for subsequent publications. he metric used in this paper gfiY = diag(1, - 1,- 1, - 1) is different from that of ref. [33].


*

( N = rn and pqz = pl i p z ) which do not diverge in the massless limit. Two of the massless functions (with h = f1) are equal to zero when the particle, described by this field, is moving along the third axis (pl = = 0, p3 # 0). The third one ( h= 0 ) is

and at the rest (Ep = p3 -+ 0 ) also vanishes. Thus, such a field operator describes the "longitudinal photons" which is in complete accordance with the Weinberg theorem B - A = h for massless particles (let us remind that we use the D(l.12, L/2) representation). Thus, the change of the normalization can Lead to the change of physical content described by the classical field (at least, comparing with the well-accepted one). Of course, in the quantum case one should somehow fix the form of commutation relations by some physical principles.1 [f one uses the dynamical relations on the basis of the consideration of polarization vectors one can find fields:

and

'I am very grateful to the anonymous referee of my previous papers ("Foundation of Physics") who suggested to fix them by requirements of the dimensionless nature of the action (apart from the requirements of the translational and rotational invariancies).


where we denoted, as previously, a normalization factor appearing in the definitions of the potentials (and/or in the definitions of the physical fields through potentials) as N. E(p, 0) and B(p, 0) coincide with the strengths obtained before by different method [9a.28], see also (35). B ' ( ~ ,Ot) = E * ( ~ , O ~= ) 0 identically. So, we again see a third component of antisymmetric tensor fields in the massless limit which is dependent on the normalization and rotation of the frame of reference. However, the claim of the pure "longitudinal nature" of the antisymmetric tensor field and/or "Kalb-Ramond" fields after quantization still requires further explanations. As one can see in [5] for a theory with L = the application of the condition (A!+)(x),~)~P>= 0 (in 2-7 our notation 8, ffiv = 0 ) , see the formula (18a) therein, leads to the above conclusion. Transverse modes are eliminated by a new "gauge" transformations. Indeed, the expanded tagrangian is

-BF~F~

Thus, the Ogievetski'i-Polubarinov-Hayashi Lagrangian is equivalent to the Weinberg's Lagrangian of the 2(2j + 1) theory [36] and [7a-el: which is constructed as a generalization of the Dirac Lagrangian for spin 1 (instead of bispinors it contains bivectors). In order to consider a massive theory (we insist on making the massless limit in the end of calculations, for physical 1 2 fpu f pu as in (37). quantities) one should add + zm The spin operator of the massive theory, which can be found on the basis of the Nother formalism, is

2 ~ h formal e difference in Lagrangians does not lead to physical difference. Hayashi said that this is due to the possiblity of applying the Fermi method mutatis mutundis.


179

In the above equations we applied dynamical equations as usual. Thus, it becomes obvious, why previous authors claimed the pure longitudinal nature of massless antisymmetric tensor field after quantization, and why the application of the generalized Lorentz condition leads to equating the spin operator to zero.' But, one should take into account the nomalization issues. An additional mass factor in the denominator may appear a) after "re-normalization" 1: -+ 1:/m2(if we want to describe long-range forces an antisymmetric tensor field must have dimension [energy12 in the c = h. = 1 unit system, and potentials, [energy]' in order the corresponding action would be dimensionless; b) due to appropriate change of the commutation relations for creationlannihilation operators of the higher-spin fields (including -- 1/m);c) due to divergent terms in E. B. A in m -+ 0 under certain choice of N. Thus, one can recover usual quantum electrodynamics even if we use fields (not potentials) as dynamical variables. The conclusions are: - While first experimental verifications gave negative results, the B ( ~ )

-

-

-

-

construct is theoretically possible, if one develops it in a mathematically correct way; The B ( ~ model ) is a relativistic covariant model. It is compatible with the Relativity Theory. The B ( ~field ) may be a part of the 4-potential vector, or (if we change connections between parts of Helmoltz bivector) may be even a part of antisymmetric tensor field; The B ( ~ model ) is based on definitions which are particular cases of the previous considerations of Ogievetski'i and Polubarinov. Hayashi and Kalb and Ramond; The Duffin-Kemmer-Proca theory has two massless limits that seems to be in contradictions with the Weinberg theorem (B- A = h); Antisymmetric tensor fields after quantization may describe particles of both helicity h = 0 and h = &lin the massless limit. Surprisingly, the physical content depends on the normalization issues and on the choice of the frame of reference (in fact, on rotations).

Acknowledgments. I am thankful to Profs. A. Chubykalo, E. Comay, L. Crowell, G. Hunter, Y. S. Kim, organizers and participants of the Vigier2K, referees and editors of various journals for valuable discussions. I acknowledge many internet communications of Dr. M. Evans (1995-96) on the concept of the field, while frequently do not agree with him in many particular questions. I acknowledge discussions (1993-98) with Dr. D. Ahluwalia (even though I do not accept his methods in science). 3 ~ 1is still interesting to note that division of total angular momentum into orbital part and spin part is not gauge invariant.

180


I am grateful to Zacatecas University for a professorship. This work has been supported in part by the Mexican Sistema Nacional de Investigadores and the Programa de Apoyo a la Carrera Docente.

References 1. Evans M. W. (1992) Physicu B182, 227; ibid. 237; (1993) Modern Non-linear Optics. [Series Adv. Cherlr. Phys. Vol. 85(2)], Wiley Interscience, NY; (1994-1999) in Evans M. W., Hunter G.. Jeffers S., Roy S. and Vigier J.-P. (eds.), The Enigmatic Photon. Vols. 1-5, Kluwer Academic Publishers, Dordrecht. 2. Dvoeglazov V. V. (1997) Found Phys. Lett. 10, 383. This paper presents itself a comment on the debates between E. Comay and M. Evans and it criticizes both authors. 3. Dvoeglazov V. V. (2000) Found.Phs. Lett. 13, 387. 4. Ogievetski'i V. I. and Polubarinov I. V. (1966) Yud Fiz. 4. 216 [Translation: (1967) Sov. J. Nucl. P h p 4, 1561. 5. Hayashi K. (1973) Phys. Lett. B44, 497. 6. Kalb M. and Ramond P. (1974) Phys. Rev. D9, 2273. 7. Dvoeglazov V. V. (1994) Rev. Mex. Fis. Sqpl. 40, 352; (1997) Helv. Phys. Acta 70, 677; ibid. 686: ibid. 697; (1998) Ann. Fond L. cle Broglie 23. 116; (1998) Int. 1. Theor. Phys. 37, 1915; (1999) ibid. 38, 2259; (1998) Electrorrlagnetic Phenorrrena 1, 465; (2000) Czech J. Phys. 50, 225; (2000) in V. V. Dvoeglazov (ed.), Photon:Old Problerns in Light of N m Ideus, Nova Science Publishers, Huntington. 8. Sankaranarayanan A. and Good R. H., jr. (1965) Nuovo Cirri. 36, 1303; (1965) Phys. Rev. 140, B509; Sankaranarayanan A. (1965) Nuovo Cile 38, 889. 9. Ahluwalia D. V. and Ernst D. 1. (1993) Int. J. Mod. Phys. E2, 397; Ahluwalia D. V., Johnson M. B. and Goldman T. (1993) Phys. Lett. B316, 102. lo. Oppenheimer J. R. (1931) Phys. Rev. 38, 725. 11. Good R. H., Jr. (1957) P h s . Rev. 105, 1914 (see p. 1915); (1959) in Lectures in theoretical physics. Universio of Colorado. Boulder, Interscience, p. 30 (see p. 47); Nelson T. J. and Good R. H., Jr. (1%9) Phys. Rev. 179, 1445 (see p. 1446). 12. Gianetto E. (1985) Lett. Nuovo Cim. 44, 140; (1985) ibid. 145. See pp. 142 and 147, respectively. E. Recarni et ul. were very close to re-discover this solution in the papers (1974) Lett. Nuovo Cirn. ll, 568 and (1990) in M. Mijatovic (ed.), Hdronic r~rechunicsand nonpotential interactions, Nova Science Pubs., New York, p. 231 but in the analysis of the determinant of the Maxwell equations they put additional constraint of transversality p . = 0 on the solutions of Eq. (1,2). 13. Ahluwalia D. V. and Ernst D. J. (1992) Mod. Phys. Lett. A7, 1967; Ahluwalia D. V. (19%) in G. Hunter et ul. (eds.), Proceedings of The Present Status of Quunt~alrTheory of Light: A Sy?~zposiurnto Honour Jeun-Pierre Vigier. York Universif?: Toronto, Aug. 27-30. 1995, Kluwer Academic Publishers. p. 443. 14. Dvoeglazov V. V., Tyukhtyaev Yu. N. and Khudyakov S. V. (1994) 13. WZ:& 37, 110 [Translation: (1994) Russ. Phys. J. 37, 8481. This is a version of the Saratov University preprint of 1992, which was revised in order to include references to the works appeared in 1991-1994. 15. Barron L. D. (1993) Physica B190, 307; Lakhtakia A. (1993) Physicu B191, 362; (1995) Found. Phys. Lett. 8, 183; Grimes D. M. (1993) Physica B191, 367. 16. Comay E. (1996) chert^. Php. Lett. 261, 601; (1996) Physica B222, 150; (1997) Found Phys. Lett. 10, 245: (1997) Physicu A242, 522; (1999) Apeiron 6. 233. 17. Hunter G. (1999) Chern. Phys. Lett. 242, 331; (2000) Apeiron 7, 17. 18. Akhtar Raja M. Y., Sisk W. N., Yousaf M. and Allen D. (1995) Appl. Phys. Lett. 67, 2123; (1997) Appl. Phys. B64. 79. 19. Dvoeglazov V. V. et a1 (1997) Apeiron 4, 45.

+

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181

20. Jackson J. D. (1980) Electrodinhrzica Clhsica. Spanish edition, Alhambra S. A. 21. Ahluwalia D. V. (1998) Mod. Phys. Lett. A13, 3123; see also V. V. Dvoeglazov's comment on this paper, to be published. 22. Dvoeglazov V. V. (1997) in J. Keller and Z. Oziewicz (eds.), Advunces in ClifSord Algebrus - Proceedings of the Int. Conference on the Theory of the Electron. Cuautitlan, Mexico, Sept. 27-29. 1995. Vol. 7C, UNAM, Mkxico. pp. 303-319. 23. In (1993) A. Lakhtakia (ed.), Essays on the Foi.trla1 Aspects of Electroinugnetic Theory, World Scientific, Singapore. 24. In (1995) T. W. Barrett and D. M. Grimes (eds.), Advanced Electroirzugnetisiiz: Foundations, Theory and Applications, World Scientific, Singapore. 25. S. Weinberg, Phys. Rev. B133 (1964) 1318: ibid B134 (1964) 882: ibid 181 (1969) 1893. 26. Dvoeglazov V. V. (1997) Weinberg Formalism and New Looks at the Electromagnetic Theory, in J.-P. Vigier et al. (eds.), The Enigirzutic Photon. Vol. IV, Kluwer Academic Publishers, Dordrecht, Chapter 12, and references therein. 27. Kirchbach M. (2000) Rarita-Schwinger Fields without Auxiliary Conditions in Baryon Spectra, in A. Chubykalo, V. Dvoeglazov et al. (eds.), Lorent. Group, CPT and Neutrinos. Proceedings of the Internationul Workshop, World Scientific, Singapore, pp. 212-223. 28. Dvoeglazov V. V. (1998) Photon-Notoph Equations, physics/9804010, to be published. 29. Ahluwalia D. V. and Sawicki M. (1993) Phys. Rev. D47, 5161. 30. Strazhev V. I. and Kruglov S. 1. (1977) Acta Phys. Polon. B8, 807; Strazhev V. I. (1978) ibid. 9, 449: (1977) Int. J. Theor. Phys. 16. 111. 31. Bargmann V. and Wigner E. P. (1948) Proc. Nat. Acud Sci. (USA) 34, 211. 32. Lurii D. (1968) Purticles and Fields, Interscience Publisher. New York, Chapter 1. 33. Weinberg S. (1995) The Quuntuirt Theory of Fields. Vol. I. Foundutions, Cambridge University Press, Cambridge. 34. Novozhilov Yu. V. (1975) Introduction to Eleirtentury Particle Theoty. Pergamon Press, Oxford. 35. Itzykson C. and Zuber J.-B. (1980) Quanturrl Field Theory, McGraw-Hill Book Co., New York. 36. Dvoeglazov V. V. (1993) Hudronic Journal 16, 459.

NON-ABELIAN GAUGE GROUPS FOR REAL AND COMPLEX AMENDED MAXWELL'S EQUATIONS E.A. RAUSCHER Tecnic Research Laboratory, 3500 S. Tomahawk Road, Bldg. 188, Apache Junction, AZ 85219, USA, Email: [email protected]

Abstract. We have analyzed, calculated and extended the modification of Maxwell's equations in a complex Minkowski metric, M4in a C2space using the SU2gauge, SL(2,c) and other gauge groups, such as SUn, for n>2 expanding the U, gauge theories of Weyl. This work yields additional predictions beyond the electroweak unification scheme. Some of these are: 1) modified gauge invariant conditions, 2) short range non-Abelian force terms and Abelian long range force terms in Maxwell's equations, 3) finite but small rest of the photon, and 4) a magnetic monopole like term and 5) longitudinal as well as transverse magnetic and electromagnetic field components in a complex Minkowskimetric R/i4 in a C4space. 1. Introduction

We have developed an eight dimensional complex Minkowski space M4(1) composed of four real dimensions and four imaginary dimensions which is consistent with Lorentz invariance and analytic continuation in the complex plane [1]. The unique feature of this geometry is that it admits of nonlocality consistent with Bell's theorem, (EPR paradox), possibly Young's double slit experiment. the Aharonov - Bohm effect and multi mirrored interferometric experiment Additionally, expressing Maxwell's electromagnetic equations in complex eight space, leads to some new and interesting predictions in physics, including possible detailed explanation of some of the previously mentioned nonlocality experiments [2]. Complexification of Maxwell's equations require a non-Abelian gauge group which amend the usual theory, which utilizes the usual unimodular Weyl U1 group. We have examined the modification of gauge conditions using higher symmetry groups such as SU2, SUn, and other groups such as the SL(2,c) double cover group of the rotational group SO(3,l) related to Shipov's Ricci curvature tensor [3] and a possible neo-aether picture. Thus we are led to new and interesting physics involving extended metrical space constraints. the usual transverse and also longitudinal, non Hertzian electric and magnetic field solutions to Maxwell's equations, possibly leading to new communication systems 183 R.L. Amoroso et a1 (eds.).Gravitation and Cosmology From the Hubble Radius to the Planck Scale, 183-188. (C)2002 Kluwer Academic Publishers.Printed in the Netherlands

E.A. RAUSCHER

184

B

and antennae theory, non zero solutions to 8 , and a possible finite but small rest mass of the photon. Comparison of our theoretical approach is made to the work of J. P. Vigier, [4] T.W. Barretl [5] and H.F. Hannuth's [6] work on amended Maxwell's theory. We compare ) of Vigier, and our Nonour predictions such as our longitudinal field to the B ( ~term Abelian gauge groups to that of Barrett and Harmuth. This author interprets this work as leading to new and interesting physics, including a possible reinterpretation of a neoaether with nonlocal information transmission properties.

2. Complexified EM Fields In The MqMinkowski Space And Nonlocality We expand the usual line element metric dg = g,UyG?X"Gk in the following manner. We consider a complex eight dimensional space, and likewise for

Z V where the indices V and

11

constructed so that ZU= X, ,U run 1 to 4 yielding

+ i XIm II

(1, 1, 1, -1).

Hence, we now have a new complex eight space metric as ds = r ] , d z "dz'

.

We have

developed this space and other extended complex spaces (1) and examined their relationship with the mister algebras and asymptotic twister space and the spinor calculus and other implications of the theory [7]. The Penrose twister SU(2,2) or U4 is 'V

constructed from four space -time, Uz63)U 2where Uz is the real part of the space and

o2is the imaginary part of the space. this metric appears to be a fruitful area to explore. The twister Z can be a pair of spinors U* and nAwhich are said to represent the twister. The condition for these representations are 1) the null infinity condition for a

2.

= 0 ,2) conformal invariance and 3) independence of the origin. zero spin field is 2" The twister is derived from the imaginary part of the spinor field. The underlying concept of twister theory is that of conformally invariance fields occupy a fundamental role in physics and may yield some new physics. Since the twister algebra falls naturally out complex space. Other researcher have examined complex dimensional Minkowski spaces, In reference [lo], Newman demonstrates that IV4 space do not generate any major "weird physics" or anomalous physics predictions and is consistent with an expanded or amended special and general relativity. In fact the Kerr metric falls naturally out of this formalism as demonstrated by Newman [l 11. As we know twisters and spinors are related by the general Lorentz conditions in such a manner that all signals are lurninal in the usual four N Minkowski space but this does not preclude super or trans luminal signals in spaces where N>4. H. Stapp, for example, has interpreted the Bell's theorem experimental results in terms of trans luminal signals to address the nonlocality issue of the Clauser, et. al and Aspect experiments. C.N. Kozameh and E.T. Newman demonstrate the role of non local fields in complex eight space [16]. We believe that there are some very interesting properties of the &'I space which include the nonlocality properties of the metric applicable in the non-Abiliass algebras

GAUGE GROUPS FOR MAXWELL'S EQUATIONS

185

related to the quantum theory and the conformal invariance in relativity as well as new properties of Maxwell's equations. In addition, complexification of Maxwell's equations in M4 space yields some interesting predictions. yet we find the usual conditions on the manifold hold [28]. Some of these new predictions come out ofthe complexification of four space 2 and appear to relate to the work of Vigier, Barrett, Harmuth and others [4, 5, 61. Also we find that the twister algebra of the complex eight dimensional, M4 space iq mapable 1 to 1 with the twister algebra, C4 space of the Kaluza-Klein five dimensional electromagnetic - gravitational metric [17, 181. Some of the predictions of the complexified form of Maxwell's equations are 1) a finite but small rest mass of the photon, 2) a possible magnetic monopole, V fi # 0, 3) transverse as well as longitudinal B(3) like components of_Eand E3, 4) new extended gauge invariance conditions to include non-Abelian algebras and 5) an inherent fundamental nonlocality property on the manifold. Vigier also explores longitudinal _E and B components in detail and finite rest mass of the photon [19]. We consider both the electric and magnetic fields to be complexified as E = E,, = iE, and = i&, for ERe,E , ,BR, and Bh are real quantities. Then substitution of these two equations into the complex form of Maxwell's equations above yields, upon separation of real and imaginary parts, two sets of Maxwell-like equations. The first set is

B BRe

the second set is

The real part of the electric and magnetic fields yield the usual Maxwell's equations and complex parts generate "mirror" equations; for example, the &vergence of the real component of the magnetic field is zero, but the divergence of the imaginary part ofthe electric field is zero, and so forth. The structure of the real and imaginary parts of the fields is parallel with the electric real components being substituted by the imaginary part ofthe magnetic fields and the real part of the magnetic field being substituted by the imaginary part of the electric field. In the second set of equations, (2), the l's, "go out" so that the quantities in the equations are real, hence V Bh = 4np,,and not zero, yielding a term that may be associated with some classes of monopole theories. See references in ref. [2].

186

E.A. RAUSCHER

We express the charge density and current density as complex quantities based on the separation of Maxwell's equations above. Then, in generalized form

p = pe = ipm and J=J,+iJ,

where it may be possible to associate the imaginary complex charge with the magnetic monopole and conversely the electric current has an associated imaginary magnetic current. The alternate of defining and using, which Evans does E = + iBI, and B = + iEImwould not yield a description of the magnetic monopole in terms of complex quantities but would yield, for example

V ( i ~ ), = 0 in the second set of equations.

-&

and for Using the invariance of the line element s2 = 3 - c2t2 for r = d = x = x2-ky2 -C z2for the distance from an electron charge, we can write the relation, 2

3. New Gauge Conditions, Complex Minkowski M4Space And New Implications F o r Physics In a series of papers. Barrett. Harrnuth and Rauscher have examined the modification of gauge con tions in modified or amended Maxwell theory. The Rauscher approach. as briefly explained in the preceding section is to write complexified Maxwell's equation in consistent form to complex Minkowski space [2]. The T.W. Barrett amended Maxwell theory utilizes non-Abelian algebras and leads to some very interesting predictions which have interested me for some years. He utilizes the non commutive SU2gauge symmetry rather than the Ul symmetry. Although the Glashow electroweak theory utilizes UI and SU2, but in a different manner, but his theory does not lead to the interesting and unique predictions ofthe Barrett theory. T.W. Barrett, in his amended Maxwell theory, predicts that the velocity of the propagation of signals is not the velocity of light. He presents the magnetic monopole concept resulting from the amended Maxwell picture. His motive goes beyond standard Maxwell formalism and generate new physics utilizing a non-Abelian gauge theory.[5] The SU2group gives us symmetry breaking to the UI group which can act to create a mass splitting symmetry that yield a photon of finite (but necessarily small) rest mass which may be created as self energy produced by the existence of the vacuum1. This finite rest mass photon can constitute a propagation signal carrier less than the velocity of light. We can construct the generators ofthe SU2 algebra in terms of the fields I Z,l3, and A. The usual potentials.

A,

is the important four vector quality

A, = (&,4)

where the index runs 1 to 4. One of the major purposes of introducing the vector and scalar potentials and also to subscribe to their physicality is the desire by physicists to

di

GAUGE GROUPS FOR MAXWELL'S EQUATIONS avoid action at a distance. In fact in gauge theories

A,

187

is all there is! Yet, it appears

that, in fact, these potentials yield a basis for a fundamental nonlocality! Let us address the specific case of the SU2group and consider the elements of a non-Abelian algebra such as the fields with SU2 (or even SU,) symmetry then we have the commutation relations where XY-YX # 0 or p , Y ] # 0. Which is reminiscent of the Heisenberg uncertainty principle non-Abelian gauge.2 Barrett does explain that SU2 fields can be transformed into UI fields by symmetry breaking. For the SU2 gauge amended Maxwell theory ad tional terms appear in term of operations such A E, A B and A x B and their non Abelian converses. For example V B no longer equals zero but is given as

V B = -jg( A B - B A) # 0

where

[A,B] # 0 for the dot product of A and B and hence we have a magnetic monopole term and j is the current and g is a constant. Also Barrett gives references to the Dirac, Schwinger and G. t Hooft monopole work. Further commentary on the SU2 gauge conjecture of H.F. Mamuth [6] that under symmetry breaking, electric charge is considered but magnetic charges are not. Barrett further states that the symmetry breaking conditions chosen are to be determined by the physics of the problem. These non Abelian algebras have consistence to quantum theory. In this author's analysis, using the SU2group there is the automatic introduction of short range forces in addition to the long range force of the U1 group. UI is one diis three dimensional and is non-Abelian. U, is also a dimensional and Abelian and SU2 subgroup of SU2. The UI group is associated with the long range 1 / r 2 force and SU2, such as for its application to the weak force yields shorl range associated fields. Also SU2 is a subgroup of the useful SL(2,c) group of non compact operations on the manifold. SL(2,c) is a semi simple four dimensional Lie group and is a spinor group relevant to the relativistic formalism and is isomorphic to the connected Lorentz group associated with the Lorentz transformations. It is a conjugate group to the SU2group and contains an inverse. The double cover group of SU2 is SL(2,c) where SL(2,c) is a complexificationof SU2. Also LS(2,c) is the double cover group of SU3related to the set of rotations in three dimensional space [3]. Toplogically, SU2 is associated with isomorphic to the three dimensional spherical, 03* (or three dimensional rotations) and U, is associated with the O2 group of rotations in two dimensions. The ratio of Abelian to non Abelian components, moving from U1to SU2, gauge is 1 to 2 so that the short range components are twice as many as the long range components. Instead of using the SU2gauge condition we use SL (2,c) we have a non-Abelian gauge and hence quantum theory and since this group is a spinor and is the double cover group of the Lorentz group (for spin %) we have the conditions for a relativistic formalism. The Barrett formalism is non-relativistic. SL (2,c) is the double cover group of SU2but utilizing a similar approach using twister algebras yields relativistic physics. It appears that complex geometry can yield a new complementary unification of quantum theory, relativity and allow a domain of action for nonlocality phenomena, such as displayed in the results of the Bell's theorem tests of the EPR paradox [22], and in which the principles of the quantum theory hold to be universally. The properties of the nonlocal connections in complex four space may be mediated by non -or low dispersive loss solutions. We solved Schriidinger equation in complex Minkowski space [25].

E.A. RAUSCHER

188

In progress is research involving other extended gauge theory models, with particular interest in the nonlocality properties on the S pact-time manifold, quantum properties such as expressed in the EPR paradox and coherent states in matter. Utilizing Coxeter graphs or Dynkin diagrams, Sirag lays out a comprehensive program in terms of the A,, Dn, and E6,E7, and E8 Lie algebras constructing a hyper dimensional geometry for as a classification scheme for elementary particles. Inherently, this theory utilizes complexified spaces involving twisters and Kaluza-Klein geometries. This space incorporates the string theory and GUT models [27].

4. Conclusion It appears that utilizing the complexification of Maxwell's equations with the extension of the gauge condition to non-Abelian algebras, yields a possible metrical unification of relativity, electromagnetism and quantum theory. This unique new approach yields a universal nonlocality. No radical spurious predictions result from the theory, but some new predictions are made which can be experimentally examined. Also, this unique approach in terms of the twister algebras may lead to a broader understanding of macro and micro nonlocality and possible transverse electromagnetic fields observed as nonlocality in collective plasma state and other media.

References 1. P. Penrose and E.J. Newman, Proc. Roy. Soc. A363, 445 (1978). 2. E.T. Newman. J. Math. Phys. l 4 , 774 (1973). 3. R.O. Hansen and E.T. Newman, Gen. Rel. and Grav. 6, 216 (1975). 4. E.T. Newman, Gen. Rel. and Grav. 7, 107 (1976). 5. H.P. Stapp Phys. Rev. A47. 847 (1993) and Private Communication. 6. J.S. Bell, Physics 1,195 (1964). 7. J.F. Clauser and W.A. Horne Phys. Rev. 526 0D, (1974) and private communication with J. Clauser 1977. 8. A. Aspecl et. al. Phys. Rev. 4 3 1804 (1982) and private communication. 9. E.T. Newman and E.T. Newman, third MG meeting on Gen. Rel., Ed. Ha Nang, Amsterdam Netherlands, North-Holland, pgs 51-55 (1983). 10. Th. Kaluza, sitz. Berlin Press, A. Kad. Wiss, 968 (1921). 11. 0. Klein Z. Phys. 37- 805 (1926) and 0. Klein. Z Phys. 4L 407 (1927). 12. J.P. Vigier, Found. Of Phys. 2-l 125 (1991). 13. M.W. Evans and J.P. Vigier "the enigmatic photon" l and 2_ "Non-Abelian Electrodynamics", Kluwer Acad Dordrecht (1994. 1995. 1996). 14. E.A. Rauscher, Bull. Am. Phys. Soc. 2 L1305 (1976). 15. EA. Rauscher, J. Plasma Phys. 2. 16. T.T. Wu and C.N. Yang, "Concepts of Nonintergreble phase factors and global formulation of gauge fields", Phys. Rev. 3845 (1975). 17. EA. Rauscher, "D and R Spaces, Lattice Groups and Lie Algebras and their Structure", April 17, 1999. 18. E.A. Rauscher "Soliton Solutions to the Schradinger Equation in Complex Minkowski Space", pps 89-105, proceed in^ of the First International Conference , 19. A. Einstein. B. Podolsky and N. Rosen. Phys. Rev. 47- 777 (1935). 20. EARauscher, Complex Minkowski Space & Nonlocality on the Metric & Quantum Processes, in progress. 21. S.P. Sirag "A Mathematical Strategy for a Theory of Particles", pps 579-588, The First Tucson Conference, Eds. S.R. Hamemff, A.W. Kasniak and A.C. Scott, MIT Press, Cambridge, MA (1996). 22. T.T. Wu & C.N. Yang, (1975), Phys. Rev. D12, 3845. 23. N. Gisin, Phys. Lett. 143, 1 (1990). 24. W. Tittel, J. Bredel, H. Zbinden & N. Gisin, Phys. Rev. Lett. 81, 3563. 25. E.A. Rauscher, Proc. 1" Int. Conf. , Univ. of Toronto, Ontario, Canada, pp. 89-105, (1981). 26. T-RLove, Int. J. ofTheor. Phys. 32, 63 (1993).

1

m,

EXPERTMENTAL EVIDENCE OF NEAR-FIELD SUPERLUMINALLY PROPAGATING ELECTROMAGNETIC FIELDS WILLIAM D. WALKER Royal Institute of Technology, KTH-Visby Department of Electrical Engineering CrarrIkrgatan 3, 9 6 2 1 57 Visby, Sweden Expanded paper at: http://xxx.lanl.gov/abs/physics/0009023 bill @visby.kth.se

1.

Theoretical analysis of a n oscillating electric dipole

Numerous textbooks present solutions of the electromagnetic (EM) fields generated by an oscillating electric dipole. The resultant electrical and magnetic field components for an oscillating electric dipole are known to be [l, 21:

Figure 1. Spherical co-ordinate system used to analyze electric dipole and resulting EM field solutions

It should be noted that all of the above solutions are only valid for distances (r) much greater than the dipole length (d,). In the region next to the source (r do), the source cannot be modeled as a sinusoid: sfdot). Instead it must be modeled as a sinusoid ) ] . solution of this problem can be inside a Dirac delta function: b f r - d , ~ i ~ ( ~ The calculated using the Lienard-Wiechert potentials [3. 4. 51. It is noted From the above analysis that the Eield solutions of the electric dipole can be written as a sum of sinuioidal waves, which travel away from the dipole source at the speed of light. Even if the waves are generated by unique physical mechanisms. only the superposition of the waves is observable at any point in space. These wave components in effect form a new wave which may have different properties from the original components, Only the longitudinal and transverse wave components are real since they can be decoupled by proper configuration of a measurement antenna. The following analysis derives general relations that are used to determine the instantaneous phase and group speed vs distance graphs for the field components.

-

189 R.L. Amorow et aL fed^.), Gravitation and Cosmology: From the Hubble Rcidius to the Planck Scale, 189-196. O 2002 KLu~verAcaderitic Publishers. Printed in the Netherlunds

190 1.1.

WILLIAM D. WALKER DERIVATION OF PHASE SPEED AND GROUP SPEED RELATIONS

In several papers written by the author [3,4, 61 and in reference PI, the instantaneous phase speed has been determined to be: c,h = w/(d8/&). Using the following relations: w = 2nf, re!= rlh, h f = c,, and 2~ = 360 deg the instantaneous phase speed becomes: -

1

nj eindeg This relation (ref. Eq. 4) shows that the phase speed is inversely proportional to the slope of the phase vs distance curve. Note, zero slope would imply infinite phase speed. In addition, in the same papers [3,4, 6, 7] it is determined that the instantaneous group speed is: cg = [(a2e/(h8r)]-'. This relation can be made a function of (kr) by multiplying the numerator and the denominator by (IF w/c,) yielding: cg = c,[(d/aw){~ae/a(kr)}]-' = c,[w (a/aw){ae/a(kr))+ aO/a(kr)]-'. Using the relation: w = c,k, and the group speed becomes: cg= C, [kr(d/a(kr)) {ae/a(kr)}+ %/a(kr)]". Using the following relations: re! = rlh = kr/(2n), and that 27r = 360 deg, the instantaneous group speed becomes:

The above relation (ref. Eq. 5) indicates that the instantaneous group speed is inversely proportional to both the curvature and the slope of the phase vs distance curve. Note that if the denominator of the above equation is zero, the group speed will be infinite. Also note that if the curvature is zero, the group speed equation (ref. Eq. 5) will be the same as the phase speed equation (ref. Eq. 4). The group speed is thought to be the speed at which wave energy and wave information propagate [8, 14 (p.26811. 1.2. TRANSVERSE ELECTRIC FIELD (Ee) Applying the phase [cph= cok/(%/8r)] and group speed [c, = [c,,(@e/(&ak)~] relations to the transverse electrical field (Ee) component (ref. Eq. 2) yields the following results

161: t9=kr-cos-'

[]*j

l-(hY cfi ='[-2W'

"

c =

axl

(61

EB Plrm?[t?) vs Rr

m

+(AT)'

+ty)

co(I - (b)'+ (b)' - 6(b)' + 7(kr)' - (krI6+ (kr)' Figure 2. Eg phase (8)vs kc

SUPERLUMINAL EM FIELDS

Figure 3. Eo cphvs kr

Figure 4. Eg cg vs kt

Plots of the longitudinal electrical field (Ee) and transverse magnetic field (H4) are found at the following reference [6].

1.2.1. Zntep-etation of theoretical results The above theoretical results suggest that longitudinal electric field waves and transverse magnetic field waves are generated at the dipole source and propagate away. Upon creation, the waves (phase and group) travel with infinite speed and then rapidly reduce to the speed of light after they propagate about one wavelength away from the source. In addition, transverse electric field waves (phase and group) are generated approximately one-quarter wavelength outside the sowce and propagate toward and away from the source. Upon creation, the transverse waves travel with infinite speed. The outgoing transverse waves reduce to the speed of light after they propagate about one wavelength away from the source. The inward propagating transverse fields rapidly reduce to the speed of light and then rapidly increase to infinite speed as they travel into the source. In addition, the above results show that the transverse electrical field waves are generated about 90 degrees out of phase with respect to the longitudinal waves. In the near-field the outward propagating longitudinal waves and the inward propagating transverse waves combine together to form a type of oscillating standing wave. Note that unlike a typical standing wave, the outward and inward waves are completely different types of waves (longitudinal vs transverse) and can be separated by proper orientation of a detecting antenna. In addition, it should also be noted that both the phase and group waves are not confined to one side of the speed of light boundary and propagate at speeds above and below the speed of light in specific source regions. The mechanism by which the electromagnetic near-field waves become superlurninal can be understood by noting that the field components can be considered rectangular vector components of the total field (ref. Fig. 5). For example, the vector diagram for the longitudinal electric field (E,) is (ref. Eq. 1):

-I"

3

.;,-i*kr

kr small i-e-r Sd(Z)+Zd(S)} 1A A Frame Matrix F can be generated by the coefficients of the 4 independent 1-forms. such that Det F = ((X)2+ ( v 2 + (212 + (s)')~ /A. It is some interest to examine the properties of the adjoint 1-form, A, defined hereafter as the Hopf 1-form. For A = 1 it follows that the Hopf 1-form is of Pfaff dimension 4. It is also of interest to consider factors A that are of the format of the Holder norm, where n and p are integers, and (a,b,k,m) are arbitrary constants.

A= (a~~+b~~+kzp+rn~~){"'P} The exponents n and p determine the homogeneity of the resulting 1-form, which is given below in an ambiguous format (the plus or minus sign) A* = k + /A = {_YYd(X)-Xd(Y))Sd(Z)+Zd(S)}/ A. For example, for n = p = 2, the scaling factor becomes related to the classic quadratic form o f a vector. The scaled Hopf 1-form, A, is then homogeneous of degree zero. For arbitrary n and p, the 3-form of topological (Hopf) torsion becomes: A* A dAk = i(T4)dX/\dYAdZAdS,where the Torsion 4 vector is equal to T4 = [X,Y,Z,S]/ A. The Torsion vector, T4, for the Hopf map is proportional to the position vector fiom the four dimensional origin and represents an expansion or a contraction process. The factor A depends upon the integers n and p as well as the constants ( a,b,k,m). The Topological Parity 4-form, whose coefficient is the 4 divergence of the - 2 )AdYAdZAdS. It is most Torsion vector, T4, becomes: d 4 "dAf = - 4 ( ~ ) - ~ ~ / * ( ndX remarkable that for n = 2, any p and any (a,b,k,m), the topological parity 4 form vanishes; the scaled Hopf 1-form is of Pfaff dimension 3, not 4. In such cases the ratios of integrals of the topological torsion 3-forms over various closed manifolds are rationaland the closed integrals are topological deformation invariants. (coherent structures). Also note that if the scaling factor is restricted to values such that n = 4, p = 2, a = b = k = m = 1, then the Frame matrix is unimodular, and the scaled Hopf 1-form is homogeneous of degree -2, relative to the substitution X 3 y X. (A somewhat different definition of homogeneity relative to the volume element will be given below.)

+

PHOTON SPIN AND TOPOLOGICAL FEATURES

203

Emphasis here will be placed on examples for which n = 4. p =2. a = b = k = 1, m = &1.

7. Spinors As Linear Combinations Of Hopf Maps The 3D isotropic (null) complex position vector, [zl, 22, z3] can be decomposed into a real and a imaginary part, such that both parts have the same magnitude and are orthogonal. In short, the Cartan Spinor, [16] can be represented as the complex sum of two Hopf vectors. The spinors come in two triples of the form a12= HI -1- i Hz, 0 2 3 = H2 + i H3, and 031 = H3 't i HI. The inner product of each element of a triple (and the complex conjugates) is an isotropic vector as aija qj= 0 for i # j. These complex combinations of Hopf vectors can be used to generate solutions for which the topological torsion vanishes, and yet the topological spin is finite and quantized. 7.1 ELECTROMAGNETISM OF INDEX ZERO HOPF 1-FORMS Guided by prior investigations, it is of interest to use the scaled Hopf 1-form as the generator of electromagnetic field intensities. When the number of minus signs in the quadratic form is zero (index O), and the exponents are n = 4. p = 2, define b2= ((X)2+ + (2)' -k (cT)'} and A = b4.The coefficients of the scaled Hopf 1-form can be put into correspondence with the classic vector and scalar potentials, [A, @], using S = CT where C is a constant. The Action for the first example is then ofthe format, A* = A f /(b)' = (f(Y d(x)-x~(Y))-cT~(z)+cz~(T)}/(&,)~ For this choice it is remarkable that the derived 2-form F = dA has coefficients E and B that are proportional to the same Hopf Map. The adjoint 1-form generated from one Hopf map has a limit set which is another Hopfmap. Using the minus ambiguity (parity) sign, leads to the classic result that E a E = c2B B but with the not-usual result that the E field is anti-parallel to the B field (If the positive ambiguity (parity) sign is used, the E and B fields are parallel.). The explicit formulas for the field intensities are: B = curl A = [~(CTY+XZ),-~(-YZ+CTX),(-X~-Y~-~-Z'-~(CT)~)(~/&,~)

(v2

E = -grad

= [~(CTY+XZ),-~(-YZ+CTX),(-X~-Y~+Z~+(CT)'~

It is natural to ask if these E and B fields admit a Lorentz symmetry constitutive constraint such that vacuum state is charge current free. Recall that a constitutive constraint is a relationship between a 2-form, F and a 2-form density G such that the coefficients of G(D,H) are related to the coefficients of F{E,B). A Lorenz vacuum condition implies that the fields are solutions ofthe vector wave equation. The question becomes, "If is presumed that D = E E and B = p H,do the Maxwell Ampere equations generate a zero 3 form of charge current? ". Direct computation for the field intensities generated by the index zero Hopf I- form indicates that dG = J # 0,unless (E p c2+ 1) = 0. Hence the scaled Hopf Action, where the scaling is of signature zero, does NOT describe a charge current free vacuum, for real positive values of E, p, and C. On the other hand, if it is presumed that the domain is such that say E or p is negative, then the Hopf Map, scaled as above, does generate charge-current free wave solutions. Negative epsilon appears to hold in metals and the Earth's ionosphere. Recent announcements in cate constructions that yield negative p [17]. However, for situations where Eor p are negative, the Hopf wave solutions imply that the Topological Spin angular

momentum, AAG is not a deformation invariant (hence Spin angular momentum of the field is not conserved.) 7.2 ELECTROMAGNETISMOF INDEX ONE HOPF 1-FORMS When the number of minus signs in the quadratic form is one (index I), and the + (z12- (~t)~)and A = kI4 (lower exponents are n = 4, p = 2, define k12 = ((XI*+ (JJ)~ case letters will be used for Index one Hopf 1-forms). For his choice, it is remarkable that the derived 2-form has coefficients E and B that are proportional to different Hopf Maps. The Action 1-form is the same as above, but with a different denominator. This fact leads to the classic result that E E = c2 B B, but now the E field is not co-linear with the B field. Using negative ambiguity (parity) sign leads to fields (from F - dA = 0): B = curl A = [2(~ty+xz),-2(-yz+~tx),(-x~-f + 22 -(ct12)21h;

-

E = -grad 4 aA/& = [2(~ty-x~),2(-~z-~tx),(-x~ -y2 + 2 -(ct12]21k~~

The Spin current density for this first non-transverse wave example is evaluated as: S4= [x(3 h2- 4(f + X ~ ) ) J h2 ( ~- 4(y2+x2)), z(h2- 4(3? +x2)),t(h2- 4Q2 +x2))] f i l l o , and has zero divergence. The Torsion current may be evaluated as T4 = [~,~,z,t]/h~, and has a non-zero divergence equal to the second Poincare invariant The solution has magnetic helicity as A B + 0 and is radiative in the sense that the Poynting vector, E x H z 0. It is possible to construct the 3-form of Topological Torsion, and its exterior derivative defined as Topological Parity. The Topological parity can be either positive, zero, or negative. For the example Hopf 1-form given above (using the negative ambiguity sign), the Topological Torsion is represented to within a factor by a position vector [-x-y,-z,-t] in 4 dimensions, and has a negative divergence or parity. If the positive sign of the ambiguity factor is changed, then the parity of the form changes sign. )I kI4, the 4 -form of For example, for the 1-form, A = {+yd(x)-xd(y)-Ctd(z)+zCd(t) topological parity is positive, and the topological torsion is represented by an outbound position vector (to within a Factor). Similar to the investigation described above for the index zero Hopfvectors, it is natural to ask if these E and B fields admit a Lorentz symmetry constitutive constraint such that vacuum state is charge current free. Again, such a condition implies that the fields are solutions of the vector wave equation. Direct computation of the Maxwell Ampere equations indicates that dG = J = 0 ifthe phase velocity constraint vanishes, E p c2 1 = 6. Hence the scaled Hopf Action, where the scaling is of index one, does describe a charge current free vacuum. for real positive values of E, p, and C. It is some interest to give the charge current solutions to show how the "phase factor "E y c2- 1 3 0" establishes the vacuum charge free conditions. The example results for the components of the current density are (note p = 0):

-

E

It is conjectured that fluctuations of the "perfect" vacuum phase relations, where p c2- 1 # 0, are associated with ZPF. Note that there are possible chargecurrent free (

PHOTON SPIN AND TOPOLOGICAL FEATURES

205

singular wave solutions) that are governed by curves in space time. There curve are generated by the intersection of the three surfaces created by setting each of the coefficients of the current density equal to zero. These solutions are valid for any phase velocity and could be a source of "needle" radiation. The solution given above is not free of Topological Torsion, AAF, and there is a non-zero value of the second Poincare invariant, (E B) # 0.However, the Spin 3-form, AAG is also non-zero, but it has, subject to the phase constraint, a zero 4divergence.(the first Poincare invariant is zero.) The divergence of the Spin 3-form, has 2 parts. The first part is twice the conventional Lagrange density of the fields, (B H D E). The second part is the interaction between the potentials and the charge currents, (A J p$). When the divergence of the 3-form is zero, then the closed integrals of Topological Spin are deformation invariants, and have closed integrals with rational (quantized) ratios. That is, with regard to any singly parameterized vector field, V, describing an di(PV)(A/Y;) = 0. The evolutionary process, L(PV)Dfi(A/Yi) = Oz3i(PV)d(A/Yi)+ function p is an arbitrary deformation parameter.

-

-

7.3 LACK OF TIME REVERSAL SYMMETRY It should be noted that if the Action 1-form in the above example is subjected to the time reversal operation in its coefficients, t 3 -t, then the new Action 1-form does NOT generate a charge current free vacuum (for real positive values of s, p, and C and the Lorentz constitutive constraint). 7.4 TWISTORS COMPOSED BY SUPERPOSING TWO INDEX 1 HOPF 1-FORMS By superposing (adding or subtracting) two different, index 1, Hopf 1-forms (which can be shown to be similar to a Penrose twistor solution) it is possible to construct a vacuum (charge current free wave) solution to the Maxwell system, subject to the constraint that the phase speed satisfies the phase velocity equation. s p C*- 1 = 0. As an example consider another Hopf 1-form of index 1 formulated as

Similar formulas for the field intensities can be determined as in the example above. Note that the parity of the Hopf forms to be superposed can be the same or different. If the parity of the two superposed Hopf 1-forms are opposite, then without consideration of the phase constraint, the Topological Torsion of the "twistor" 1-form vanishes, AAF=O. Yet the quantized topological spin 3-form, AAG, does not vanish, and moreover, subject to the phase constraint, the closed integrals of the Spin 3 form are conserved. This result implies that such a construction yields "quantized values for the Spin integrals. These formulations can be compared with the Penrose twistor defintions in terms of differential forms [18]

8. The Classical Photon When the spinor solution of two Hopf vectors of opposite parity are combined, the

206

R M. KIEHN

resulting wave solution is transverse magnetic (in the topological sense that A B = 0). Not only does the second Poincare invariant vanish under the superposition, but so also does the Torsion 4 vector. Conversely, there can exist transverse magnetic waves which can be decomposed into two non-transverse waves. A notable feature of the superposed solutions is that the Spin 4 vector current does not vanish, hence the example superposition is a wave that is not transverse electric (in the topological sense that A D # 0) For the superposed example, the first Poincare invariant vanishes, which implies that the Spin integral remains a conserved topological quantity, with values proportional to the integers. The non-zero Spin current density for the combined examples is given by the formula:

-

Spin S4 = iy+ct)( [2x(y+ct), (x2 - 9 + z2 - 2cty ~%'),-2z(~+ct), (x2 - yZ + 2 - 2cty 22 c t )]41 ph'o, while the Torsion current is a zero vector, A^F = 0. In addition. for the superposed example, the spatial components of the Poynting vector are equal to the Spin current density vector multiplied by y, such that

E x H = y S with y = -(x2+ y2 t 2 + 2cty + ~2t'/2c(~+ct)h~ . These results seem to give classical credence to the Planck assumption that the vacuum state of Maxwell's electrodynamics supports quantized angular momentum, and that the energy flux must come in multiples of the spin quanta. In other words, these combined solutions to classical electrodynamics have some of the experimental qualities of the quantized photon.

References [1] R L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffths, Exterior Differential Systems(Springer Verlag. 199 1). [2] S. Lipschutz, General Topology (Schaum, New York, 1965) p. 72. [3] R M. Kiehn and J. F. Pierce, "An Intrinsic Transport Theorem" Phys. Fluids 12,1971 (1969) [4] R M, Kiehn, "Topological Torsion, Pfaff Dimension and Coherent Structures", in: H. K. Moffatt and T. S. Tsinober eds, Topological Fluid Mechanics (Cambridge University Press, Cambridge, 1990) p. 449. [5] E. J. Post. Quantum Reprogramming. (Kluwer. Dordrecht 1995) [6] R M. Kiehn, G . P. Kiehn, and R B. Roberds, "Parity and Time-reversal Symmetry Breaking, Singular Solutions", Phys Rev A 43, 5665 (1991). [7] http://www.cartan.pair.com, especially http://www22.pair.com/csdc/pdf/helical6.pdf [8] H. Bateman, Electrical and Optical Wave Motion, (Dover, New York, 1914, 1955) p.12. [9] J. E Marsden and T. S. Riatu, Introduction to Mechanics and S y m m e t r(Springer-Verlag, y 1994) p. 122 [10] G. deRham, Varieties DDifferentiables(Hermann, Paris, 1960). [11] D. Van Dantzig, Proc. Cambridge Philos. Soc. 30, 421 ( 1934). Also see: D. Van Dantzig, "Electromagnetism Independent of metrical geometry", Proc-Kon. Ned. Akad. v. Wet. 37 (1934) [12] R M. Kiehn, "Are there three kinds of superconductivity" Int. J. Mod Phys B 5 1779 (1991) [13] R M. Kiehn, NASA Arnes NCA 2 -OR-295-502 (1/20/76) [14] E. Cartan, Lecons sur les invariants integraux (Hermann, Paris, 1958) [15] http://www.cartan.pair.com, especially http://www22.pair.com/csdc/pdf/irrevl .pdf [16] E. Cartan, The theory ofSpinors, (Dover, New York, 1966) [17] Physics Today, p17 May 2000. [18] R. Penrose, "The Central Program of Twistor Theory", Chaos, Solitons, Fractals, 10 2-3, p581-611, 1999)

THE PROCESS OF PHOTON EMISSION FROM ATOMIC HYDROGEN MARIAN KOWALSKI York University 4700 Keele Street, Petrie Bldg, R~r1206 Toronto, On M3J 1 P3, Canada

Abstract, Ultra-fast lasers generating pulses as short as one or two photon periods raise the question: "How long does the process of photon emission take"? This question has been answered by modeling photon emission by an atom in terms of classical radiation theory. This theory involves the Coulomb force and a radiation resistance force. The change of energy and angular momentum and the transition of the electron between atomic states is considered. The transition time is treated as a distinct concept from the lifetime of the excited state. In this semiclassical model the calculated transfer of the electron's energy and angular momentum is in accord with spectroscopic data. The emitted radiation is monochromatic. 1. Review Of Atomic And Radiation Theories And Data With Conclusions

The energy (E). radius (r), circumference (I). velocity (v). orbital period ( T ) and orbital angular momentum (L), ofthe orbit for quantum number n are:

En= E 1 / n 2 , vn

=

vl/n

r n = rin

2

T . = Tln3r

,

2

l n = l,n , Ln=

(1)

L I ~ ,

where

-

El = e2/(2a~ ) = -1 3.60eV, rl = as = ii2/(me3=5.29x10-" m, TI= TB=27c aB/(21~~l/m)'" = 0.15 18 x~o-'~s, lI=2mB~ 3 . 3 2X I O - ' ~ ~ , vl =vs = c p 9 x I 0 m s ( a = e2/(Ac)= 1/137.036), LI= A are the values for the first Bohr orbit.

The energy and orbital period ofthe nfhelliptical orbit, the semi-major axis an,semiminor axis bn, the radii r,,,,, r,,,,, and the electron's acceleration of the 'n orbit are:

+

207 R.L. Amoroso et a1 (eds.),Gravitation and Cosmdog?.: From the Hubble Radius to the Plcrnck Scale, 207-222. O 2002 Kluwer Academic Publfihers. Printed in the Netherlands.

MARIAN KOWALSKl

208

where

n

and

n,

are the radial and angular quantum numbers and

Q,

is the

eccentricity of the ellipse; the circular Bohr orbits correspond to E,,,= 0 (i-e. n, =n). The orbital angular momentum of the electron in the elliptical orbit is where 1 is orbital angular momentum number: Z=0, 1, 2,

.,

L,, =n, R = (Z+l)A,

n-1, and n,=Z-!-1 and

n=nr+Z+l. In the n i O states ((n, I)=Is, Zp, 34 45 ...) the electron is moving in circular orbits(a, = bn); the remaining orbits (2s, 3s, 4s, .., 3p, 4p, Sp, .., 46, 5d, .., 5J; 6f;... ) are elliptical. 1.3. REVIEW OF EXPENMENTAL FEATURES OF PHOTONES EMITTED FROM ATOMIC HYDROGEN

The experimental energy and the angular momentum of photons are:

Em. = hv,, = hc /A,. = h/T,. = ~ h c ( l / n ' ~/ n- l2 ) ;

L,,,,, = P1,

(4)

where R is Rydberg constant, Rhc= JE,J= e2/(2aB)=13.60eV is the ionization energy of atomic hydrogen. 1.4. CONCLUSION FROM COMPARISON O F EXPERIMENTAL DATA WITH BOHR-SOMMERFELD THEORY

Comparison of photon's wavelengths A,,# and periods T,- (see tables 1 and 2.).

TABLE 1. The experimental wavelengths of photons 1,- [ l ~emitted ] from H atom.

THE PROCESS OF PHOTON EMISSION FROM ATOMIC HYDROGEN 2 3 4 5 6 7

365.63 365.630 308.414 292.425 285.720

1973.94 1462.05 1305.52 1233.71

5639.35 3856.62 3290.64

209

12184.1 22433.8 37196.1

The electron's orbital circumferences In and periods Tn:

The photon's experimental periods T,,i=il,,, /c are in Tab. 2:

We can see that photon's and electron's periods are in relation: Tm..,=(Tn+T'-,)!2 (7) for 2p-ls, 3d-2p, 4 f 3 d ,... see diagonal values of Tab. 2 and 2a. For non-diagonal values of n and n' (see Tab. 2 and 2a) the average of T, and Trio increases rapidly, whereas T,,. decreases slightly. However, the transition time of the electron is close to the photon's period T,. if the transition is assumed to start when the electron on its elliptical orbit (n, I) crosses the lowest Bohr orbit with the same 1, and with lower value of n equal to n%z '+ 1.

210

MARIAN KOWALSKI

In order to compare the lengths, we estimate the distance traveled by the as the product of the average of its velocities in the electron during a time equal to Tnnr initial and final states and the photon's period: vnn=(vn+vn~)/2 = v l (n+n')Dnn' ~ , AZnns=vnnl*TM (8) The relation between the distance traveled by the electron and the wavelength of the photon (see tables 3 and 3a) is:

.

E 3 . The approximate distances AI,,,,.{7) baveledby the electron during one phom period T,.

[

2naB=3.324fl]

(n'9

TABLE 3.a. Thc average values ofBohr circumferences ({,+I,,-)/2.

(1,+1,.)/2 [ il=2nae= 3.324Aj

Is

2~

3d

(HI,

4f

p) 5g

6Fr

It is seen from the tables 3 and 3a that the distance traveled by the emitting electron duriig one photon period are close to the average values of Bohr circumferences (especially for Rydberg states (n, I)+ a)Cin a case of elliptical orbits the situation is similar): AIml s (Zn+Zn-,)D =hn-, * 1 I137.0 (10) Thus, a photon can be emitted from a hydrogen atom during one complete rotation of the electron around the nucleus.

1.5. REVIEW OF CLASSICAL ELECTRODYNAMICS OF RADIATION THEORY. Around a charge moving with velocity v there exists electromagnetic fields connected

. ..

THE PROCESS OF PHOTON EMISSION FROM ATOMIC HYDROGEN

2 13

with the acceleration a= P of this charge1): E ([% a)]/(I R 1 c z , B=[ % x El, where R is the electron radius to the observer. This electromagnetic field causes the electron's deceleration The emitted radiation power1' is obtained from the Poynting vector: dP - - 1 c -dlR

E

x

47r

B

e' IZ= IR

47rc

x (R x

+}I2=

e 1 6 l2 sin 47rc

28,

(1 1)

where B is the angle between v and the radius of the observer R. The total emitted power is equal to: The radiation resistance force ,F is associated with the average work done by this force upon the electron (dE/clt = -P= F,,. v) in the time interval T:

In the initial and the final states of the transition process, the electron is in BohrSommerfeld periodic orbits, for which v -v=O (i.e.P Iv), hence1':

= m rv,

F,

(14)

where m is the mass of the electron and z=2e2/(3mc?) = 6 . 2 6 ~ 1 0 -is ~ ~the s characteristic time. However, there are known difficulties connected with self-dispersed solutions of the Abraham-Lorentz equation: for m (a - zi;) =,,, F =O, There is defined in the literat~re,',~) y'=z& called the damping constant, where 0 ,is the angular frequency of the electron oscillating along the x axis, under a restoring (not Coulomb force), the radiation resistance force is force F~~~ = - m 0 F,

=

,,ym.

The resistance force with damping constant') I-" for non-radiative

dissipative processes is F dis with the equation:

rn ( x

+

=

,

-

1

i

. The electron is changing energy in accord

r'x - r'x'+ m w : x )

=

F d ( w ),

r

where total damping constant = T '+ (u / u ,) y '. The values of parameter are very small and time of the radiation is very big.

(15)

r' and y'

1.6. REVLEW OF QUANTUM THEORY. The total electromagnetic field for the radiation and the particles is:

E(x,t)=-llc~A(x,i)li%-VV, B(x,t)=VxA.,

V=1/2xeiej/lx,-rjl

(16)

i,J,i# j

and the total momentum is introduced with the gauge invariance fulfilled with the minimal coupling: p - e , A (xi , t ) l c. The interaction operator between the particles

,

MARIAN KOWALSKI

213 and the electromagnetic field is:

The first part induced transitions for which one photon is either created or annihilated:

The total system (atom

,,.

I i > = I a i >[..,n

>

+

radiation field) can decay from the initial state:

to the final state:

If

>= 1 a /

>I .., n,, + 1,.. >

by

emitting a photon. Ifno photon is present at the beginning (nka=O) the lifetime rof the initial atomic state I izi >: is:

E , - E , ) 1 is the transition frequency. This relation is the quantum-mechanical analogue to the classical Larmor equation (15) considered in this paper. The lifetime ofthe pp (m=O)> state in the hydrogen atom with respect to decay ) ,is equal to: into IIs> state in spherical coordinates and hw1,,, = E,, - E,, = 3e2 where

=(

Knowing that a = e /(hc) = 1 /137.036

and

a, = f i Z lme2 we obtain that

rpl.6~1~~gs. This lifetime is much larger than the photon period T'! =0.405x1iJi5sand much larger t h very s h r t laser pulses obtained recently ( t < l ~ x l i J ~ ~ass )well , as much larger than very short X-ray pulses (shorter than 10 optical cycles) emitted from the atoms after excitation by these short laser pulses14). This experimental evidence indicates that the transition can take place in a time much shorter than the lifetime ofthe exited state.

2. Classical Calculations Of Photon Emission From Hydrogen During the action of the non-conservative force F,,, on the electron it is losing energy E and angular momentum L according to formulae:

where E=Ek+V is the total energy of the electron, , r is its radius centered at the nucleus and v is its velocity. We used equation (13) for Frrs= m Zii , where in the cylindrical coordinates we have:

THE PROCESS OF PHOTON EMISSION FROM ATOMIC HYDROGEN The loss of the energy and orbital angular momentum by the radiated electron is: dE e i= -- m ( i L'+ r i p i 2 + r 2 p i Y ) + T F r i + F,rpi,

dt

r

213

(238)

and we obtain the Newtonian form in cylindrical coordinates:

ma.

= m( 2 i p i

+

r @) = F ,

,

(25)

which has the vector form: m a + e Finally. using (22) we obtain equations:

2

l r 3 r = Fres .

(28) y ' = [ 2 r ' @ + r $ * - 3 z ( i m g i + i $ ' ) +z r p " ] / ( r r ) . However, these equations are self-dispersed, as for the Abraham-Lorentz equation. The approximation which provide non-dispersive solutions of the resistance force is: r.. g i S z q m g G 0 . (29)

.

F,

+.

n rn z [ - 3 i g i 2 , - r g i 3 ] ,

(30)

It was found numerically that the factor '3' is not so important. This suggests that the radiation resistance force can be anti-parallel to the electron's velocity. as in dissipative processes: . 3 I)ZC 7 [ - 3 e 2 , - r p ] = - m C rm ' v , (31) where = + w v/r is angular velocity of the radiating electron. Parameter C ought to be close to 1. We can define damping parameter y = ~ 7 &w CZ(V/~)~ l i e y' in (15). From the equation: m a + ( e 2 / r ) r = F,, we have in cylindrical coordinates:

Assuming that Cml, according to known radiation theory1), we obtain correct values of the electron's radius, energy and orbital angular momentum in the final state. However, because of small value of the 7 parameter the electron travels many times (SS 103 around the nucleus during the transition process. This result is not satisfactory because the time needed by the radiating electron for one orbit is changing continuously from Atl=Tn =T,/n3 in the initial state to At ',=T,,

MARIAN KOWALSKI

214

=TI l(n-1)3 in the final state. This implies that the emitted radiation would not be monochromatic (not have gaussian line's shape) contrary to what is observed, Our estimates show thutfor Rydberg states the photon's period is close to the Bohr orbital period and for lower states the photon's period is close to the average of the Bohr periods of the initial and final states. We can obtain a monochromatic photon if the electron makes only one orbit around the nucleus during the transition; i.e. within a time equal to the photon's period At, =TnnS Thisproposed wchanism is supported by a recent experiment with an ultra-fast B laser that generates light pulses shorter than two photon wavelengths 9 14'. When we assume that C =lo6we obtain only one electron orbit, a transition time equal to the photon's period and the correct energy and angular momentum in the final state. It is also possible to use the resistance force to be anti-parallel to the velocity of the electron v F, = - m r v , (34) but with average resistance parameter for transitions between the two neighboring (n, w l ) atomic states. )2 two Equation (34) suggests that can be similar to average of y = ~ r d = ~ z ( v / rbetween atomic states, see table 4. The orbiting electron (in transiting between the n and n' levels) looses energy and angular momentum (see (21)) equal to

r

r

where t is the time ofthe electron transition. The QED zero-point electromagnetic radiation can balance the radiation resistance force on orbits and as a result stable orbits are created. see H. M. Franca et a15). There are two ways of treating the transition processes: by using the radiation resistance force or the circular electromagnetic field (in the (x, y ) plane):

where rEis the resistance parameter and q(t) = arccos(vy /v) =2n t/T,# is the angle between the electron's v,, and v velocities. A similar electric (electromagnetic) field is carried by the photon itself. Using the resistance force (34) and equations (21) we have,

Remembering that there exists the Coulomb conservative force also acting on the electron (the electron's energy is: E =&-e /r), we obtain two equations:

L

= L .eU", where Lo is the initial value of the electron angular momentum. In angular coordinates (r, p), for the energy and angular momentum

(39)

THE PROCESS OF PHOTON EMISSION FROM ATOMIC HYDROGEN

qj=-

Lo 2e

215

-rt

mr Then, (40-42) leads to the equation:

which is difficult to solve analytically. The formula r(t)=r,e-wt is only an approximate solution. Accurate solution was obtained by integrating (43) numerically. In previous treatmentslA7'(see (15)) the electron's motion is oscillatory in the radiating atom. This is possible, because both the centrifugal and Coulomb forces acting on the orbiting electron may be written approximately as mv 2

e

2

]? s- m o 2 xr, * (44) (Y,+x)~ where ro =e2 /m3 is the equilibrium distance of the electron from the nucleus, m=mv3/e2 is the angular velocity of the radial oscillating electron and x=r-r, is its displacement from equilibrium. This rcrdial approximation neglects circular aspects oftheprocess - the angular momentum of the electron and the emittedphoton. The electron must lose energy En,. given by equation (4) and must lose angular momentum equal to L, =A during the emission process during one circulation around the nucleus. Knowing the energy of the radiating electron in the initial state En we stop the numerical integration when the electron has the final state energy equal to En.. Stopping the numerical integration in this way yields a time for the emission process, At,,,., the distance traveled by the electron A],,: and the loss of angular momentum, Lnn:by the electron: F=[--

q)+x

In accord with equation (39) the duration of the process depends upon the value of the damping parameter I-,,.. With damping constant used in the literature1': y'=z&, the electron completes many orbits around the nucleus and the transition time is many times the photon period, Tnn., However, this model of radiation is physically incorrect, because the radiation is not monochromatic as is observed. In order to obtain monochromatic radiation the electron has to make only one orbit during the transition process, since the period of the radiation is equal to the period of rotation of the electron. In order to fulfill this requirement equation (39) was used to calculate the value of the stifling parameter, T,,,, with the time t, equal to the photon's period: t=At,-, =Tnmlfor the emission between the n and n-1 states, with Lo and L equal to the

MARIAN KOWALSKl

216

-

7

electron sangular momentum in the initial and final states: Lo = L,, L = L,,-, = Ln A :

where the latter expressions are obtained from equation (4). The values of this stifling parameter Tm-lderived from (46) between every pair of neighboring (n,n-1) states are shown in table 4. TABLE 4. The radiation resistance parmeter r0.,[l/r,].

r,,_l[lh04

1 -39073-io"15 I/S] n- 1

r is in principle a function of v and r like in y=~zd=~z(v/r)2. In the numerical calculations we introduced a parameter k equal to the number of electron orbits around the nucleus, and then solved the equations with a damping parameter T',,=Tn, ,/k. However, k had to be greater or equal to l(k2l) in order to obtain a good value for the change in angular momentum consistent with the known transition energy. For k> and u bumps n in the covariant density distribution of a real average covariant Dirac aether built with extended random elements filling flat spacetime. Some possible experimental tests are also discussed.

1. Introduction The problem of the unification of gravity and electromagnetism into a single theory is as old as Modern Science itself and it has not been solved until now. Despite the initial discovery of similar forms ofthe Newton and Coulomb potential the two theories are still developping independently. Until the present, unification has been attempted mainly (as a consequence of Einstein's discoveries) by Einstein himself [I], following Schrodinger [2], Maxwell [3] (and their present successors) within a frame associating electromagnetism with new geometrical properties of spacetime. The aim of the present paper is different. Following MacGrkgor [4], Puthoff 151, and others, both fields are represented by fourvector field densities Ap ; and one considers both types of phenomena as different types ofmotions within the same real physical zero-point field in flat spacetime, i.e. as two different c( aether w types of collective perturbations carried by a single (< aether n field moving in such a space. Since this approach suggests new types of experiments and yields an interpretation of unexplained new effects it will (perhaps), if confirmed, help to disantangle the present theoretical discussion. This model has the following experimental basis : The first basis (observational) is that the observable universe apparently does 1) not change with distance [15] (as it should with big-bang type theories) and the ratio of 24 1 R.L. Amoruso et a1 feds.). Gravitation aiul Co.snudogy: From the Hubble Radius to the Planck Scale, 241-258. O 2002 KluwverAcudemic Publisl~ers.Printed in tlze Netlzedaiuls.

242

J-P VIGIER & R. L. AMOROSO

the local 2.7" microwave radiation is only isotropic in a specific absolute inertial frame lo: so that the velocity of light not only changes with its direction (which suggests a non-zero photon mass my $0) but is also isotropic in I, , in time. The second basis is that our essential instrument of (distant) observation (i.e. 11) electromagnetic waves) is more complex than its initial discoverers (Maxwell and Ampere) thought. Newtons initial guess that light was both waves and particles (photons) was later confirmed by Einstein in 1905. The discovery by Fresnel that these waves were essentially transverse (i.e. with possible zero mass and invariant velocity of propagation) was later completed by de Broglie's and Einstein's discovery that one could write E = h v = mc2(with m = mo (1 - v 2 / c2)-'I2 ) so that individual massive photon's can be considered as piloted by real non zero-mass Maxwellian waves i.e. by new properties of the Sagnac effects in a recent experiment of Levit et al. [7] which shows that the electromagnetic field should be represented by a vector density A,. As shown by Aharonov-Bohm effect, this implies that the electromagnetic field is not completely represented by the pv fields [6,7]. 111) The third basis has its theoretical origin in the introduction by Dirac et al. of a real covariant chaotic physical 0 and gravitation to 8m < 0 one can describe EAS-P. such subelements as holes (6m < 0) around a point 0 around which rotate two point-like charges rotating in opposite directions as shown in Figure 1 below.

Figure I . Conceptual diagram of two oppositely charged subelements rotating at v 2 c around a central point 0 behaving like a dipole (< bump 1) and 0. This ensures that (a$2+ be2 + ~ $ 8>) 0 and that W' is real in the classically forbidden regions (V > E). The motion in phase space is specified by Eq. (2) and is a function of the set of coefficients (a, b, c ) . In general, the conjugate momentum expressed by Eq. (2) is not the mechanical momentum, i.e., W' # mx. Actually, mx = mdE/aW' [1,2]. The solution for the generalized reduced action, W, is given by

w = h arctan

(:!@$$)

+

where K is an integration constant that we may set to zero herein. The reduced action is a generator of motion. The equation of motion in the domain [x,t] is rendered by Jacobi's theorem (often called the Hamilton-Jacobi transformation equation for constant coordinates). The procedure simplifies for coordinates whose conjugate momenta are separation constants. Carroll has shown that for stationarity Jacobi's theorem applies for W' is a Legendre transform [9]. For stationarity, E is a separation constant for time. Thus, the equation of motion for time, t , relative to its constant coordinate T , is given as a function of z by

where the trajectory is a function of a set ofcoefficients (a,b, c) and epoch.

T

specifies the

TRAJECTORY REPRESENTATION OF QM

403

The set ($,8) can only be a set of independent solutions of the SSE. Direct substitution of Eq. (2) for W' into Eq. (I) gives

For the general solution for W', the realcoefficients (a,b, c ) are arbitrary within the limitations that a, b > 0 and from the Wronskian that ab - c2/4 > 0 . Hence, for generality the expressions within each of the three square brackets on the left side of Eq. (5) must vanish identically. The expressions within the first two of these square brackets manifest the SSE. so the expressions within these two square brackets are identically zero if and only if $ and 8 are solutions of the SSE. The expression within third bracket vanishes identically if and only if the normalization of the Wronskian is such that w 2 ( $ ,6 ) = 2m/[h2(ab- c 2 / 4 ) ] . For W ( $ , 8 ) # 0 , 4 and 8 must be independent solutions of the SSE. Hence, and 8 must form a set of independent solutions of the SSE. Equation (5) is independent of any particular choice of ansatz. When comparing trajectories to Copenhagen and Bohm, we have broad selection for choosing a convenient ansatz to generate the equivalent wave picture (nothing herein implies that the trajectories need waves for completeness; only convenience). By Eq. (2), W' is real in the classically forbidden zone. Inside barriers, W' still manifests a trajectory. For bound states of finite V(3;),the trajectories go to turning points at .z. = fmwhere W' + 0 regardless of the ( a ,b, c ) because at least one of ( 4 , 8 ) becomes unbound as x + fm. This is a nodal point singularity. For a given energy eigenvalue. E. of the SSE, there exist infinitely many microstates (trajectories or orbits with turning points at fm ) specified by (a, b, c ) . These microstates are not distinguishable from the SSE for bound states [2,7].

3. Different Predictions between Trajectories and Copenhagen First, we examine impulsive perturbations. Trajectories and Copenhagen render different predictions for the first-order change in energy, El due to a small, spatially symmetric perturbing impulse. XV(x)d(t), acting on the ground state of a infinitely deep, symmetric square well [3]. In the trajectory representation, El is dependent upon the particular microstate ( a ,b, c ) . This has been investigated under a Copenhagen epistemology even for the trajectory theory, where complete knowledge of the initial conditions for the trajectory as well as knowledge of the particular microstate are not necessary to show differences for an ensemble suffi-

404

EDWARD R. FLOYD

ciently large so that all microstates are individually well represented. In the trajectory representation, the first-order change in energy, El,is due to the location of the particle in its trajectory when the impulse occurs. The trajectory representation finds that the perturbing impulse, to first order, is as likely to do work on the particle as the particle is to do work perturbing system. Hence, the trajectory representation evaluates (El),,,,,, = 0. On the other hand, Copenhagen predicts El to be finite as Copenhagen evaluates El by the trace ground-state matrix element XVOoG(O)at the instant of impulse. Due to spatial symmetry of the ground state and V(x), Voo# 0. In an actual test, we do not need perturbing impulses, which were used for mathematical tractability. A rapid perturbation whose duration is much shorter than the period of the unperturbed system would suffice [3]. We also consider a redundant set of constants of the motion. For a square well duct, we have proposed a test where consistent overdetermination of the trajectory by a redundant set of observed constants of the motion would be beyond Copenhagen [4]. The overdetermined set of constants of the motion should have a redundancy that is consistent with the particular trajectory. On the other hand, Copenhagen would predict a complete lack of consistency among these observed constants of the motion as Copenhagen denies the existence of trajectories. Such a test could be designed to be consistent with Copenhagen epistemology [4].

4. Other Differences between Trajectories and Copenhagen As the trajectory exists by precept in the trajectory representation, there is no need for Copenhagen's collapse of the wave function. The trajectory representation can describe an individual particle. On the other hand, Copenhagen describes an ensemble of particles while only rendering probabilities for individual particles. The trajectory representation renders microstates of the Schrodinger wave function for the bound state problem. Each microstate is sufficient by itself to determine the Schrodinger wave function [7]. Thus, the existence of microstates is a counter example refuting the Copenhagen assertion that y!~ be an exhaustive description of nonrelativistic quantum phenomenon. The trajectory representation is deterministic. We can now identify a trajectory and construct, for convenience, its corresponding q!J witb sub-barrier energy that tunnels through the barrier with certainty [lo]. Tunneling with certainty is a counter example refuting Bern's postulate of the Copenhagen interpretation that attributes a probability amplitude to I$. As the trajectory representation is deterministic and does not assign a probability amplitude to q!J, it does not need a wave packet to describe or localize a particle. The equation of motion, Eq. (4), for a particle (monochromatic wave) has been shown to be consistent with the group velocity of the wave packet [I I]. Though probability is not needed for tunneling through a barrier [lo], the trajectory representation for tunneling is still consistent with the Schrodinger representation


405

without the Copenhagen interpretation [lo]. Trajectories differ with Feynman's path integrals in three ways. First, trajectories employ a quantum reduced action while a path integral is based upon a classical reduced action. Second, the quantum reduced action is determined uniquely by the initial values of the QSHJE while path integrals are democratic summing over all possible classical paths to determine Feynman's amplitude. While path integrals need an infinite number of constants of the motion even for a single particle in one dimension, motion in the trajectory representation for a finite number of particles in finite dimensions is always determined by only a finite number of constants of the motion. Third, trajectories are well defined in classically forbidden regions where path integrals are not defined by precept. By the QSHJE, knowledge of a set of initial conditions [W, W W'Jat some point xi is necessary and sufficient to specify E and the quantum motion while for the classical stationary Hamilton-Jacobi equation the set of initial conditions is reduced to [W'J at xi to specify E and classical motion [6]. Thus, the Heisenberg uncertainty principle assumes a subset of initial conditions (x,p) that is insufficient to specify E and quantum motion [the SSE operates in (x,p)-domain; the QSHJE in (x, t)-domain by a canonical transform]. The Heisenberg uncertainty principle is premature since Copenhagen uses an insufficient subset of initial conditions (x,p) to try to describe quantum phenomena [12]. Bohr's complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument's specific properties. But Faraggi and Matone [2] have derived the QSHJE from an equivalence principle without evoking any axiomatic interpretation of the wave function. Furthermore, Floyd [Iand Faraggi and Matone [2] have shown that the QSHJE renders additional information beyond what can be gleaned from the Schrodinger wave function alone.

5. Trajectories vis-a-vis Bohmian mechanics

The trajectory representation differs with Bohmian representation [13,14] in many ways despite both representations being based on equivalent QSHJEs. We describe the various differences between the two representations in this section. These differences may not necessarily be independent of each other. The two representations have different equations of motion. Jacobi's theorem, Eq. (4), gives the equations of motion for the trajectory representation, Meanwhile, Bohmian mechanics eschews solving the QSHJE for a generator of the motion, but instead assumes that the conjugate momentum be the mechanical momentum, mk, which could be integrated to render the trajectory. But the conjugate momentum, as already shown herein, is not the mechanical momentum [1,2,9,11]. Bohmian mechanics considers to form a field that fundamentally effects the quantum particle. The trajectory representation considers the SSE to be only a phenomenological equation where Il, does not represent a field. To date, no one has ever measured such a $-field. Bohm postulates a quantum potential, Q, in addition to the standard potential,

+

406

EDWARD R. FLOYD

that renders a quantum force proportional to -VQ. [The negative of Bohm's Q in one dimension appears on the right side of the QSHJE, Eq. (I).] But Q , by the QSHJE, is dependent upon E and the microstate (a,b, c) of a given eigenvalue energy E because

Therefore, Q is path dependent and cannot be a conservative potential. Consequently, -VQ does not generally render a force. While Bohmian mechanics postulates pilot waves to guide the particle, the trajectory representation does not need any such waves. Bohmian mechanics uses an ansatz that contains an exponential with imaginary arguments. The Bohmian ansatz in one dimension is I!,J = (w')-Il2exp(iW/h). As Eq. ( 5 ) is valid for any set ( 4 , 8 ) other , ansatze including trigometric forms are acceptable [1,2,5]. Bohmian mechanics asserts that particles can never reach a point where TJ vanishes. On the other hand, trajectories have been shown to pass through nulls of TJ with finite conjugate momentum, W' [I,?]. Bohmian mechanics asserts that bound-state particles have zero velocity because the spatial part of the bound-state wave function can be expressed by a real function. On the other hand, the QSHJE, Eq. (I) is still applicable for bound states in the trajectory representation. For bound states. the trajectories form orbits whose action variables have the Milne quantization J = $ W' dx = nh,n = 1,2,.. . independent of the microstate (a,b, c) [1,7,15]. Bohmian mechanics asserts that a particle will follow a path normal to the surfaces of constant W. Meanwhile, trajectories in higher dimensions are not generally normal to the surfaces of constant W [4,11]. In higher dimensions, trajectories are determined by Jacobi's theorem, Eq. (4) rather than by V W . Bohmian mechanics asserts that the possible Bohmian trajectories for a particular particle do not cross. Rather. Bohmian trajectories are channeled and follow hydrodynamic-like flow lines. On the other hand, the trajectory representation describes trajectories that not only can cross but can also form caustics as shown elsewhere in an analogous, but applicable acoustic environment [16]. The two representations differ epistemologically whether probability is needed. The trajectory representation is deterministic. Bohmian mechanics purports to be stochastic and consistent with Born's probability amplitude [14]. Let us consider three dimensions in this paragraph to examine the familiar stationary auxiliary equation

to the three-dimensional QSHJE. Bohm and Hiley [14J identify R as a probability amplitude and Eq. (6) as the continuity equation conserving probability. Hence, R2VW must be divergenceless. The trajectory representation can now show a non-probabilistic interpretation of R ~ V W Let . us consider a case for which the stationary Bohm's ansatz, = Rexp(iW/h),is applicable. Bohm [13] used R~ = $J


407

U 2 + v2and W = tiarctan(V/U)where U = ?I= ? R, cos(W/ti) ($)and V = S($)= Rsin(W/fi). Hence, by the superposition principle, U and V are solutions to the SSE. Upon substituting U and V into Eq. (6). we get the intermediate step ( R 2 V W ) = UVV - VVU which is a "three-dimensional Wronskian". Whether or not this "three-dimensional Wronskian" renders a constant, it is divergenceless [5]. Therefore, the trajectory representation finds that the auxiliary equation contains a "three-dimensional Wronskian" that satisfies Eq. (6) without any need for evoking a probability amplitude. Bohrn had expressed concerns regarding the initial distributions of particles. Bohm [13] had alleged that in the duration that nonequilibrium probability densities exist in his stochastic representation, the usual formulation of quantum mechanics would have insoluble difficulties. The trajectory representation has shown that the set of initial conditions may be arbitrary and still be consistent with the Schrodinger representation [6]. Stochastic Bohmian mechanics, like the Copenhagen interpretation, uses a wave packet to describe the motion of the of the associated $-field. As previously described herein, the deterministic trajectory does not need wave packets to describe or localize particles. Holland [17] reports that the Bohm's equation for particle motion could be deduced from the SSE but the process could not be reversed. On the other hand, the development of Eq. (5) is reversible. In application, the two representations differ regarding tunneling. Dewdney and Hiley [18] have used Bohmian mechanics to investigate tunneling through a rectangular barrier by Gaussian pulses. While Dewdney and Hiley assert consistency with the Schriidinger representation, they do not present any results in closed form. Rather, they present graphically an ensemble of numerically computed trajectories for eye-ball integration. On the other hand, the trajectory representation exhibits in closed form consistency with the Schrodinger representation. Also, every Bohmian trajectory that successfully tunnels slows down while tunneling even though Steinberg et a1 [19] have shown that the peak of the associated wave packet speeds up whiie tunneling. Our trajectories that successfully tunnel speed up [4.10] consistent with the findings of others [20-231.

Acknowledgement 1 am pleased to thank M. Matone for many discussions. 1 also thank D. M. Appleby, G. Bertoldi, R. Carroll, and A. E. Faraggi.

1. Floyd, E. R.: "Modified Potential and Bohm's Quantum Potential", Phys. Rev. D 26 (1982), 1339-1347. 2. Faraggi, A. E. and Matone. M.: "The equivalence postulate of quantum mechanics", Inf. J. Mod Phys. A 15 (2000), 1869-2017, hep-thB809127.

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3. Floyd, E. R.: 'Which causality? Differences between tajectory and Copenhagen analyses of an impulsive perturbation", Int. J. Mod Phys. A 14 (1999), 11 11-1 124, quant-phl9708026. 4. Floyd, E. R.: "Reflection time and the Goos-Hiinchen effect for reflection by a semi-infinite rectangular barrier". Found. Phys. Lett. 13 (2000). 235-251. quantphl9708007. 5. Floyd, E. R.: (2000) "Extended Version of 'The Philosophy of the Trajectory Representation of Quantum Mechanics' ", quant-phl0009070. 6. Floyd, E. R.: "Arbitrary initial conditions of hidden variables", Phys. Rev. D 29 (1984), 1842-1844. 7. Floyd, E. R.: "Where and why the generalized Hamilton-Jacobi representation describes microstates of the Schriidinger wave function", F o u t d Phys. Len. 9 (1996), 489497, quant-phl9707051. 8. Floyd, E. R.: 'Closed-form solutions for the modified potential", Phys. Rev. D 34 (1986), 3246-3249. 9. Carroll, R.: "Some remarks on time, uncertainty and spin", J. Can. Phys. 77 (1999), 319-325, q~ant-~h/9903081. 10. Floyd, E. R.: "A trajectory interpretation of tunneling", An. Fond. Louis de Broglie 20 (1995). 263-279. 11- Floyd, E. R.: "A trajectory interpretation of transmission and reflection". Phys. Essays 7. (1994) 135-145. 12 Floyd. E. R.: "Classical limit of the trajectory representation of quantum mechanics, loss of information and residual indeterminacy", Int. J. Mod. Phys. A 15 (2000), 136S1378, quant-ph/9907092. 13. Bohm, D.: "A suggested interpretation of the quantum theory in terms of 'hidden' variables. I", Phys. Rev. 85 (1952), 166-179. 14. Bohm D. and Hiley, B. J.: "An ontological basis for the quantum theory", Phys. Rep. 144 (1987), 323-348. 15. Milne, W. E.: "The numerical determination of characteristic numbers", Phys. Rev. 35 (1930), 863-867. 16. Floyd, E. R.: "The existence of caustics and cusps in a rigorous ray tracing representation", J. Acous. Soc. Ant. 80 (1986), 1741-1747. 17. Holland, P. R.: The Quanmn Theory ofMotion, Cambridge U . Press, Cambridge, UK, 1993, p. 79. 18. Dewdney, C. and Hiley, B. J.: "A quantum potential description of the onedimensional time-dependent scattering from square barriers and square wells". Found. Phys. (1982), 12,2748. 19. Steinberg, A. M., Kwiat, P. G. and Chiao, R. Y.: "Hidden and unhidden information in quantum tunneling", Found. Phys. Lett. 7 (1994), 223-237. 20. Olkhovsky, V. S. and Racami, E.: "Recent developments in the time analysis of tunnelling processes", Phys. Rep. 214 (1992), 339-356. 21. Barton, G.: "Quantum mechanics of the inverted oscillator potential", An. Phys. (New York) 166, (1986), 322-363. 22. Hartmann, T. E.: "Tunneling of a wave packet", J. Appl. Phys. 33 (1962), 34273433. 23. Fletcher, J. R.: '"Time delay in tunnelling through a potential barrier", J. Phys. C 18 (1985), L55-L59.

SOME PHYSICAL AND PHILOSOPHICAL PROBLEMS OF CAUSALITY IN THE INTERPRETATION OF QUANTUM MECHANICS B . LANGE Departamen2 for Logic, Methodology and Philosophy qf Science, University of Gdarisk, ul. Bielahka 580-951, Gdarisk, Poland

1. Introduction The International Conference of Physicists in Warsaw took place from 3dhMay to 31dJune 1938. The Conference was organised by the International Institute of Intellectual Cooperation, and the Polish Commission as its associate body. The talks were chiefly devoted to philosophical issues that emerged along the development of quantum theory and theory of relativity, and other basic issues related to interpretation of new discoveries and theories. The need for organising such meetings resulted from the necessity ofdiscussing and confronting the ideas ofmost prominent scholars in relation to the most basic problems emerging in the period of rapid development of contemporary physics. It was the fmt of the whole pre-planned series of conferences. and it should be pointed out that the very first conference took place in Poland, which is a proof of a hlgh world rank of Polish physics. The conference was initiated and then chaired by an eminent Polish physicist, Professor Czeslaw Bialobneski. The conference was also attended by other eminent scholars. like Niels Bohr. Leon Brillouin, Charles Darwin, Arthur Eddington, George Gamow, Hendrik Anthony Kramers, Paul Langevin, John von Neumann, Eugen Wigner, and others. Poland was represented by Professors Wojciech Rubinowicz. Szaepan Szczeniowski. Jan Weyssenhoff. Ludwik Wertenstein. and F. J. Wisniewski. Complicated political situation of that time, however, was the reason why some invited scholars failed to arrive. During the conference eight papers were delivered, and they were afterwards published together with the abstracted discussion in the volume titled h s Nouvelles Theories d e la Physique [ll. The outbreak ofthe War was the cause why the information and the conference materials did not reach most of the world centres of physics. Therefore the conclusions drawn from the papers and discussion did not become a source offurther debates. Thus the Conference did not have some major impact on further development of the foundations and interpretation of quantum physics. It seems, however, that there is a need to present chief ideas discussed during the conference, particularly since some of them - like the proposition of Bialobrzeski, for instance - were attempts to overcome basic problems in interpreting formalism ofquantum theory. Those issues have not been successfully solved so far, and they still are a point of interest to physicists and philosophers. The following paper aims at presenting the discussion during the Conference, which was joined by Bohr, von Neumann, Bialobrzeski, and Kramers. The discussion developed over the paper of Bohr, whose intervention seemed to have stirred the highest interest among the participants. The proposal of interpretation of the reduction process of the wave function Y presented during the discussion by C. Bialobrzeski [l], seems to be of particular interest here.

409 R.L. Ainuroso et a1 (eds.),Gravitation rrr~dCa~mcdogv:From the Hubble Radius to the Plunck Scale, 409-412. O 2002 Kluwer Actulenlic Publishers. Printed in the Netherlaiuls.

6.LANGE 2. A Dispute Between N. Bohr, J. Von Neumann, Cz. Bialobrzeski, And H. A. Kramers On Causality In Atomic Physics While analysing a problem of the use of the laws of mechanics and thermodynamics in explaining atom's stability Bohr stated that we should make additional assumptions, i.e. that any well defined change in the status ofelectron within the atom depended ofthe shift ofthe atom from one stationary position into another, similar one. According to Bohr, quantum postulates are not only completelyalien to the notions ofclassical mechanics, but they also assume divergence from any casual description of such processes. This particularly refers to possible shift of the atom from a stationary position into another one. Bohr claimed that "the atom faces a choice with no determining circumstances. Situation like that, any forecasts may refer only to the probability of various possible directions of atomic processes, which are susceptible to direct observation" [I]. Further on Bohr claimed that the "utilisation of purely statistical consideration is the only instrument which allows for generalisation of usual description, necessary for explaining individual character expressed by quantum postulates..., (and) this is how we limit classical theory to an extreme case, where the changes ofaction are large as compared to the quantum ofaction. The only reason for formulating such a generalisation was the need for utilisation ofclassical notions in the widest possible scope which could be reconciled with quantum postulates" [I]. Basic assumptions of the Bohr's intervention can be presented as follows: measurement cannot mean anything else but unbiased comparison ofa given property ofthe object under research, with a corresponding property of another system used as a measuring instrument. Properties of the measuring instrument directly result from the definition taken from classical physics. The above mentioned comparison ofthe properties must be convergent with the definition expressed in a common language. Whereas in the field of classical physics such a comparison may be performed without substantial intervention into the object itself, in the quantum theory, however, the interaction between the object and the measuring instrument will basically influence the phenomenon itself. We have to be particularly aware ofthe fact that the interaction cannot be exactly separated from the behaviour ofthe undisturbed object. Thus the necessity of basing our description on classical notions implies disregarding all the quantum effects in the very description, particularly disregarding more precise control over the object's reaction than it might result from the uncertainty principle. Adopting the complementarity point of view we may avoid unnecessary discussion on determinism. There is no need to discuss indeterminism when we state a general scheme ofcausative idea which may aim at the synthesis ofthe phenomena which can be described within the function of the behaviour of the objects irrespective of the mode of their observation. Bialobrzeski, in turn, thought that there were doubts concerning logical coherence of quantum theory. The interpretation of Bohr and Heisenberg presented two, complementary sides of elementary creations of the material world: the corpuscular and the wave sides. It is rightly said that the formalism of quantum mechanics gives synthesis which unifies the two images of the single reality. We must refer to these images to describe natural phenomena within space-timecontinuum. Bialobrzeski thinks, however, that it is possible to find dualism of a different nature within the notions of quantum mechanics. This dualism questions uniformity ofa doctrine, unless it can be explained within the chief postulates of the doctrine. The dualism means - as Bialobrzeski stated - that on the one hand we have a casual seyuence of phenomena governed by the Schrodinger equation, on the other hand, however, when we measure a certain volume of "A", the status represented by function Yis subject to rapid non-casual change. The sudden change infunction Y results from the fact, that we can obtain a result ofa measurement in the form of any eigenstate 5,Y2,Y3... which would correspond to measured value. "We assume," the scholar says, "that we are dealing with a discontinuous system of these states. States YNare

PROBLEMS OF CAUSALITY IN QUANTUM MECHANICS

411

independent from the initial state, decisive only for probability of their realisation expressed by the [...I. If we consider a large number of the systems in the same state of then the formula ( WNj2 measurement ofthe volume 'XA~performed on the set of all these identical systems turns the set into a mixture, which is a proof of non-casual character of the process. The shift from the state Y into one of the states YNmust be either instant or brief, which on the one hand leads to indeterminacy of energy related to time by one of the Heisenberg's uncertainty relations, on the other hand it clearly separates time from the co-ordinates defining position. which is in contradiction to the postulate of relativity" [I]. Bialobrzeski asked a basic question: "What is the mechanism of realisation of the state YNfrom the state Y naturally ifthere exists describable mechanism of the change?" [I]. Bialobrzeski's proposal looks as follows: after accepting the thesis, that von Neumann's analysis defined impossibility(within quantummechanics)of eliminating indeterminismbyintroducing hidden parameters for preserving the description of the measuring process with the use of Schriidinger equation, we musl assume that coupling of the measuring instrument with the object under research is the cause of distortion. The distortion is discontinuous and changeable in such a way, that a probability of realisation of state Y, is strictly defined by the formula (YYN)2. Discontinuity and indeterminacy, which were eliminated from the measurement, reappeared in the distorted operator of energy, which ough~to change in a discontinuous and indeterminable way. As a result, we obtain nothing which could explain the mechanism under discussion. Bialobrzeski concludes his comments with the assumption that the action of taking measurement performed upon the system in which its natural alternation of state is governed by Schrijdinger equation causes another type of alteration of indeterminate character, as it is solely governed by statistical laws. Anyway, it seems to be necessary - as Bialobrzeski believes - to assume the existence ofa basic postulate which refers to the very act oftaking measurement: "quantized physical quantity is basically indeterminate, and we can only recognise the probability of realisation of edch of eigenvalues ofthe quantity. It is just determined by the formula ( w ) ~ . " [I] J. von Neumann noticed that we must have an observer placed somewhere within a given system, therefore it is necessary to draw the borderline between the observer and the observed. It is not necessary, however, that the border overlapped with physical body limits ofthe observing person. He believes that it is possible to "shrink" or "blow up" the observer. We may include, for instance, everything that is happening in the observer's eye and include it all into the part under observation, which is described in a quantum manner. The observer would begin then behind the retina of the eye. We could also include some of the equipment used for physical observation, e.g. a microscope, into the observer. The principle of psycho physical parallelism defines a situation, where the abovementioned border can be shifted as deeply into the body ofthe observer as we wish. As far as the issue discussed by Bialobrzeski was concerned, Bohr thought that the dualism mentioned by him was simply a problem of selecting most adequate description of an experiment. As for von Neumann's comment on differentiation between the phenomenon and the observer, Bohr thought that the differentiation was a natural process with the phenomena we are dealing with within quantum theory, however we try to explain a given phenomenon in a classical way. Kramers in turn briefly referred to Bialobrzeski's comments, and expressed his own opinion on them. He thought that if we adopted calculation scheme utilised, as a matter of fact, in many real physical problems, it would be dubious if we could really talk about independent postulate referring to measurements. Kramers believed that "demanding such a postulate is a consequence of too a classical approach to the problem. We cannot speak about Schrodinger equation and about the ways we build up Hamiltonian unless we think about experiments at the same time. In other words, we cannot reject results of measurements which allowed us to make use ofwave functions. and we cannot neglect the way in which the value of mechanical quantity can be linked with the wave image. To measure is nothing but confirming details in the instrument's state. These details carry the sense

B. LANGE

412

directly derived from the notions of classical physics. According to the notions, the condition of a given system may be either like this or like that. There is neither indefiniteness nor distribution of probability among various possible states. Natural laws, however, say that the result frequently cannot be predicted even if the starting stage of an experiment is defined in the most precise way." [l] Kramers saw in this the role for the quantum ofaction in stabilising Nature. On the other hand, the role of the quantum of action is of the type that does not allow us to talk about some 'mechanism' regulating the selection pertormed by an instrument when we want to measure something. To put forward a question like this means - according to Kramers - a shift into the scheme ofclassical determinism. This determinism, however, had been rejected the moment we accepted wave function. In the further part of discussion Bialob~eskistood by his opinion by saying, that he had not properly understood Kramers's explanation. He noticed that the way of thinking in the field ofquantum theory was the very way we are used to in natural sciences. The situation could always be presented as an interaction between an object and an instrument. The system could be studied - the scholar thought - with the use of additional expression in the hamiltonian formula. [fthe expression exists, the condition changes. The emergence of a spectral line on a photographic plate may be an example here. We have phenomenon which can be described with the expression defining disturbance. Then the whole process of the phenomenon can be defined. Thus we can use this particular postulate, because - as von Neumann said - a division line within a measuring system can be drawn anywhere. In his reply Kramers stated that we could adopt this procedure for more precise analysis of a measuring instrument, we could introduce to hamiltonian words that would define interaction ofthe instrument with variables characteristic for the state ofthe instrument. We could prove by this that we understood the function of measuring instruments well. According to Kramers, however, this did not constitute the 'mechanism' of making choice. In his summary, Bohr said that there were a few ways ofexpressing relations between classical and quantum physics. In the theory of classical electromagnetism, for instance, theoretical description is inseparable from certain experimental methods used to define certain basic notions. In quantum mechanics the situation is different in the sense that interpretation of various measurements which could be performed, whatever their precision be, do not allow for classical definition of the system. According to Bohr it would be very difficult to describe such a situation by saying that it would take additional mathematical postulate to link a result of a measurement with the other principles of the theory.

Bialobrzeski's idea that interpretation difficulties forced us to recognise a choice during measurement of eigenvalue of the measured quantity, as an independent postulate which was mathematically expressed by the formula defining probability of the choice of any of eigenvalues, did not receive recognition. Bialobrzeski did not give up, however, the development of his idea, and made it more comprehensible in his book entitled Cognitive Foundations of the Physics of the Atomic World, [2].

References 1. 2.

Les Nouvelles Theories de Ea Ph.sique, (1939) International Institute of Intellectual Cooperation, Paris. Bialobrzeski C., 11956) Podstaw? poznawczefiqki swiata atomowego, [Cognitive Fouildations of the Physics of the Atomic World,), PWN Warsaw.

THE FORCE c 4 / ~ THE , POWER c5/6 AND THE BASIC EQUATIONS OF QUANTUM MECHANICS

LUDWIK KOSTRO Department for Logic. Methodology and Philosophy of Science, University qf Gdalsk, ul. Bielalska 5, 80-951 Gdalsk, Poland E-mail: fizlk@ univ.gda.pl

1. Introduction

In two recent papers [I-21 the quantity C~/G was interpretated as the greatest possible force in Nature. In a third paper [3] following LR. Kenyon [4] the quantity c5/G was interpretated as the greatest possible power. In the three mentioned papers I have limited myself to classical considerations. I have shown eg. that the classical Newton law and the classical Coulomb law can be rewritten in the following way:

Newton force when m, = m2

when ml#m2

Coulomb force

It was also indicated that the quantities c4/Gand CS/G and their inverses appear in the equations of General Relativity [1,2,3] and Kenyon's interpretation [4] of this fact was presented. Ln my considerations I use the following constants and constant coefficients: c velocity of light in vacuum; G - the gravtational constant; ...- Planck's constant; e - the elementary electrical charge; m - the mass of an elementary particle; K = 1 / 4 n ~..I take into account also the units of lenght, time and mass determined by the following set of constants (c,G,nt) (c,G,e) (c,G, ..) (c, G,ggJ. (c, G, gw). Where in, e, g ~,gw , are the

respective charges of four fundamental interactions. Using the dimensional analyse we obtain the fallowing units: 413 R.L. Anloraw e.? a1 feds.).Gravitation and Cosnlolog?.:From tlw Hzibblt.Rcxius to tltr Planck Scale, 413-418. @ 2002 Kluwer Acude~nicPublislters. Printed in tlte Nethei-luitds.

L. KOSTRO

414

(a) gravitational lenght ZG ,time tG,and mass m~

(b) J.G. Stoney' s lenght Is ,time ts , and mass ms introduced by him in 1874 [5-6]

(c) M. Planck's lenght ZP ,time tp,and mass mp introduced by him in 1899 [7]

ZP

3 112

= (..G/c)

,.

tp= (..G/c)5

I12

,.

ms = (..c/G)

'n

(d) Lenght Is,, time ts, and mass ms,r connected withe the strong interactions

(e) Lenght lw, t i e tw and mass mw connected withe the weak interactions

It is interesting to note that forces F and powers P connected with these units all are equal:

These forces and powers appear especially when the density p of the matter is the greatest possible one. The following formulae present the limiting density of the gravitationally, electrically, strongly and weakly charged matter and the Planck's density.

6 2 ps= (11~2) (c /G )

p,, = (I/. .c)(c6/&)

(Stoney's density)

(Planck's density)

As we can see the respective limiting density depends upon the inverse of the square of the respective charge multiplied by the factor c6/G 2. In other words, it constitutes the product ofthe square ofthe respective charge and the factor c6/G 2

BASIC EQUATIONS OF QUANTUM MECHANICS

2. The Quantities Equivalence

C'/G And C%

And Einstein's Principle Of Mass And Energy

It is interesting to note that the Einstein's Principle of mass and energy equivalence E= me2 can be rewritten in the following way:

This fact shows, may be once again, the dynarnical nature of the matter. [f an elementary particle could deliver its total energy E = me2 acting on the path equal to lG and during the time equal to fc then it could show its greatest force (c4/G) and power (c5/G). If it could happen then the extremal force {c4/e) and the extremal power {c5/G) would be hidden in every particle. Perhaps in the future mankind will find out the circumstances in which it will be possible. At the present day, however, we can ose only to interpretate the two quanties as exremal ones.

3. The Quantities 6'E And C%

And Schriidinger Equation

As is well known Schrodinger equation is the basic equation of the non relativistic Quantum Mechanics. In textbooks it is written in the following way

-

where V can be the Coulomb potential V = ~ 2 / r .In these equations we find the constants .., n ~e, and the coefficient K. Since the constants used in physics and the units determined by them are correlated and interconnected therefore it is not difficult to rewrite the Schrodinger equation in such a way that the quantities c4/G and C'/G and the considered units appear in it. The Coulomb potential can be rewritten as follows

and the Schrodinger equation in the following way

Since lsts = lP t p a (where a = K ~ / . . Cis the fine structure constant) we obtain also

As we can see, in the Schrodinger equation written in such a way, threre appear not only the quantities C'/G and cS/G but also the Planck's lenght and time and Stoney's

L. KOSTRO

416

lenght and time.We see also that Planck's constants ...is related to the quantities c4/Gand 2/G as follows:

When we divide the both sides of the eq. (8) by (I,ip)we obtain

We must be aware, however, that such a division changes the numerical value and the dimensions of the both sides ofthe equation. The threedimensional Hamilton operator

-

H = (..?/2m)[(d 2 r ~ / d x 9 +(aZyl/d$)+ ( d 2 y l / d 2+ V

(1 1)

can be rewritten introducing c4/Gand C'/G as follows

H = - (.l2m)(c4/G)(lPt p )[(&/.x?+

(dzyl/d$)

+(

2

)+ V

(12)

-

When V = - ~ e ' / r= - (c4/G)(1,2/r)= (c5/G)(lStS/r) then we can write

4. The Quantities

And

P/GAnd The Klein-Cordon Equation

Let's consider the Klein-Gorden equatiom written e.g. for the mesons ~t..

- ..z(@*J)

=

- ..~CZ(a2v ~ t ? ~ 2 + ~+d ~ d~y /z

d 2 ) C# + v~ ~ ~

113)

Taking into consideration thequantities C?G and cS/Gthe Klein-Gordon equation can be rewritten:

As we can see the Planck's charge raised to the second power (..c) is related to (c'/G) as follows

Since rn,

2 =(c4/G)IG,=(cS/~)tG, the eq. (15) can be also written as follows

BASIC EQUATIONS OF QUANTUM MECHANICS

417

Since lc, tc, = (lp tP)%x (where %n = Gmn %.c is the coupling constant of gravitational interactions between two particles of the same mass i our case the coupling constant of grawitational interactions between two mesons n) the eq. (16) can be rewritten as follows

Dividing the both sides of eq. (17) by (1,t p )we obtain

We must be aware. however. that such a division changes the numerical value and the dimensions of the both sides ofthe equation. 5. The Quantities &/G And C?/G And The Dirac Equation

The Dirac equation can be written as follows:

where the matrix yP have the following properties

(where

t

means the hermitonian coupling)

Taking into consideration the quantities c4/G and cS/Gthe Klein-Gordon equation can be rewritten:

'

~ gravitational time where t, = (.G/cS) is the Planck's time and tG= ~ m / iscthe

L. KOSTRO 6. Conclusion It was very easy to introduce the quantities (c4/G)and ( c 5 / ~ into ) the basic equations of Quantum Mechanics (and we can even say that such an introduction constitutes a very trivial operation) but it is very dufficult to interpretate their part played in these ) into the classical equations. When we introduce the quantities (c4/G) and ( c 5 / ~e.g. equations of Newton and Coulomb we see immediately their meaning as limiting quantities i.e. the greatest force and the greatest power but when we introduce them into the basic equations of Quantum Mechanics we do not see clearly their physical meaning. They work, however in these equations and therefore we can think that also here they play not only a role of constants but also as Limiting quantities, as the greates force and the greatest power.

References 1.

Kostro L. and Lange. B. (1999) Is c4/Gthe greatest possible force in nature? Physicv Essays. 12. No 1, 182

2-

Kostro L. and Lange B., (1998) The force c4lGand Relativity Theory, in M.C. Duffy (ed), Phj.sical Znterpretatioris of Relativie Tlieo~y.(Proceedings) British Society for Philosophy of Science, Impaial College, London, pp. 183-193

3.

Kostro L. and Lange B., (19%) The Power c5/G and Relativity Theory, in M.C. Duffy (ed), Physical Interpretations of Relativi~Tlleory. (Later Papers) British Society for Philosophy of Science, Imperial College, London, pp. 150-155 Kenyon I.R. (1990) General Relativity, Oxford University Press Barrow J.D. 11983) Q.JlR. Astr. Soc. 24.24 Stoney GJ. (1881). Phil. Mag.. 5,381 Planck, M. (1899) Sit:ungsberichte d. Preus. Akad. Wiss./Mitteilung/S, 440 and Planck, M. (1900) Ann. Phys. 1, 69

4. 5.

6. 7-

PROGRESS IN POST-QUANTUM PHYSICS AND UNIFIED FIELD THEORY JACK SARFATTI Internet Science Education Project & International Space Sciences Organization [email protected]

Abstract

Progress in extending the de Broglie- Bohm-Vigier (AKA dBBV) quantum ontology to the experimental mind-matter problem is reported in Part I. Progress in extending Einstein's classical orthodox holonomic topology-conserving general relativity of 1915 to the unified field theory including topology-changing anholonomic torsion fields from the "hyperspace" of M-theory is reported in Part L1. I also make a conjecture that the empirical duality in the Wesson compared to the Sirag data plots. noted by Gray in 1988, is actually showing the M-theory T-duality R ~u l / 2 ~ n, ff m The Wesson "Regge trajectory" (Kaluza-Klein excitation n -winding number m) mass scale

'

- &M,, +1018~ e -vI, /& - 1 o - cm~ .~ The mass scale - M, /& + 1 0 ~ ' ~ e -v &IP M, =

+ 10" ~

-

e vl p I

-

''dual" Blackett-Sirag magneto-gyro

cm, where a = e2/hc

n: 11137

and

cm. What is clear is that we now

have a new "telescope" directly into the quantum gravity scale showing strong anholonomic unified field effects beyond Einstein's 1915 theory. This is as important as the Hubble flow, the cosmic microwave background, missing mass, gravity waves, and the anomalous acceleration of the universe. A completely new conception of COSMOS is now emerging from the actual data.

Part 1: Progress In Post-Quantum Physics Orthodox quantum theory has many "degenerate2 informal interpretations that appear to have no crucial experimental tests to "lift the degeneracy". The experimental situation is now changing dramatically and quickly with my recognition of the real meaning of data lying around in journals unread for twenty years. All of the contemporary competing

' 11.6 eq. (11.6.2) p. 477 "Intro to Superstrings and M-Theory", M.Kaku, Springer-Verlag, 1999.

* In the sense of atomic spectroscopy with "degenerate energy eigenfunctions" of the Harniltonian operator. 419 R.L. Arnoroso et a1 (eds.),Gravitation and Comolog?.: From the Hubble Rudius to the Planck Scale, 419430. O 2002 Kluiver Academic Publi~her.~. Printed in the nether land^.

420

J. SARFATTI

interpretations3. save dBBV. only have the quantum wave and not the particle in their models of quantum reality. The "particle" or "Bohm point" moving on the landscape formed by the quantum potential and other forces is "e~chewed".~This immediately causes confused thinking, e-g., in the recent articles claiming to divide the charge e ofthe electron into two equal pieces.5 This confusion comes from falsely assuming that the quantum wave "is the essence of an electron" carrying the charge because there is no particle at the micro level. The experimental result of increased current is easily understood intuitively in the dBBV interpretation in which the complete indivisible electron particle, of screened charge e, is completely localized in only one of the two bubbles that the physically real pilot wave divides into. One of the paired bubbles has a real but empty branch. The smaller bubbles move faster through the liquid helium explaining the observed increase of electric current without having to cut the electron in half. Therefore, this experiment seems to lift the degeneracy.6 Bohm's ontology works better for this experiment. In another recent development, Henry stapp7 has proposed a model of ontological collapse8 of the quantum wave with infinite speed in the preferred cosmological rest frame of the Hubble flow9 in the standard cosmological model. This is theory on a classical curved space-time geometry not full blown quantum gravity. Stapp's conjecture is similar to Bohm's and Hiley's in which the quantum potential Q ;its instantaneously in this same preferred frame. There is no reason to suppose that classical Diff(4) local gauge symmetry10 of the 1915 general theory of relativity should be valid when quantum nonlocality is important. However, experiments by is in" et-a1 in Geneva seem to rule out this idea, although Stapp and I are in serious disagreementI2 on how to think about this problem. I am only giving my biased perspective here. Special relativity still works locally. According to the Einstein E.g., Stapp's "'ontological collapse", Penrose's "R" and "'OR", "many worlds" in all of its variations such as David Deutsch's "multiverse", Gell-Mann/Hartle decohering histories, John C ~ m e r ' s"transactional" with weak backward causation consistent with "signal locality" AKA "passion at a distance" (Abner Shimony). "Bohmian Mechanics and Quantum Theory: An Appraisal" Ed. J.T. Gushing, A. Fine, S. Goldstein (Kluwer,

1996) "ew Scientist magazine, 14 October 2000 "This sounds harmless enough, but the implications are staggering. Ifthe bubble split, half of the electron's wave function would be trapped in a c h of the two daughter bubbles .... As the wave function is the essence of an electron, the electron would be split into two. The indivisible would have been divided. ... 'There were more bubbles, and being smaller they were more mobile,' says Maris. Although the total charge in the system remained the same, the smaller bubbles felt less drag in the helium, and thus moved faster. Consequently, the current went up," My solution here is reminiscent of King Solomon's when asked to divide the baby in half. The electron, like the baby, has been thrown out with the bathwater in all the alternative interpretations save dBBV. Discussion by e-mail among Stapp, Stan Klein and myself. "speed ofquantum information" (Gisin et-al, ref 11) In which the cosmic microwave black body background radiation is isotropic to about one part in ld. 10 Integrable holonomic general coordinate transformations that are global 1-1 conserving topology of the 4d s cetime manifold. 1pa@ant-ph/OOolo384 July 2 0 T h e Speed ofQuantum Information and the Preferred Frame: Analysis of Experimental Data" V. Scarani, W. Tittel, H. Zbinden, N. Gisin l2 There is no preferred frame in global special relativity, consequently whether or not nonlocal EPR correlations are observed cannot depend upon the common state of uniform motion of the two detectors relative to any other frame. However, global special relativity breaks down in general relativity where it can only be used locally. Therefore, this issue is an experimental one. Stapp inconsistently tries to maintain global special relativity and the preferred frame ofthe Hubble flow together in my opinion.

'

POST-QUANTUM PHYSICS AND UNIFIED FELD THEORY

421

addition of velocities, an infinite speed in one frame is a finite superluminal speed

v = C'/U

where u is the subluminal speed of the moving "G-frame" relative to the allegedly preferred global rest frame of the Hubble flow. For example. the rotating Earth's motion has a 24 hr periodic projection u. of its CM motion on to the flight line between the G-detectors, of amplitude 300 to 600 kmlse~.'~Therefore, except for 5 seconds every 43,200 seconds, the G-frame qubit speed is less than the minimum 13,

-

-

2

213 x 10' c required dipping down to v = (3 x 1 0 5 ) /(4 x lo2) = 2.5 x 10' km/s

= 103 c

reaching a peak less than 213 x 10' c for most of the data run. In contrast, Gisin cites "a lower bound for the speed of quantum information in this "G-frame" at 2/3 x 10' c " for the parameters of the actual experiment'5 in Geneva of EPR photon pair correlations over a distance 10.6 km. Therefore, G-frame detectors, each with speed u relative to the Hubble flow, separated by a distance 10.6 krn requiring a qubit speed of at least

213 x107 c to travel between them in the 5 picosecond time uncertaintyI6 in the detections, should not show the actually observed EPR correlations most of the time. In fact, the EPR correlations are seen all of the time. This argument assumes that the qubit speed in the Hubble frame is infinite, so that objective collapse with a real qubit speed is a useful way to picture how the nonlocal EPR correlations are maintained. Therefore, Stapp's conjecture here is falsified by the actual experiment. Does this also shoot down the Bohm-Vigier conjecture that the quantum potential Q acts instantly in the preferred Hubble flow frame? Yes, the model that does survive is that of "backward causation" l7 originating in the Wheeler-Feynman 1940 program for classical electrodynamics eliminating independent dynamical degrees of freedom for the electromagnetic field with "Everything is particles."18 However, it appears that Q = (fi/2rn)v2~/~can be redefinedI9 in terms of backward causation. Note that the term "ret" for "retarded" means "from the past", or what Aharonov calls the "history" state vectw. Similarly, the term "adv" for "advanced" means "from the future", or what Aharonov calls the "destiny" state vector. Feynman already used this idea in his original paper on the path integral in nonrelativistic quantum theory.20

Narlikar. p.301 (Cambridge, 1993) Time resolution of 5 picoseconds in footnote [19] of ref 11. l6 Photon pulse temporal width l7 "Feynman zig zag" (0. Costa de Beauregard), "transactional interpretation"(J. 0. "history and destiny state vectors" (Y. Aharonov et-al), Hoyle-Narlikar"advanced response of the universe" in "Cosmology and Action at a Distance Electrodynamics" World Scientific, 19%. 'Time's Arrow and Archimedes Point", Huw Price, Oxford, 1996. Is "Geons, Blackholes & Quantum Foam", Wheeler's autobiography with Ken Ford. l 9 There is no collapse in the Bohm ontology. hence no "qubit speed" for ~mllapse. We have seen that the standard backward causation models without the particle cannot explain the apparent splitting of the charge on the electron as naturally as the Bohm ontology can. 20 Rev. Mod. Phys. 20,267, "Space-Time Approach to Non-Relativistic Quantum Mechanics" (1948) 15

Bohm showed that the pilot wave is a physical field of qubits in the configuration space of the piloted material. The ideas of quantum probability21are not fundamental. God does not play dice with post-quantum r ty. Therefore. unlike the orthodox statistical interpretations relying fundamentally on ensembles of identically prepared simple atomic systems such as particle beams in scattering experiments, Bohm's ontology is ideally suited to explain unique complex highly entangled systems such as the living human brain. There was only one shakespeare2 and to invoke shadow Shakepeares in a multiverse of parallel worlds is "excess metaphysical baggage".'3 This is not to deny the possibility of ''other worlds" close by in material hyperspace, less than a millimeter away, as in M-theory with "3D membranes" folded by anholonomic torsion fields.24One can even imagine traversable wormholes connecting these worlds to each other. Bohm and ai Hiley also emphasize that the quantum pilot field is "nonmechanical" and "organic" with no "preassigned interactions between the parts". In this sense, the pilot field is not at all like a classical machine. The pilot field in configuration space for entangled subsystems is form-dependent and intensity-independent totally opposite to classical fields in z eali and undiminished with increasing separation ordinary space. It is intimate 5. immediate. unlike the classical dynamical force fields of electromag~etism gravity. and torsion confined to ordinary space. These are all desiderata for the "mental field" out of which our thoughts, feelings, and perceptions arise in consciousness. Indeed, the pilot field idea immediately explains how thought can move matter. What Bohm's and Vigier's "causal theory" cannot qu l tatively explain in principle is how matter influences thought to create the inner conscious experience. This is because of an argument Bohm and Hiley ?iveZ6 that the standard statistical predictions of quantum theory for ensembles of identically prepared simple independent unentangled systems" require that there can be no direct reaction or "back-action" of matter on its pilot field. Such a compensating postquantum reaction to quantum pilot action would result in "signal nonlocality" violating the Stapp-Eberhard no-go theorem. The latter forbids the use of quantum nonlocality as a d i i t communication channel for what Einstein called "spooky telepathic action at a distance" in violation of the retarded causality postulate of the classical theory of special relativity. The modem theory of quantum computing, cryptography and teleportation would fall apart if the signal-locality of orthodox quantum theory could be violated. Yet, Feynrnan's phenomenological rules: add complex amplitudes before squaring for indistinguishable alternatives. Square amplitudes before adding for distinguishable alternatives. 22 Whether or not the real Shakespeare was really the brothers Francis and Tony Bacon at the Scriptorium in London with Ben Johnson (Sirag "Shakespeare's doublet reversed in First Folio picture") not the issue. 23 Wheeler refuting "many worlds" he once endorsed. 24 August 9000, Scientific American "The Universe's Unseen Dimensions" p.62, N. Arkani-Hamed et-aL 25 Literally attached to a material configuration like the private mind of an individual living brain. 26 P.30 and Ch. 14 in "The Undivided Universe" (1993) 27 We can call this the "'actuary's limit" of sub-quantal equilibrium of the nonlocal hidden variables, AKA "subquanta1 heat death" in Antony Valentini's Ph.D. dissertation unda Dennis Sciama at Cambridge. 2'

POST-QUANTUM PHYSICS AND UNIFED FIELD THEORY

423

this is precisely what the human mind does when it experiences and knows. The quantum potential Q characterized by action without reaction is "fragile".28 It is this Vragility" that maintains irreducible uncontrollable local quantum randomness29even in nonlocally entangled systems. It is not possible to control a quantum probability at a distance in orthodox quantum theory. Yet, this is precisely what happens in the brain, indeed, in the entire living body beyond the neural transmission of electrical signals and the transport of chemical messenger molecules as important as they are. The pilot Geld of the Bohm-Vigier "causal theory", with deterministic particle trajectories, is an "absolute physical object"3o. Therefore, this pilot wave is just like Newton's absolute space and absolute time before special relativity and just like the absolutely flat spacetime of special relativity before Einstein introduced the direct back-reaction of matter on spacetime geometry to bend it into gravity. The unified field theory goes further twisting the spacetime geometry and changing the topology of the 3D membranes to create and destroy traversable wormholes as one example. Why does quantum theory work so well? 1 propose the following model. Imagine two barriers. One barrier is for the action of the quantum pilot field on the matter it is piloting of height hper qubit31.The other barrier is for the reaction or "backaction" of the matter on its pilot field. When the quantum action is balanced in strength by this new post-quantum reaction, one forms a self-organizing feedback control loop between pilot field and its matter suppressing quantum randomness with "signal nonlocality". Orthodox quantum theory is strongly violated in this situation. The BohmVigier "causal theory" breaks down completely in this new regime. The particle paths are no longer deterministic, rather, they are self-determining . The entangled individual particle paths, inseparable in configuration space, exhibit strangely synchronized motions beyond the local contact forces from classical signal connections when viewed in ordinary space. This is the essence of the "self in biological life. Indeed, the particle paths co-evolve with the changing shape of the landscape formed by the now postquantum potential Q*. Both the particle paths and the shape of the landscape they ride on in configuration space are tweaked by external perturbations from the non-self environment. There is a further mathematical generalization in that the co-evolutionary flow of the nonlocally entangled particle paths on the landscape of their common pilot field is no longer an integrable holonomic path-independent flow characterized by exact differentials. One now has a more complex dynamically changing self-determining topology of closed inexact differential nonintegrable anholonomic flows on the landscape. Indeed one must use the Pfaffian theory of nonintepbility of R. Kiehn32 wo) ,

We find that the ZPF mode with the higher frequency, that is w l , actually has its intensity reduced below the vacuum level. I propose to call this Parametric Depletion of the Vacuum (PDV); it is somewhat different from squeezing, since in this case both quadrature modes are reduced in intensity, while in squeezing one of them is reduced at the expense of the other. Such a phenomenon will not easily be demonstrated experimentally, but the other mode, that is w , is amplified. So we need to know where to look for wz, and what intensity to expect. I have published sufficient details of this on

ENGINEERING THE VACUUM

467

the Los Alamos archive, as cited above, so all you have to do is look at the predicted angles and you will see the phenomenon. 7. New technology?

I think that, once the ZPF has been fully accepted as a red physical object, it will have an engineering, as well as a scientific impact. A prime candidate for this is the field of laser design. To emphasize the relevance of the ZPF in a Laser, I point out that a laser is a device which selectively amplifies certain modes of the ZPF. Indeed the acronym "lasef' stands for light ampl[fication by stimulated emission of radiation, and it is precisely the ZPF which does the stimulating! It is no accident of history that effectively the first working laser (called, at that time a maser) was the apparatus used by W. E. Lamb and colleagues in 1947 to measure the Lamb shift. As we have seen, Willis Lamb is outstanding, among the pioneers of Quantum Electrodynamics, in his recognition both of the importance of the ZPF and of the inadequacy of the Photon concept. Lasers are extremely inefficient heat engines. They convert incoherent (that is thermal) Light energy from the pump into a coherent output ("work?'), but typically, in a narrow-band continuous-wave laser, about 10 kilowatts of pump power is required to produce 500 milliwatts of coherent output. Good coupling of the pumped atoms to the ZPF, both outside and inside the laser, is achieved by good cavity design, and I think the standard Fabry-Pemt cavity can be immensely improved, once we have understood how the ZPF may be engineered. There is a historical parallel here. Heat engines played a central role in the Industrial Revolution between 1770 and 1850, but scientists at that time accepted a deeply incorrect theory of heat; it was thought to be a material substance called "caloric". Then Thermodynamics was correctly formulated and heat engines became even better, culminating in the automobile. We have made a lot of progress with lasers since 1947, even though our theory of Optics has become dominated by mythical objects called "photons". The time has come, when we should recognize, in the zeropoint field of Max Planck, a new branch of Thermodynamics, that of zero degrees Kelvin. 8.

Theses on Magic

- No competent magician believes in magic. - A competent magician is really a scientist pretending to be a magi-

cian. - A competent magician is one who knows how the rabbit got into the

hat ...

468

TREVOR W. MARSHALL

- . . . and to distinguish between a scientist and a competent magician is a problem in ethics rather than science. References 1. L. de la Pefia and A. M. Cetto, The Quantum Dice, (Kluwer, Dordrecht, 19%) 2. P. W. Milonni, The Quantum Vacuum, (Academic, San Diego, 1993) 3. T. W. Marshall and E. Santos. The myth o f the photon in The Present Status o f the Quantum T h e o ~ yof Light, eds. S. ~iffersletal'(Kluwer, Dordrecht, 1997) pages 67-77. T. W. Marshall, www.demon.co.uk and homepages.tesco.net/-trevor.marshal1 W. E. Lamb, interview in La Nueva EspaAa, Oviedo, Spain (July, 23, 19%) D. Bouwmeester and A. Zeilinger. Nature 388, 827-828 (1997) G. Weihs. T. Jenneswein, C. Simon, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett, 81, 5039 (1998) A. Pais, Subtle is the Lord, page 384 (Clarendon, Oxford, 1982) W. E. Lamb, Appl. Phys. B, 60, 77-82 (1995) R. P. Feynman. You must be joking Mr. Feynman L. J. Wang, X. Y. Zou and L. Mandel, Phys. Rev. A, 44, 4614 (1991) A.Casado, T.W.Marshal1, and E.Santos, J. Opt. Soc. Am. B, 14, 494-502 (1997). A.Casado, A.FernBndez Rueda, T.W.Marshal1, R-Risco Delgado, and E-Santos, Phys.Rev-A, 55, 3879-3890 (1997). A.Casado. A.Ferndndez Rueda, T.W.Marshal1. R.Risco Delgado. and E-Santos. Phys.Rev-A, 56, R2477-2480 (1997) A. Casado, T. W. Marshall and E. Santos, J. Opt. Soc Am. B, 15, 1572-1577 (1998) A.Casado, A.Fern&ndez Rueda, T.W.Marshal1, J. Martinez, R.Risco Delgado, and E-Santos, Eur. Phys. J., D11,465 (2000) K.Dechoum T. W. Marshall and E. Santos, J. Mod. Optics. 47, 1273 (2000) K. Dechoum, L. de la Peiia and E. Santos, Found. Phys. Lett., 13, 253 (2000) D. M. Greenberger, M. A. Home and A. Zeilinger, Phys. Today, 46 60.8,22 (1993) P. A. M. Dirac, Principles of Quantum Mechanics, page 9 (Clarendon, Oxford, 1958) B.E.A.Saleh and M.C.Teich, Fundamentals of Photonics, (John Wiley, New York, 1991) T.W. Marshall, http://xxx.lanl.gov/abs/quant-ph/9803054.

THE PHOTON AS A CHARGE-NEUTRAL AND MASS-NEUTRAL COMOSITE PARTICLE Part I. The Qualitative Model

HECTOR A. MUNERA Department of Physics Universidad National de Columbia A.A. 84893, Bogota, Columbia

Keywords: Photon, Aether, Composite photon model, Maxwell equations, Advanced fields, Charge neutral particles, Mass neutral particles.

Abstract. In the context of a 4D aether model, where rest mass is associated with a flow of primordial mass (preens), the photon was described as an electron-positron pair. Such a composite particle is then a charge-neutral and mass-neutral entity; thus accounting for photon standard properties: zero charge and null restmass. The electromagnetic field of such photons contain both advanced and retarded components, without any causality breach. The model obeys conventional Maxwell equations.

1. Introduction

The idea that the photon may be a composite particle is not new. Long ago De Broglie (1932) suggested that the photon was a composite state of a neutrino-antineutrino pair; such pair, however, did not obey Bose statistics. To avoid this difficulty Jordan (1935) introduced neutrinos with different momenta. Over the years, additional adjustments were made by Kronig (1936), Pryce (1938), Barbour et al. (1963), Ferreti (1964), Perkins (1 9651, Bandyopadhyay and Chaudhuri (1971). Since the photon rest-mass is zero, or very small (Vigier, 1997), neutrinos are chosen as its components. However, 469 R.L Amoroso et a1 (eds.),Gravitation aircl Co.snlology: From the Hubble Radius to the Planck Scale, 469-476 O 2002 Kluwer Acaclemic Publishers. Printed in the Netherlm~cls.

470

HECTOR A. MUNERA

one would nziively expect that if the photon is a composite particle it may under some conditions decay or be separated into its components. Indeed, photon pair production leads to an electron-positron pair, but not to a neutrino-antineutrino one. However, from the viewpoint of total rest mass, an electron plus a positron can not be the components of a low energy photon. There is a clear difficulty for composite models along this line of thought. On the other hand, the idea that the modern "vacuum" (= aether in this paper) is of hydrodynamic nature is a recurrent one. Recent examples: a superfluid of particle-antiparticle pairs (Sinha et al., 1976), a fluid of "stuff' particles (Di Marzio. 1977), and a variety of fluids (Shekhawat. 1976: Widom and Srivastava, 1990; Winterberg, 1997; Ribaric and Sustersik, 1997). From such fluids, some authors derive electrodynamic and particle models: Thomson (1931), Hofer (1998), Marmanis (1998), Dmitriyev (1999). This author recently proposed a four dimensional (4D) hydrodynamic model that allows for a variable component of the 4-velocity along the time axis (Munera, 1999), which leads to a LCD-force as the gradient of the 4pressure; the 3D-electromagnetic force is a particular case (Mdnera, 2000). Also, we have argued elsewhere (Chubykalo. Mirnera and Smirnov-Rueda. 1998) that, in the context of Maxwell's equations, the concept of zero charge in vacuum may be interpreted as neutrality of charge almost everywhere, rather than as complete absence of charge. In this note we take one step further. The photon is modelled as a source-sink pair (into and out of our 3D world), having a zero net mass flow into our 3D world. thus accounting for the photon zero rest mass. Next section 2 summarizes the 4D aether model, and section 3 sketches the photon modeL A final section 4 closes the paper.

2. A Four Dimensional Fluid

Let us assume the existence of a fourdimensional (4D) flat Euclidean where the time dimension w=vwtbehaves exactly the space space Z=(w,xSySz), same as the 3 spatial dimensions (Mdnera, 1999; 2000). Further, let Z be filled with a fluid of preons (= tiny particles of mass m and and Planck length dimensions). These particles are in continual motion with speed ~ ( V ~ , V ~ , V ~ VN~O ~a V priori , , , ,limits V ) . are set on the speed vw of preons along the w-axis. (Notation: 4D-concepts and vectors are represented either by calligraphic or by Greek uppercase letters, while 3D-vectors are in the usual bold face). Note that the Limitations of the special theory of relativity (STR), if applicable, refer to v = (v;+v,?+v, 2) 112 (= the speed of particles in 3D) not to vw= the projection of the 4D-velocity '1/ on the w-axis. Here, we extend the

PHOTON AS COMPOSITE PARTICLE

notion of absolute space to 4D (= (ct,x,y,z), i.e.

471

R~),whereas the spacetime of STR is

R'.~.

Motion of individual preons in T: is governed by a 4D-equation of motion, given by the matrix expression (Miinera, 1999):

where p = nm is the preonic fluid mass density, n is the number of preons per unit 3D-volume, the column vector V = [v,,vx,vy,vJ is the 4D-velocity of individual preons, Va = [cDvX,vy,v,l refers to the time-arrow, the vector is a 4D gradient. the 4x4 matrix T~~ is the 4Doperator dp[&,V],&dl& stress tensor, and P =P (w,x,y,z)is the pressure generated by the preonic fluid; the Greek index p = (w,x,y,z).Finally, the energy-momentum tensor pWa(a 4x4 matrix) results from the dyadic product Wa. Consider now a 3D-hypersurface formed by a projection of the 4Duniverse onto the w-axis, say w = wo= v,oto (Fig. 1 ) . The plane w-r may be interpreted in two complementary ways: Interpretation 1 (Fig. la). At a fixed time to (say the present), the Line w=wo divides the plane into three classes of particles: preons moving with vw> V , O (upper region), preons moving with vw< v , o(lower region), preons moving with vw=v , o (on the horizontal line). Interpretation 2 (Fig. lb). For the class of preons moving with vw= vw,o, the line w = wo divides the plane into three periods of time: the future for t > to(upper half-plane), the past for t < to(lower half-plane), the present t = to (on the line). The conventional worldlines of STR and the space underlying Feynman diagrams belong to interpretation 2 with v ,unspecified. ~ If we postulate that we live in a 3D-hypersurface where vw,o= c, then all preons in our world move with constant speed c along the time axis. This brings in a novel intrpretation for constant c: the speed at which our hypersurface slides from the past to the future (Interpretation 2 above). The meaning of the w-r plane under Interpretation 1 can now be rephrased as: at a given to (say the present) our 3D-world separates superluminal from subluminal preons. Furthermore, as seen below, there is a continuous exchange of p r a m between our hypersurface and the two half-spaces above and below. For events inside our hypersurface, eq. (1) reduces to

472

HECTOR A. MUNERA

where the elements of T~~~associated with the 3D spatial dimensions are given by the conventional ~3~~ viscosity matrix. The elements associated with the w-dimension are

where S=(S,S,SJ is a (displacement) energy flux along axes x,y.z (dimensions: energy per unit time per unit area), and the source/sink ~ ,isfa concentrated energy flow along the w-axis (dimensions: energy per unit time), and 8 represents the position of the energy source/sink (positive/negative respectively), and a(.) is a 3D-Dirac's delta function (dimensions: (l~m~ht)-~). Eq. (3) may be interpreted as a transfer of energy by displacement from the waxis into the spatial axes (or the other way around), whereas eq. (4) is a transfer of energy along the w-axis. Therefore, the 4D-source S = (s,~,s) simply represents a "convective" tramfer of preons from one region of the 4Dfluid into another, i.e. there is conservation of energy in the whole 4Duniverse. Note that other fluid theories contain expressions similar to our eq. (2) (for instance, eq. 3 in Ribaric and Sustersic, 1998). However, our approach is fundamentally different because we allow for interaction between our world and other regions of ):with v, $ c (described by the more general eq. 1). This interaction gives rise to the 4D-source 5 described by eqs. (3) and (4). By analogy with the standard 3D-case, the 4Dpreonic fluid exerts force, and performs work along the four dimensions (w,x,y,z), via its hydrodynamic pressure P (in this sense, P is interpreted as potential energy per unit volume). This immediately leads to a component of force along the w-dimension, which is responsible for the appeareance of sources and sinks in our hypersurface, via the following mechanisms: Sources Sf are produced by the fourth component of force, which acts upon preons outside our hypersurface, via two mechanisms (Fig. 2a): preons moving with v, > c are decelerated to enter our world at t=to with v, = c, preons moving with v, < c are accelerated to enter our world at t=to with v, = c.


I subluminal

r = (x,y ,z) a) condanttirne 1 = t

= [X,Y,Z~ b) constant speed v=, vOw

0

Fig. 1

Figure I. Four-dimensional representation of universe as an r-w diagram. The projection on the w-axis is a 3D hypersurface. This horizontal line partitions the universe into two half-spaces. Part a) For a given time to (say, the present) the upper (lower) space corresponds to universes with higher (lower) speeds on the w-axis. Part b) For a given v,,, = c, the upper (lower) space corresponds to the future (past). See the text.

Fig. 2

Figure 2. The four mechanisms for producing sources and sinks (see the text).

474

HECTOR A. MUNERA

Sinks S-are produced by the fourth component of force, which acts upon preons in our hypersurface that move with vw =c for t < to. Two mechanisms (Fig. 2b): @accelerationofpreons which leave our world at t = to with vw> c, and edeceleration of preons which leave our world at t = to with vw< c. For additional details see Mlinera (1999; 2000). 3.

The Photon As A Source-Sink Dipole

It is widely known that Maxwell equations (and also the homogeneous wave equation) have two sets of solutions: retarded (= outgoing) and advanced (= incoming) ones. The latter are typically neglected on the grounds of causality violations (for instance, Panofsky and Phillips, 1962, p. 244). To account for the advanced solutions in the presence of charges, Wheeler and Feynman (1945, 1949) located an absoher ofradiation at some distance from a charge (see also Panofsky and Phillips, 1962, ch. 21). In the model for the photon described next, we also allow for advanced and retarded solutions without causality violations. In the 4D aether described in previous section, a particle (antiparticle) is a S 1 ), Energy is source (sink), whose rest mass is proportional to Sw+( 1 [ transported into (or out of) our 3D world by a flow of preons. Such representation immediately explains away the difficulties associated with infinities in potential energy (gravitational. electrical. or otherwise) Let the photon be a pair electron-positron, both of them inside a small region of diameter ro. Since the rest masses of the constituent particles are identical, then Sw+-+ Sw-= 0,which implies that the effective rest mass of the campsite particle is zero. Evidently, the net charge is also null, except inside a neighbourhood ro. A simple 3D-analogy is a water filled vessel, with a source Q+ and a sink Q- allowing equal flows of water in and out of the vessel. Consider a differential volume located at an arbitrary point P connected to Q+by a line of flow. Water particles emitted from Q+at t, reach P at a later time according to the speed of propagation of the particle, thus giving rise to a retarded field of pressure. Likewise, consider a particle that is absorbed by the sink Q- at a time t,along a line of flow coming from P. Of course, such a particle was at P at an earlier time, and gives rise to an advanced pressure field. However, there is no causality breach. Simply put, some of the particles located at a given time inside the small volume at P may come from the source, some may reach the sink at a later time, and many other particles may have quite a different fate.


475

The photon is then a rotating dipole in the normal 3D Euclidean space. Let the timedependent retarded (and advanced) electric field associated with the electron (positron) be N (P). It can be shown that fields N and P obey the symmetric Maxwell's equations that we described some time ago (Mdnera, 1997). Such symmetric system is tautologically equivalent to the conventional Maxwell equations, provided that the standard electric and magnetic fields be defined as:

P- N E = -----,B = P + N

.

2

2

4. Concluding Remarks

In the context of a 4D aether model, where rest m a s is associated with a flow of primordial mass ( p n s ) , the photon was described as an electronpositron pair. Such composite particle is then a charge-neutral and massneutral entity, thus accounting for the photon standard properties: zero charge and null rest mass. The electromagnetic field of such photon contains both advanced and retarded components, without any causality breach. The model leads to a symmetric system of Maxwell's equations (Mdnera. 1997). containing two sources (electron and positron). Such system immediately leads to the conventional Maxwell equations (the details will be published elsewhere). Falaco solitons were recently reported (Kiehn, 2000) as pairs of solitons that exist on the surface of a fluid (water), and are interconnected through the third spatial dimension. Our model for the photon is a pair of 3D solitons interconnected through the fourth dimension. An open question to be pursued at a later stage is the connection between the equation of motion describing the 4D aether proposed here and the BohrnVigier (1958) relativistic hydrodynamics.

References Bandyopadhyay, P. and P. R. Chaudhuri, "The photon as a composite state of a neutrinoantineutrino pair'', Phys. Rev. D 3, No. 6 (1971) 1378-1381. Barbour, I. M., A. Bietti, and B.F. Toushek, Nuovo Ciirrento28 (1963) 453. Bohrn, D. and J.-P. Vigier, "Relativistic hydrodynamcis of rotating fluid masses", Phys. Rev. 109, NO. 6(1958) 1882-1891.

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Chubykalo, A. E., H, A. Miinera, and R. Smirnov-Rueda, "Is the free electromagnetic field a consequence of Maxwell's equations or a postulate?", Found. Physics Lett. 11, No. 6 (1998) 573-584. De Broglie, L, Compt. Rend. 195 (1932) 862; Compz. Rend. 199 (1934) 813. Di Marzio, E. A., "A unified theory of matter. I. The fundamental idea", Fotmd Physics 7 (1977) 511-528. "11. Derivation of the fundamental physical law7', ",Found. Physics 7 (1977) 885-905. Dmitriyev. V. P.. "Turbulent advection of a fluid discontinuity and Schriidinger mechanics". Galilean Electrodynatnics 10, No. 5 (1999) 95-99. Ferreti, B., Nuovo Ci~nento28 (1964) 265. Hofer, W. A., "Internal structures of electrons and photons: the concept of extended particles revisited", PhysicaA 256 (1998) 178-1%. Kiehn, R. M., paper at Vigier 2000 Symposium, University of California, Berkeley, USA (August 2000). Kronig, P. Physica 3 (1936) 1120. Marmanis, H., "Analogy between the Navier-Stokes equations and Maxwell's equations: Application to turbulence", Phys.Fluids 10. No. 6 (1998) 1428-37. "Erratum". Phys.Fluids 10, No. 11 (1998) 3031. Miinera, H. A. "A symmeric formulation of Maxwell's equations", Mod Phw. Len. A 12, NO. 28 (1997)2089-2101. Miinera, H. A. "A realistic four-dimensionalhydrodynamic aether interpreted as a unified field equation", presented at the International Workshop Lorentz Group, CPT and Neutrinos, Universidad Aut6noma de Zacatecas, Zacatecas, Mexico (June 1999). Published in the Proceedings edited by A. Chubykalo, V. Dvoeglazov, D. Emst, V. Kadyshevsky and Y.S. Kim. Miinera, H. A. "An electromagneticforce containing two new terms: derivation from a 4D aether7',Apeiron 7, No. 1-2 (2000) 67-75. Panofsky, W. K. H., and M. Phillips, Classical Electricity andMagnetisrn , 2nd edition, Addison-Wesley Publishing Co. (1962) 494 pp. Perkins, W. A., Phys. Rev. 137 (1965) B1291. Pryce, M.H.L., Prm. Roy. Sm. (London)165 (1938) 247. Ribaric, M. and L. Sustersik, "Transport theoretic extensions of quantum field theories", Eprint archive: hep-tW9710220 (Oct. 97) 36 pp: "Framework for a theory that underlies the standard model". LANL electronicfile hep-th/9810138 (Oct. 1998). Shekhawat, V., "Some preliminary formulations toward a new theory of matter", Found. Physics 6 (1976) 221-235. Sinha, K. P., C. Sivaram, and E. C. G. Sudarshan, Found. Physics 6, No. 1 (1976) 65-70. Thomson, J. J., "On the analogy between electromagnetic field and a fluid containing a large number of vortex fdaments", Phil. Mag.. Ser. 7 12 (1931) 1057-1063. Vigier, J.-P., "Relativistic interpretation (with non-zero photon mass) of the small ether drift velocity detected by Michelson. Morley and Miller", Apeiron 4. No. 2-3 (1997) 71-76. Widom, A. and Y. N. Srivastava. "Quantum fluid mechanics and quantum electrodynamic^^^, Mod Phvs. Lett. B 4 (1990) 1-8. Wheeler, J. A. and R. P. Feynman, Revs. Modern P h y 17 (1945) 157, and Revs. Modem Phys. 21 (1949) 425. Winterberg, F. "Planck aether", Zeistch fur Naturforsch. 52a (1997) 185.

PREGEOMETRY VIA UNIFORM SPACES W.M. STUCKEY and WYETH RAWS Departr~lentof Physics & Engineering Elizabethtown College Elizabethrown, PA 17022

Abstract. We begin by motivating a pregeometric approach to quantum gravity. A pregeometry is then introduced over denumberable sets which employs the discrete uniform space Dx and a uniformity base UB induced by a topological group. The entourages of Dxprovide a non-metric notion of locality consistent with the open balls of a differentiable manifold, thereby supplying a pregeometric basis for macroscopic spacetime neighborhoods. Per the underlying group structure, entourages of U, provide a pregeometric model of quantum non-localitylnon-separability, thereby supplying a pregeometric basis for microscopic spacetime neighborhoods. In this context, a robust pregeometric correspondance between microscopic and macroscopic spacetime structures is enumerated. Finally, we show how the pseudometric generated by a uniform space may be used to provide the M4 geodetic structure. This affine structure may produce a metric structure by requiring the covariant derivative annihilate the spacetime metric per standard Riemannian geometry. An example on s2 is provided.

1. Introduction There is as yet no definitive course of action in the search for a theory ofquantum gravity (QG). Since QG must satisfy the correspondence principle with general relativity (GR) and quantum mechanics (QM). it must provide a basis whence the locality of GR and the non-locality of QM. Demaret, Heller. and Lambert write [I]. In this Section we analyse fundamental concepts of quantum mechanics. We show that they lead to some problems which force us to modify the usual notion of spacetime. ... The second problem is related to the famous E.PR. paradox which introduces the idea of non-locality or more precisely of non-separability with respect to space. In fact, in quantum mechanics space cannot be viewed as a set of isolated points. These problems lead to a deep modification of our representation of "quantum" space-time. Stuckey has argued [2] that a non-local, reductive model of reality precludes the fundamental use of trans-temporal objects. Since trans-temporal objects are fundamental to kinematics and kinematics is fundamental to dynamics, concepts such as mass, momentum, and energy are excluded from the foundation of a rational reductionist theory of QG. It is difficult to imagine where to begin modeling reality without reference to the 477 R.L. Amoroso et al (eds.),Gravitation and Cosmdogx: From the Hubble Radius to the Plunck Scale, 477-482. O 2002 Khwer Acadeiriic Publishers. Printed in the Netherlands

478

W.M. STUCKEY AND WEYTH RAWS

concepts of dynamics. Weinberg writes [3]. "How can we get the ideas we need to formulate a truly hndamental theory. when this theory is to describe a realm where all intuitions derived from life in space-time become inapplicable?' Accordingly, QG may require an approach a la Wheeler's pregeometry. That is [4], "the features of the conventional space-time, such as its continuity, dimensionality, and even causality and topology, should not be present from the beginning, but should emerge naturally in the transition process from pregeometry to the usual space-time dynamics of our conventional physical theories." Further, it may behove us to base the program in discrete mathematics. Butterfield and Isham write [5], "For these reasons, a good case can be made that a complete theory of quantum gravity may require a revision of quantum theory itself in a way that removes the a priori use of continuum numbers in its mathematical formalism." Au writes 161, "One can see how a discrete theory could reduce to a continuum one in the large scale limit, but to shed light on a discrete theory while working from the perspective of a continuum one seems difficult to achieve." And Sorkin writes 171, "The dynamical principles learned from quantum mechanics just seem to be incompatible with the idea that gravity is described by a metric field on a continuous manifold." Thus, we are motivated to search for QG using pregeometry based in denumerable sets.

2. The Model We are guaranteed that a uniform space U may be constructed over any denumerable set X by introducing a group structure and the discrete topology over X. And. the discrete uniform space Dx induces the discrete topology over X while its entourages provide a conventional, but non-metric, definition of a ball centered on x E X. Thus. Dx induces the topology required for U while providing a pregeometric definition of macroscopic spacetime neighborhoods. Given that the introduction of a group structure over X underlying Dx provides a uniformity base U, for U. we have the means to define microscopic spacetime neighborhoods independently of. but consistently with. macroscopic spacetime neighborhoods. Specifically 181, for x and y elements of X. a symmetric entourage V is a subset ofX x X such that for each (x, y) E V, (y, x) is also an element of V. Dx is the collection of all symmetric entourages. For (x, y) E V the distance between x andy is said to be

less than V . The ball with center x and radius V is {y E X

1 (x, y)

E

V) and is denoted

B(x. V). A neighborhood of x in the topology induced by Dx is Int B(x.V). so all possible balls about each x E X are established. This is precisely in accord with the conventional notion of locality, i.e., open balls about elements of the spacetime manifold. Therefore, B(x. V) is a perfect pregeometric definition of a macroscopic spacetime neighborhood of x for denumerable X (cf. Sorkin's finitary topological spaces [9]). In section 4, we will show that this definition of macroscopic spacetime neighborhoods accommodates the topological priority of causal chains over metric balls per Finkelstein [lo]. To show that the introduction of a group structure G over X underlying 4( allows for the construct of U, we construct its uniformity base Us via neighborhoods of the identity e ofG in the following fashion [l 11. The entourage A, of U is ((x, y) E X x

PREGEOMETRY VIA UNIFORM SPACES

X

I

479

xym' E 2) where 2 is a neighborhood of e in the topology over X. When X is

denumerable of order N, ((w, y) E X x X entourages A, (x

E

X such that x

1

w # y) is partitioned equally into the

# e) for the N - 1, order-two neighborhoods of e, i-e.,

I

Ax is generated by {e. x). The entourages A, and C / {(x, x) x E X) constitute a base U, for U. Entourages generated by larger neighborhoods of e are given by members of U,,i.e., {e, x, y) generates A, U A,, etc. While for some group structures all members of U, are elements of Dx, e.g. the Klein 4-group [12], this is not true in general. In fact, A, E Dx # x V X such that x = x-I. This, since for (y, z) E A, such that y z, yz-' = x and therefore, zy-' = x" = x j (z, y) E A,. For the base members A, and A,, such that x = y-l, we have A;' = Ay where

A-'

=

((w, z)q

(z, w) E A). This, since for (w, z) E A, such that w + z, wz-'

=

x and

therefore, zw-'= x-l = y =3(z, w) E Ay We may now construct the largest element of Dx via multiplication of the members of Us. With A a subset of any entourage (uniquely and axiomatically), we have in general for entourages A and B that A C A B and B C AB where AB / {(x, z)

1

(x, y) E

A and (y, z) E B). Next. consider {(& y), (y, z)l (x, y) E % and (y, z) E A, with x Z y and y # z). In addition to c these account exhaustively for the elements of A, and A,. For any such pair (x, y) and (y, z), (x, z) E A:A, by definition and (x, z) E As,, since sw = ( ~ ~ - ' ) ( ~ z= " )x i ' . The N pairs (x.z) with C account exhaustively for the elements of kW and. excepting the impact of C on AsAy, the N pairs (x. z) account exhaustively for the elements of &A,. Again, the impact of C on Ask, is to render A, c A,A, and A, c A,Aw. Therefore, A,Aw = A, U A, V A., So, ifG is cyclic with generatorx, A '-: = Ax U Ay U ... U A, where y = 2 and z = xN-'. Ax U Ay U ... U A, is of course the largest element,V of Dx. [This is of particular interest, since the cyclic group structure Z,.. exists for all N E N and is the unique group structure for N prime.] IfG is not cyclic, one may produce V,via -1 A, u U A, U AwA, U ... U A,A, for x = x-', ..., y = 6', w = z , ...,s = v-I, since &A, = A, v 4 when w = z-'. We are also guaranteed to produce V, via some variation of A,A,...A,where {x, y, ..., z ) = X, according to G. It should also be noted that, as implied supra, the entourage A, U A, V ... U A, of U is generated by the entire set X

...

3. Consequences Should we define microscopic spacetime neighborhoods with the members of U, analogously to macroscopic spacetime neighborhoods per the symmetric entourages of D,, we note the following interesting consequences.


1. A, = A;' for s = w-',so when s f w the distance between elements of A, is non-separable from that ofA,,,, lest we compromise the symmetry of our pregeometric notion of distance. For s = w, i.e., S = s-l, A, E DX and our microscopic spacetime structure accommodates locality. Thus, the degree to which our spacetime is to accommodate quantum non-separability is determined by the choice ofG. 2. The choice of G over X underlying Dx is all that is needed to produce the microscopic spacetime structure embedded in the macroscopic spacetime structure. 3. The members of the base Us ofthe microscopic spacetime structure U may be combined via entourage multiplication to yield the largest element V,, of the macroscopic spacetime structure Dx. Complementing this, , V is equivalent to the entourage of U generated by the entire set X. Thus, a robust pregeometric correspondance between the microscopic spacetime structure and the macroscopic spacetime structure is provided. 4.

Nexus to Physics

First, we show how a pseudometric induced by 4( yields the geodetic structure of M4. We borrow from a proof ofthe following theorem [13]:

...

For every sequence Vo, V1, of members of a uniformity on a set X, where Vo = X x X and (vi+,13 c Vi for i = 1.2, ..., there exists a pseudometric > on the set X such that for every i 2 1

( ( X ~ Y )p~ ( x , ~ ) < ( l ~ )c~ vi I c((x,Y)

I

p &y)~(lf2)'1-

To find >(x, y), consider all sequences of elements ofX beginning with x and ending with y. For each ad~acentpair (h, xMl) in any given sequence, find the smallest member of (Vi) containing that pair. [The smallest Vi will have the largest i, since (vi+d3 C Vim] Suppose V, is that smallest member and let the distance between x, and x,,+~be (1/2)m. Summing for all adjacent pairs in a given sequence yields a distance between x and y for that particular sequence. According to the theorem, >(x,y) is the smallest distance obtained via the sequences. While this pseudometric is Euclidean rather than Minkowskian, it may be used to define geodesics €or either space, since their affine structures are equivalent. Thus, we define a geodesic between x and y to be that sequence yielding >(x, y). Since some sequences might contain 'distant' adjacent pairs, our definition is suitable only for M4.[TO consider curved spacetimes, we would have to restrict our attention to sequences harboring only 'local' adjacent pairs.] The finest resolution would result when considering sequences Vo, V1, of maximal length. The construct ofmaximal sequences is possible with X denumerable and finite. Of course, this immediately suggests a pregeometric foundation for the path integral formulation of QM. And, should one consider various sequences of entourages satisfying (Vi+1)3 C Vb SO that the combinatorics of both element and entourage sequences are considered, then a pregeometric foundation for quantum field theory is also

...

PREGEOMETRY VIA UNIFORM SPACES

481

intimated. Since each sequence of entourages produces a pseudometric, a combinatoric formalism over entourage sequences is analogous to the path integral formulation of quantum cosmology per Hartle and Hawking [14]. The transition from pregeometry to classical spacetime dynamics might be initiated by constructing an affine definition of 4-momentum Let a worldline be

5.

defined by a sequence of elements of X. Then, 4-velocity i? is defined by adjacent pairs in the sequence with direction specified via ascending order ofthe sequence. Thus, for a particle of mass m we have j j l m? . To accommodate curved spacetimes the definition of >(x,y) would have to be restricted to sequences constructed of 'local' adjacent pairings. Speculatively, an equivalence relation might be used to partition X into cells providing this restriction. These local affinely M4 frames would then be pieced together so that the spacetime metric is consistent with the 4-momentum distribution a la Einstein's equations. Since this demands a relationship between affine and metric structures, a reasonable axiom is that of Riemannian geometry as noted by Bergliaffa et al. [15], i.e., the covariant derivative annihilates the spacetime metric. We have

V i (&0'@ O k ) = & * i O~J f yields gsw,i - g .

r :,- gsk r , k

j ) ~ - & k :lr 0'

Zj

Zl

=

d and

ae

0' @ m S = 0 w h i c h

= 0-

As an example, consider coordinates. Let

@ a -kg j k r :

s2with the standard polar (8)and azimuthal (p)

d z2 = . The affine structure yields V

3~

1

Zl

=

r {I

= 0 and the other Christoffel symbols can be computed by using parallel transport and

rk(viz,)= r:

,where r 1 = d 6 a n d r 2 = d p . w e k n o w

I E2,sog12=g2,

= 0. Combined with the annihilation condition, we have gll,]= 0 and glI2= 0.Thus. gll is a constant. The annihilation condition also gives gll(cose sine) = g22cot@,or gz2= gl Isin2@, thus completing our example.

Acknowledgement This work was funded in part by a grant from AmerGen Energy CompanyIThree Mile Island Unit 1.

References J. Demaret, M. Heller, and D. Lambert, Foutd Sci. 2, 137 (1997). W.M. Stuckey, "Pregeometry and the Trans-Temporal Object," to appear in Snrciies on the structure of tittle: Frollt physics ro psycho(patho)logy, R Buccheri, V . Di Ge& and M. Saniga, eds., (Kluwer, Dordrecht, 2001). 3. S. Weinberg, Sci Amner. 281.72 (1999). 4. Ibid I. 5. J. Butterfield and C.J. Isharn, "Spacetimeand the Philosophical Challenge of

482


Quantum Gravity," grqcf9903072 (1999). 6. G.K. Au. 'The Quest for Quantum Gravity," gryc/9506001 (1995). 7. R Sorkin in i7w Creation of Ideas in Physics, J. Leplin, ed, (Kluwer, Dordrecht, 1995) 167. 8. R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989). 9. R. Sorkin, Inr. J. Theor. Phys. 30,923 (1991). 10. D. Finkelstein. Phys. Rev. 184. 1261 (1969). 1I. R. Geroch, Mathematical Physics (Univ of Chicago Press, Chicago, 1985). 12. W.M. Stuckey, Phys. Essays 12,414 (1999). 13. lbid & 14. S.W. Hawking in 300 Years of Gravitation, S.W. Hawking and W. Israel, eds.. (Cambridge University Press. Cambridge. 1987) 631. 15. S.E.P. Bergliaffa, G.E. Romero, and H. Vucetich, Int. J. Theor. Phys. 37, 2281 (1998).

A ZPF-MEDIATED COSMOLOGICAL ORIGIN OF ELECTRON INERTIA M. IBISON Institute for Advanced Studies at Austin 4030 Braker Lane West, Suife300 Austin, TX78759, USA

Abstract Support is found for a fundamental role for the electromagnetic zero-point-field (ZPF) in the origin of inertia. Simply by requiring that that a universal noise field be selfconsistent in the presence of the lightest charge, it is shown that this field must be the ZPF, and that the mass of that charge must be close to 10"' kg. The ZPF functions as homeostatic regulator, with the electron mass decided by cosmological quantities. The calculation validates Dirac's second Large Number hypothesis.

1.

Introduction

Several speakers at this conference have been pioneers championing the cause of an electromagnetic zero-point-field (EM ZPF) origin for inertia. Notable amongst these have been Haisch, Rueda and Puthoff [l-111. Currently the implementations are classical, with a 'classicized' ZPF as conceived within the program of Stochastic Electrodynamics (SED), (see Kalitsin [12], Braffort [13] and Marshall [I41 for the original works, and Boyer [15] for a review of this field). Epistemologically, a common theme of their work is that the ZPF is the cause of resistance to acceleration. In mathematical practice though, the end result is an inertial mass-energy that attributable to the ZPF. Brwddly, the ZPF is seen as an external, energizing influence for a local degree of freedom, which, classically, is the co-ordinate of the particle whose mass we wish to explain. Thus the program has - in part - some of the flavor of Mach, because the ZPF provides a 'background' against which the acceleration can be measured. The particle, once energized, is conceived as having attributable energy, and therefore inertia. Although Haisch et a1 [I 11 in particular have made a distinction between the inertial and energetic aspects of matter, this distinction appears to be largely epistemological; since any 'localized' packet of energy is found to resist acceleration, it is sufficient to explain within the context of this program how EM ZPF energy can become localized. Even so, within this program. there are two quite different possible implementations distinguished by different degrees of non-locality for the origin of mass. To date. 483 R.L. Amoroso et a1 (eds.),Gravitation and Cosmology: From the Hubble Rudius to the Platzck Scale, 483-490. O 2002 Kluwer Acndemic Publishers. Printed in the Netherlurzds.

484

M. IBISON

despite the active role of the ZPF, current implementations result in a value for the inertial mass that is intrinsic to the particle in question. In contrast, the purpose ofthis paper is to argue for an alternative implementation, wherein the value of the inertial mass is determined entirely by external - cosmological - factors. For this reason the approach described here is much closer to the original conception of Mach (ca 1883) than previous ZPF-as-background approach. In the following is given contrasting descriptions of the intrinsic and extrinsic approaches to ZPF-originated inertia. These are followed by a calculation supporting the latter, wherein one of Dirac's large number hypotheses [16,17] is derived and interpreted as evidence of a cosmological ZPF-origin of the inertial mass of the electron.

2.

Role Of The ZPF I n A Model Of Locally Determined Inertial Mass

In a locally determined, ZPF-originated, model of inertia, there exists a local dynamical degree of freedom, such as an oscillator amplitude [I], or a resonator excitation level. This co-ordinate is conceived initially as quiescent, and having no intrinsic energy. Subsequently, if the ZPF is switched on, then the oscillator or resonator is energized, and the ZPF-originated energy that is now associated with the oscillator or resonator can be regarded as the 'rest' mass. In some work, the local properties ultimately deciding the mass of the charged particle enter as a Fourier form factor governing the spectral response to the ZPF [3,4,5]. This has the advantage of leaving open whether the response is due to the energetic resonance of an oscillator or geometric structure. But in either case it is taken to be a local property. The end result is the same in that not only is there resistance to acceleration, but there is also a localized energy density that can be associated with the particle in question. Clearly, in this approach, the object has zero true rest mass, whilst the ZPF-energized mass may nonetheless be statistically at rest due to the homogeneity and isotropy of the ZPF. Also, thanks to the peculiar k-space distribution of the ZPF - the ZPF retains the same homogeneous energy density in every inertial frame - it follows that with suitably chosen dynamics it should be possible to make the rest mass a fully invariant scalar. These original ideas have stimulated new thinking about the origin of inertia and brought forth some encouraging responses [18,19,20]. However, it is premature to claim that the origin of inertia has been found in the ZPF, because there are some unresolved and unsatisfactory aspects of the current approach: 1. The particle oscillator or resonator must contain electrical charges capable of interaction with the ZPF, so the model cannot describe a neutral elementary particle such as a massive neutrino, for example.

2. The electron also poses a problem unless it is admitted that it has some - as yet unobserved - structure.

3. The values of the intrinsic mass (of the electron, muon, and tau say), are not predicted, but must be inserted by hand. The final mass - the energy stored in the oscillator or resonator - is decided by intrinsic qualities i-e.: locally, wherein combinations of charge-field coupling, geometric form factor, and spectral form factor, must be chosen to give the desired final mass. (The existence of these energy-storing

ORIGIN OF ELECTRON INERTIA

485

'degrees of freedom' rests upon the presumed existence of a structure for the particle in question - points 1 and 2 - and neither this structure nor these coupling and form factors are explained.) 4. There is no clear path of development for the theory by which it can unite the inertial and gravitational aspects of mass. There ir the hope that it also has something to do with the EM ZPF, as first suggested by Sakharov [21,22]. But to date there have been no successful implementations of a ZPF origin for gravity. Perhaps the most attractive feature of the current thinking along these lines is that the proposed energetic source by which means both gravity and inertia may perhaps be united - the ZPF - is a 'ready-made', omnipresent, influence. In the next section is investigated a different implementation which retains this foundational feature, but which overcomes some of the above enumerated difficulties.

3.

A Model For Non-Locally Determined Inertial Mass

By non-local model is here meant that inertia is conceived not as an intrinsic, unitary property, but as arising out of a non-local rnutual interaction. Just as the mutual interaction energy of charges, current elements, and (gravitating) masses cannot be assigned to either partner in the interaction, so - it is suggested - inertial mass-energy cannot be ascribed to a single particular particle, but results from the multiple mutual (pair-wise) interactions with distant partners. Like the foregoing examples, the Casimir and van-der-Waals energies are also mutual, yet these are different in that they exist only by virtue of the ZPF. Though an apparently intrinsic Casimir energy does exist for a conducting curved surface embedded in the ZPF [23], one may regard this energy as arising out of the mutual interaction of local elements of the curved structure, just as in a Casimir cavity. To date, no one has identified a mutual yet distant interaction energy of electromagnetic origin that can explain inertia. And this is the reason why the ZPFinertia advocates have concentrated on local, 'unitary' qualities that might cooperate to localize ZPF energy. A detailed description of the distant interaction believed to be responsible will be given in a future document. Briefly, a consequence of that work is that, like the local models, the positional / motional particle degree of freedom may be regarded as 'energized' by the ZPF. But unlike local models, the energy of interaction turns out to be mutual, involving all distant particles. A good metaphor is provided by van-der-Waals binding energy, except that the rate of radial fall-off precludes it from candidature. For now, the following calculation is presented as evidence to support the claim that inertial mass is a non-local energy - with the ZPF as its rrzeans.

4.

Derivation Of The Electron Mass

In the following calculation it will be assumed that associated in some way with a charged particle is a resistance to acceleration equal to a final renormalized (not bare) inertial mass me.It will be assumed that underlying this mass is a particle in micromotion (commonly, but not very accurately termed zitterbewegung). We also need to

486

M. LBISON

assume that an EM noise field and the micro motion of the charge are consonant in that the 'in' fields impacting the source and the 'out' fields leaving the source have the same statistics. In other words, it will be assumed that the EM noise field has attained a selfconsistent state in the presence of the particle micro-motion. (A causal flow is not implied here: the particle's motion does not cause the field nor does the field cause the particle's motion. Rather, the field and motion are to be viewed as mutually consistent) So far this sounds rather like the SED program that results in a ZPF-energized, but nonetheless intrinsic-valued, inertia. However, the particle employed here is deemed to have no intrinsic structure, and therefore cannot have an intrinsic-valued inertia - with or without the ZPF. Very broadly, this can be concluded simply from the absence of any length scale that could conceivably be associated with a mass. More specifically, it can be shown that a massless classical point charge dropped into the ZPF leaves the electromagnetic spectrum and energy density unchanged from that of the charge and ZPF considered apart from each other - unless the charge is permitted to interact with other charges. It must be admitted at the outset that the particle that will be singled out by this calculation is the electron. This is because it is the lightest charged particle. As a consequence it gives the largest acceleration per unit field, and therefore the largest out field per unit in field. It follows that, provided the in and out fields are universally selfconsistent, the fields must be maintained predominantly by the electron. Therefore, in the following calculation, it will be assumed that N loR0electrons, sprinkled approximately uniformly throughout the visible universe, cooperate in the maintenance of an EM noise field. Further, it can reasonably be assumed that the micro-motion will have a coherence length somewhere between the classical electron radius and the Compton wavelength. From this it follows that, although widely varying, the local environment of electrons is to a good approximation of no consequence to the presumed micro-motion, since the coherence length of the latter is relatively so shoa In this paper the self-consistency calculation will be simplified by assuming that the micro-motion is non-relativistic. For this to be true for all electrons from the perspective of our own earthly reference frame, we must necessarily consider only a static universe - i.e. without expansion. This is because the electrons near the Hubble radius will turn out to dominate the self-consistent field calculation. (A more complicated calculation admitting expansion gives a very similar result, as discussed below.) With this restriction the electromagnetic noise-induced acceleration is approximately (c=4m0=1)

-

a ='in

(elme)

(1)

for which the outgoing radiation, in the far-field, has electric field

The corresponding 3D orientation averaged energy density from both the magnetic and electric fields, as viewed in the frame in which the particle's expected position is always at the origin, is

OFUGLN OF ELECTRON INERTIA

487

In the presumed static cosmology there are N sources approximately uniformly distributed throughout the universe of static radius R. i-e.. p(r)d3r= ( 3 ~ / 4 r r R ' ) & r . The expected energy density from all the sources is therefore (&NP)

= Jd3rp(r)(&, (r, f)) =

3N e2a2

R

4r

ldrr2 o

IVe2a2

=.

2nR2

(4)

By contrast, the energy density of the in field at the particle in question can be expressed in terms of the acceleration using Eq. (I):

For self-consistency the energy density of the in field must, at all locations, equal the energy density due to all the out fields:

Consequently, one obtains the Dirac large number hypothesis [I61

me= & ? e 2 / ~ .

(7)

With N =loE0, and R set to the Hubble radius of 1 0 ~ ~ c m this , computes to kg, i.e. 40% of the observed value of the electron mass - well within the 0.36 x tolerance set by the uncertainty in (which expressed as a factor is between about 0.5 and 2).

5.

Discussion

The above calculation establishes a linear relations p between the in and out fields. It follows directly that it does not matter how strong or weak is the noise field: the electron mass given by Eq. (7) would have the same value whatever. Another consequence of the linearity is that the computed electron mass is also insensitive to the energy spectrum of the ZPF. (A consequence of the fact that the charge-field scattering is elastic.) A relativistically correct version of this calculation performed in a flat expanding universe turns out to give, apart from a numerical coefficient of order unity, the same Dirac relation, and therefore the same electron mass as a function of cosmological constants, with the former remaining independent of time. Specifically, taking into account the special role played by the ZPF in maintaining homeostasis, the second Dirac hypothesis [I61- that N1'*e2!Ris constant - can be validated. Dirac's suggestions seem to have been rejected mostly on the basis of his first relation, which predicts a

time-dependent gravitational constant, and is considered to be incompatible with observation; see [24] and [25] for reviews. Although not directly impacting the validity of the second relation under investigation here, it is nonetheless interesting that similar ZPF-mediated arguments have led Puthoff [25,27] to claim a time-independence for the first relation. It is hoped to reproduce elsewhere the detailed calculations and qualifying cosmologies discovered to maintain constancy of the second relation. In those calculations it turns out that appeal must be made to velocity-invariant statistics of the EM noise field. That is, the self-consistent field must be, at least at the level of expectations of quadratic field operators, the electromagnetic zero-point field as furnished by second-quantized Maxwell (henceforth the ZPF). The reader may be alert to the fact that a proper relativistic treatment necessitates the use of the Lorentz-Dirac equation with non-linear radiation reaction terms, or - in the quantum domain - the corresponding Heisenberg equation of motion [26].This, and related issues concerning the bandwidth of the self-consistent field and the origin of time asymmetry, require a much more detailed treatment, and will also be addressed elsewhere. A concern expressed by some is that the cosmological distribution of matter is such that any alleged derivation of particle constants from cosmology will suffer from an unacceptable level of frame dependent, or perhaps time dependent, variability. A rough estimate of the variability in the predicted value of mass is to entertain fluctuations in N, which are likely to be of order f 1 2 . This gives rise to corresponding fluctuations in N'I2 of order unity, i.e. one part in lo4'. Therefore, at least by this mechanism, cosmological variability does not lead to a detectable variability in the mass. It must be emphasized that nowhere in the above was inertial mass 'explained'. Rather, this calculation tells us only that if the fields are to be self-consistent, the electron mass could not have any value other than the one it is observed to have, given the cosmological numbers. The calculation does not explain the mechanism of the mass. Despite these caveats, the success of the calculation provides support for the novel emphasis placed on the ZPF by Haisch and others in their work on inertial mass. This work continues that effort, but with a different role for the ZPF. Here, the ZPF is the means by which homeostasis is maintained; it is the means by which the electrons throughout the universe come into electromagnetic equilibrium with each other, whereby the electron mass attains universal consistency.

6.

Cosmological Origin Of Length-Scale

In natural units where e = 1, mass has units of (length)-'; the classical length corresponding to the electron mass (the classical electron radius) is, in S.I. units, e?(4n~~rn~~), which is about 3 x 10'15 m. In the previous calculation based upon , cosmological self-consistency this length is found from Eq. (7) to be R / ( ~ N ) ' ' ~where both R and N are cosmological constants. Despite all the talk of electromagnetism and ZPF-induced micro-motion, the final result constructs a very small length out of cosmological constants. If the cosmic somehow determines this length, as implied in this work, then there should be a direct cosmological interpretation for this very small length, without any reference to electrons. This is the focus ofthe following discussion.

489

ORIGIN OF ELECTRON INERTIA

Notice that the integral in Eq. (4) may be regarded as computing the expectation of

r-2over the Hubble volume. For a homogeneous distribution,
-R" for any n, so nearly the whole contribution to the integral comes from matter at the Hubble radius. In other words, for the purposes of computing a self-consistent field, and to a very good approximation, all the matter in the universe appears to be at the Hubble radius. It is as if all matter is projected onto the Hubble sphere, creating the appearance of a surface whereupon 2 w 2is a roximately the mean density at the Hubble radius of N/(~~cR~), nearest neighbor distance between the points. That is. 2R/N' IS the mean nearest neighbor distance on the Hubble sphere between the points that are the radial projections of all the electrons in the universe. To within a factor of order unity this is the previously computed mass-length of the electron. and therefore this distance must be the corresponding cosmological entity. and. allegedly. the origin ofthat length. It is clear from the above that a sufficiently large telescope could, in principle, be used to resolve the individual electrons in the universe if its probing radiation had a wavelength shorter than the mass-length. This means that the universe of electrons must be at least partly transparent to ZPF 'radiation' at this and shorter wavelengths. Ignoring for now the possibility of future collapse, it follows that the electrons cannot maintain a universally self-consistent noise field beyond the mass-frequency. (In a more realistic cosmology it is to be hoped that this quantity will look like a frame independent cutoff.) In other words, the mass-length is also the critical wavelength at which the universe of electrons starts to become transparent. Therefore, based upon the considerations of this and the previous section combined. one may conclude that the locally observed mass-length is authored cosmically (as RIN"') and broadcast by the ZPF (as a cutoff at that wavelength). No attempt has been made to investigate, from this cosmological perspective, the relationship between the Compton wavelength and the mass length. Since their ratio is the fine structure constant, a search for a cosmic relation is therefore equivalent to a search in this context, for a geometric interpretation of a? It is interesting that Wyler [29] (see [30] for a review in English) found an expression for a involving the ratio of projections of volume elements - especially since the mass-length calculation above also involves a projection. Obviously, the dimensionality here is wrong because we have ignored universal expansion; if the two paths do converge there remains much more work to be done.

P

Acknowledgements The author gratefully acknowledges the kind encouragement and the many productive conversations with H. Puthoff and S. Little. Thanks also to the organizers of the conference for putting together an intense and inspiring program of interesting physics.

M. IBISON

References Haisch, B., Rueda, A, and Puthoff. H. E. (1994) Inertia as a zero-point-field Lorentz force. Phys. Rev. A 49,678-694Haisch, B., Rueda, A., and Puthoff, H. E. (1997) Physics of the zero-point-field: Implications for inertia, gravitation and mass, Speculations in Science & Technology 20,99-114. Haisch, B., Rueda, A., and Puthoff, H E. (1998) Advances in the proposed electromagnetic zero-point field theory of inertia, proc. 34th AIAA/ASME/SAE/ASEEAIAA Joinr Propd.~ionConference. AIAA paper 98-3 143. Rueda, A., and Haisch, B., (1998) Contribution to inertial mass by reaction of the vacuum to accelerated motion. Found. Phys. 28, 1057-1 108. Rueda, A., and Haisch, B., (1998) Inertia as reaction ofthe vacuum to accelemted motion, Phys. Letters A 240, 115-126. Haisch B. and Rueda, A. (1998) The zero-point field and inertia, in G. Hunter, S. Jeffers & J.-P. Vigier (eds.) Causaliq and Localig in Mode171 Ph?.sics, Kluwer Academic Publishers, 171-178. Rueda, A. and Haisch B. (1998) Electromagnetic vacuum and inertial mass, in G. Hunter, S. Jeffers & J.-P. Vigier (eds.) Causalit?.and Localit?. in Modern Ph?.sics, Kluwer Academic Publishers, 179-186. Haisch, B.. and Rueda, A, 11999) Progress in establishing a connection between the electromagnetic zero-point field and inertia, in M. S- El-Genk (ed.) Proc. Space Technology and Applications Inter-rmtiormlForurn (STAIF-1999), AIP Conf. Publication 458, 988-994Haisch, B., and Rueda, A. (1999) Inertial mass viewed as reaction of the vacuum to accelerated motion, Proc. NASA Breakthrough Propulsion Phyics Workshop, NASAJCP-1999-208694, pp. 65. Haisch. B.. and Rueda. A.. (2000) Toward an interstellar mission: zeroing in on the zero-point-field inertia resonance, Proc. Space Technology a t d Applications Intertmtiorml Forztrn (STAIF-2000), AIP Conf. Publication 504, 1047- 1054. Haisch, B., Rueda, A., and Dobyns, Y. (2000) Inertial mass and the quantum vacuum fields, Annalen der Physik, in pressKalitisin, N. S. (1953) JETP 25, pp. 407. Braffort, P., Spighel, M., and Tzara, C. (1954) Acad. Sci Paris, Comptes Rendus 2?@, 157. Marshall, T. W. (1963) Proc. R. Soc. London, Ser. A 275, pp. 475. Boyer, T. H. (1980) A brief survey of Stochastic Electrodynamics. in A. 0. Barut (ed.), Foundarions of ~ Quantum Ekctro&narmics. Plenum Press, New Yolk, 49-63. Radiation T h e o and Dirac, P. A. M. (1979) The Large numbers hypothesis and the Einstein theory of gravitation, Proc.. R. Soc. Lordon, Ser. A 365, 1930. Dirac, P. A. M. (1938) Proc. R. Soc. Lordon, Ser. A 166, pp. 199. Davies, P. C . W. (1992) Mach's Principle, Guardian Newspaper, 22"' September, "http://www.physicsadelaide~du.au/itp/1994/940~2Mach.html". Jammer, M. (1999) Concepts of rmss in Contemporar?.Physics arul Philosophy, Princeton Univasity Press, Princeton. Matthews, R. (1994) Inertia: Does empty space put up the resistance? Science 263, 612-613. Sakhmv. A. D. (1968) Vacuum fluctuations in curved space and the theory of gravitation. Sov. Phys. Doklae 12 1040-1041. Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973) Gravitation, Freeman, San Francisco. Candelas, P. (1982) Vacuum energy in the presence of dielectric and conducting surfaces, Anrmls of Physics 143,241-295. Alpher, R. A. (1973) Large numbers, Cosmology, and Gamow, American Scientist 61.51-58. Harrison, E. R. (1972) The cosmic numbers, Phpics Today 25,30-34. Puthoff, H.E. (1989) Source of vacuum electromagnetic zero-point energy, Phys. Rev. A 40, 48574862. Puthoff, H.E. (1991) Reply to '%omment on 'Source of vacuum electromagnetic zero-point energy"', P k s . Rev. A 44, 3385-3386. Sharp, D. H. (1980) Radiation reaction in non-relativistic quantum theory, in A. 0. Barut (ed.), Fourldations $Radiation Theor? and Quantum Electro&narmics, Plenum Press, New York, 127-141. Wyler, A. (1969) Theorie de la Relativite - L'espace symetrique du groupe des equations de Maxwell, Acad. Sci Paris, Cornptes Rerdus 269A. 743-745. GBL (1971) A mathematician's version of the fine-structure constant. Physics Today 24, 17-19.

VACUUM RADIATION, ENTROPY AND THE ARROW OF TIME

JEAN E. BURNS ConsciousnessResearch 1525 - 153rd Avenue San Leandro, CA 94578

Abstract The root mean square perturbations on particles produced by vacuum radiation must be limited by the uncertainty principle. i.e.. < 6 X2 >" < 6 P, >'I2 = h / 2 , where < 6 X2 >" and < 6 p: >" are the root mean square values of drift in spatial and momentum c o d i a t e s . The value < 6 x2 >1/2 = ( A t / ,) I N , where rn is the mass of the particle, can be obtained both from classical SED calculation and the stochastic interpretation of quantum mechanics. Substituting the latter result into the uncertainty principle yields a fractional change in momentum coordinate, < 6 P 2 >1/2 / p , wherep is X the total momentum, equal to 2-3'2( h /Et)'12 , where E is the kinetlc energy. It is shown that when an initial change < fip;>" is amplified by the lever arm of a molecular 2 112 interaction, < 6 px> / p 1 in only a few collision times. Therefore the momentum distribution of a collection of interacting particles is randomized in that time, and the action of vacuum radiation on matter can account for entropy increase in thermodynamic systems. The interaction of vacuum radiation with matter is time-reversible. Therefore whether entropy increase in thermodynamic systems is ultimately associated with an mow oftime depends on whether vacuum photons are created in a time-reversible or irreversible process. Either scenario appears to be consistent with quantum mechanics.

>

1.

Introduction

In this paper we will see that entropy increase in thermodynamic systems can be accounted for by vacuum radiation, and then discuss the relationship between vacuum radiation and the arrow of time. The problem in accounting for entropy increase has always been that dynamical interactions which occur at the molecular level are time-reversible, but thermodynamic processes associated with entropy increase, such as diffusion and heat flow, only proceed in one direction as time increases. In the past it was often held that entropy increase is only a 491 R.L. Amoroso et a1 (eds.),Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 491-498. O 2001 Khwer Academic Pztblishers. Printed in the Netherlands.

J. E. BURNS macroscopic phenomenon, which somehow appears when a coarse-grain average is taken of microscopic processes. But no averaging of time-reversible processes has ever been shown to account for phenomena which are not time-reversible.[l] Nowadays entropy increase is often viewed as coming from effects of the environment, such as walls of a container or thermal radiation, not taken into account in the description of a system. Unruh and Zurek [2] have given examples in which entropy increase is produced in this way. However, the second law of thermodynamics s-es that entropy increase must also occur in an isolated system. So if we are to hold that entropy increase is produced by a physical process at the microscopic leveL we must also understand how it can be produced in this way in an isolated system. Any explanation must satisfy the basic assumptions of statistical mechanics. Classical statistical mechanics has only one assumption: At equilibriunl it is equallyprobable that the systenr will be in any (classical) state which satisfies the thennodynan~icconstraints. Quantum statistical mechanics has two basic assumptions. The first is essentially the same as for classical, except that states are now counted quantum mechanically. Thus: At equilibriunl it is equallyprobable that the system will be in any (quantum) state which satisfies the them~odynanricconstraints. The second assumption of quantum statistical mechanics is: At equilibriunl the relative phases ofthe eigenvectors describing the systent are random. Once these fundamental assumptions are made, one can then define entropy as klog(number of states), where k is Boltzmann's constant. It is always also assumed that the number of molecules, and therefore the number of states, is extremely large. One can then develop the physics of the microcdnonical ensemble in the usual way, by requiring that different parts of an isolated system be in equilibrium with each other at temperature T. By placing the system in equilibrium with a heat bath one can then derive the physics of the canonical ensemble, and so forth. [3] In order to talk about entropy, we must specify the context in which we refer to the ensemble of all possible states. In the coarse-grain view we would use an ensemble of states with all possible initial conditions, and then argue that because the number of states is very large, the only states we are apt to see are the most probable ones (and not ones in which all molecules are clustered in a corner of a box, for instance). Thus equilibrium merely refers to the most probable state in a large collection of systems. In the view in which entropy is produced at the microscopic level, we start with a single system which has

VACUUM RADIATION, ENTROPY AND TIME

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specified initial conditions (classical or quantum mechanical) and look for a process which produces many random perturbations and by this means places the individual system into its most probable state. In order to inquire about an isolated system, let us consider the system to be comprised of not only the interacting molecules under consideration, but also the walls of their container. any heat bath surrounding them and all the thermal radiation which might affect them It would seem that we have taken into account all interactions which could possibly affect the system What then could serve as an "environment" which would account for entropy increase? Let us ask if an interaction could take place within the Limits of the uncertainty principle which would affect molecules randomly? If this interaction could randomize the momentum of each molecule and (when quantum mechanical description is needed) randomize the quantum phases of the eigenvectors describing the system, this process would then account for entropy increase. Yet the interaction itself could not be detected in measurements of the system Vacuum radiation acts at the limits of the uncertainty principle. and clearly it would perturb molecules in a random way. But are these effects Large enough? A mermodynamic system goes to equilibrium in a few molecular collision times. [3] So in order to account for entropy increase. vacuum radiation would have to randomize the momentum of a system and the quantum phases of its eigenvectors in that short time. Let us first take up the question of momentum.

2.

Randomization Of Momentum By Vacuum Radiation

2.1. DRIFT IN SPATIAL COORDINATE It has been shown by Rueda [4] in a classical stochastic electrodynamics (SED) calculation that the coordinate drift produced on a free particle by vacuum radiation can be described by diffusion constant D = h /2m, where n z is the mass of the particle. A quantum mechanical calculation of this effect of vacuum radiation has not been done. However, when only energy and momentum transfer are involved and not anything specifically quantum about the nature of the radiation involved, it is reasonable that an SED calculation will give the same result as a quantum mechanical one.[5.6] Rueda showed that vacuum radiation moves electrons in a random walk at relativistic speeds and that this motion accounts for nearly all of their mass, with step length varying h m the Compton wavelength to the de Broglie wavelength. The radiation acts on hadrons at the quark level and moves the hadrons at sub-relativistic velocity.[4] We note that the stochastic interpretation p ] ofthe Schrijdiger equation, which has no direct connection to vacuum radiation, but attributes a quantum brownian motion to particles, yields the same diffusion constant. In a similar vein, the stochastic action of particles, with the same range of step lengths as above, can be derived directly from the

494

J. E. BURNS

uncertainty principle in the following way. Suppose that we have an ensemble of particles, labeled 1, 2,.... Each is subject to a series ofposition measurements at equal time intervals. Particle 1 is measured with resolution A X I , particle 2 with resolution A x2, and so forth. with A xI > A x2 > .... According to the uncertainty principle, as measurement resolution becomes increasingly fine, particle momentum is increasingly more uncertain, and the path is more erratic. Using this point of view, a particle can be described as following a continuous, non-differentiable path of fractal dimension two, which corresponds to brownian motion. [8] Further analysis shows that the step lengths vary from the Cornpton wavelength to the de Broglie wavelength. [9] The above diffusion constant yields a root mean square spatial drift < t? x2 >I" = (2Dt )" [lo], so

The above result can be confirmed experimentally using a tightly collimated beam of low energy electrons. For instance, if a beam of 100 ev electrons has vJv. = lo-' (where x is the forward direction oftravel), the spread in beam width due to the above process will be larger than the spread due to diffraction in the first 19.5 cm of travel.[ll] This experiment has not presently been done, however.

2.2. RANDOMIZATION OF MOMENTUM

Vacuum radiation acts at the limits of the uncertainty principle, so we write < 6 >'I2 < 6 p: >" = fl / 2, where < 6 p: > is the root mean square shift in

"

momentum component of the particle produced by vacuum radiation. It is then easily found that

where p is the total momentum of the particle and E = p2/2m is the energy. We see that -=SP,> is proportional to f V2,so momentum is conserved as time becomes large. Perturbations in momentum of a particle will change its original value, and when 2 I12 < t?px> / p > 1, momentum has been completely randomized. We wish to know how long this will take. In order to have a concrete example, let us start with air at standard conditions. At the end of one collision time (i.e., the time to travel a mean free path), 2 In < 6 p, > / p = 1.186 x 10-~.[11]However, any change in momentum is multiplied by , where A is the mean free path and r the molecular radius, during the a lever arm A = h!.

VACUUM RADIATION, ENTROPY AND TIME

495

next collision.[l 1] In air at standard conditions A = 1.005 x 104.[111 Therefore, the momentum distribution of the molecules has been randomized in two collision times. 2 112 The product A < p,> / p is proportional to fiT )" / (gp") .[I I] Therefore, momentum is randomized in a few collision t i i s for all gases except those at very high pressures (> 100 atm, or higher if the temperature is substantially more than 300 K). In solids and liquids many particles interact simultaneously, so it is reasonable to suppose that momentum will randomize within a few collision times in these also.[l I]

3. Randomization Of The Phases Of The Eigenvectors In order to fulfill the second fundamental assumption of quantum statistical mechanics. it is necessary to show that vacuum radiation can randomize the relative phases of the eigenvectors descriiing the system within a few collision times. We make no calculation here, but simply show that this is likely to be the case. First, we note that perturbation theory tells us that components of eigenvectors added to a system because of a perturbation are out of phase with the original state vector. [12] Furthermore, because vacuum radiation will produce many small, independent effects, we can see by considering either a coordinate or a momentum representation of the eigenvectors that these effects would affect different eigenvectors differently. So we would expect the relative phases of the eigenvectors to be randomized. The above does not tell us how quickly this randomization would occur. However, Unruh and Zurek [2] have shown in various examples that when an environment perturbs a system, the offdiagonal elements of the density matrix go to zero in a much shorter time scale than effects involving spatial and momentum distributions. Thus it seems likely that vacuum radiation can diagonaliize the density matrix in a shorter tune than it takes to randomize momentum.

4. The Arrow Of Time The dynarnical laws of physics are time reversible, i.e., for any given trajectory described by them, the time reversed trajectory is also a solution of the equations. And in nearly all cases, both the process described by these equations as time moves forward and the process described when time is reversed can be observed to occur. But curiously, there are a few exceptions to this rule. The decay of K-mesons violates CP and therefore (assuming CPT holds) is not time symmetric. Electromagnetic waves emanate from a source out to infinity, but do not converge from infinity to a source. Collapse of the wave function is a one-way process.[l3,14] And as Prigogine and cc-workers have shown, in systems which are so unstable that they cannot be described analytically in an ordinary dynarnical framework, process can go in only one diiectional.[l5] Such processes can be called irreversible, and they are accounted for by saying they are governed by an urrow of time. It is not known

J. E. BURNS

496

what an arrow of time is, what it has to do with the rest of physics, or whether any of the above arrows oftime have anything to do with each other. It has been shown herein that entropy increase in thermodynamic systems is produced by the interaction of vacuum radiation with matter. This interaction is time reversible. However, we can go back a step and ask how vacuum radiation is produced. Whether an arrow oftime is ultimately involved in entropy increase depends on the answer to this question. as we will see. In examining this issue, let us start with a classical (SED) analysis. Puthoff [l6] has shown that if vacuum radiation with its frequency-cubed spectrum once exists, then random interactions with matter in which radiation is absorbed and matter accelerates and reradiates maintain this frequency-cubed spectrum indefinitely. From this perspective, the random nature of the interaction of vacuum radiation with any given particle is caused by the random distribution in position and momentum of other particles the radiation previously interacted with. All interactions are time-reversible, and it is not necessary to invoke an "arrow oftime" to explain entropy increase in thermodynamic systems. In quantum mechanics photons exist in quantized units of energy h v . However, the average energy per photon of vacuum radiation is 1/2 hv For that reason it is commonplace to explain the average energy by supposing that photons spontaneously and causelessly arise out of the vacuum, exist for the time allotted by the uncertainty principle, and then annihilate themselves back into the vacuum. In this scenario information describing the state of the newly created vacuum photon arises from nothing, the photon interacts with matter and mo fies the information describing its state according to this interaction, and this modified information is then destroyed when the photon annihilates itself. The dynamical information which is introduced in the creation of virtual photons is purely random. However, the information which is removed is no longer random (or potentially is not because the virtual photons could have interacted with an ordered system). Thus the beginning and end points are inherently different, and an arrow of time is defined. According to this view, entropy increase is therefore ultimately associated with an arrow of time.[l l ] On the other hand, it would seem that quite different views of the arising and disappearance of photons are possible. The basic equations of QED and quantum field theory do not tell us how vacuum photons (or other virtual particles) arise. And creation and annihilation operators, although they have evocative names, simply describe mappings from one state to another in Hilbert space. the same as any other operators. The idea that vacuum photons arise spontaneously out of the vacuum is basically a pictorial device to account for the average energy per photon of 1/2 h v . Alternatively, one can conceive that, comparably to the classical picture, vacuum photons arise and disappear through constructive and destructive phase interference of a large number of photons traveling in different directions. To be consistent, one would have to view all other virtual particles as also arising and disappearing through constructive and destructive interference of quantum phase, perhaps through interaction with negative energy particles. But the appearance and

.

di

VACUUM W I A T I O N , ENTROPY AND TIME disappearance of virtual particles could perhaps occur in this way. Another possibility is that the seemingly random appearance and disappearance of virtual particles comes about through interactions in the extra dimensions provided by string theory. Ln each of these cases processes would be entirely time-reversible, and no arrow oftime would be involved. We can put this issue another way by asking: 1s the universe a continuous source of random dynamical information, creating virtual particles which can interact with matter and then return some of the previous dynamical information describing this matter to the vacuum? Or does the universe merely transform dynamical information, with virtual particles arising and disappearing through a process such as the above? At present there is no answer to these questions and. given quantum indeterminacy within the limits of the uncertainty principle, there may never be any conclusive answer.

5. Conclusion As vacuum radiation interacts with particles, it exchanges momentum with them. The 2> -1/2 fractional change in momentum of a particle < 6 px / p after one collision time, when multiplied by the lever arm of succeeding molecular interactions, becomes greater than one in only a few collision times. Therefore, particle momentum is randomized during that time, and vacuum radiation can account for entropy increase in thennodynamic systems. Vacuum radiation interacts with matter in a time-reversible process. Therefore, whether entropy increase in thermodynamic systems should be viewed as ultimately connected with an arrow of time depends on whether the arising and disappearance of vacuum photons should be considered as a time-reversible or irreversible process. Either possibility appears to be consistent with quantum mechanics.

References I. 2. 3. 4.

5. 6. 7.

Zeh, R-D. (1989) The Ph?*sicalBasis ofthe Direction of Tiine, Springer-Verlag, New York. Unruh, W.G. and Zurek, W.H. (1989) Reduction ofa wave packet in quantum Brownian motion, Phys. Rev. D 40(4), 1071-1094. Huang. K. (1%3) Statisticul M e c h i c s . Wiley. New York. Rueda, A. (1993) Stochastic electrodynamics with particle structure, P m I: Zero-point induced brownian behavior, Found Phys. Lett. 6(1), 75-108; (1993) Stochastic electrodynamics with particle structure, Part II: Towards a zero-point induced wave behavior. Fourrd. Phy.7. Lett. 6(2). 139-166. Milonni, P.W. (1994) The Quanfurtz Vacuurn: An Introduction to Quanturn Electrodyrmamics, Academic, New York. SED calculations are known to give the same result as quantum mechanical ones for the Casimir effect, van der Waals forces, the shape of the blackbody spectrum, and the Unruh-Davies effect. See Ref. 5. Chebotarev, L.V. (2000) The de Broglie-Bohm-Vigier approach in quantum mechanics, in S. Jeffers, B. Lehnert, N. Abramson, and L. Chebotarev (eds.), Jean-Pierre Vigier and the Stochastic Interpretation of Quanrurn Meclmanics, Apeiron, Montreal, pp. 1-17.

J. E. BURNS Abbott, L.F. and Wise. MB. (1981) Dimension ofa quantum-mechanical path. Am. J. Phys. 49(1). 37-39; Cannata, E and Ferrari, L. (1988) Dimensions of relativistic quantum mechanical paths Am J. P h s . 56(8)),721-725. Sornette, D. (1990) Brownian representation of fractal quantum paths, Ear. J. Phys. 11,334-337. Haken, I-L (1983) Syergetics, Springer-Verlag, New York. Burns, J.E. (1998) Entropy and vacuum radiation, Fomd P ~ . T28(7), . 1191-1207. Peebles, P.J.E. (1992) Qaanturn Mechanics, Princeton University Press, Princeton, NJ. Penrose, R. (1994). SIradows of the Mind. New York: Oxford University Press. pp. 354-359. It should be noted that not all interpretations of quantum mechanics assume there is such a thing as collapse of the wave function. See, e.g., Ref. 7. F'rigogine, I. (1997) From PoincaSs divergences to quantum mechanics with broken time symmetry, &itschrift fur Natur$or.~chung52a 37-47: Petrosky. T. and Rosenberg. M. (1997) Microscopic nonequilibrium structure and dynamical model of entropy flow, Foundations of Ph~sics27(2), 239-259. Puthoff, H.E. (1989) Source of vacuum electromagnetic zero-point energy, Phys. Rev. A 40(9), 4857-4862; (1991) Reply to "Comment on 'Source of vacuum electromagnetic zerepoint energy"', Phys. Rev. A 44(5), 3385-3386.

QUATERNIONS, TORSION AND THE PHYSICAL VACUUM: THEORIES OF M. SACHS AND G. SHIPOV COMPARED DAVID CYGANSKI Worcester Polytechnic Institute Worcester, MA

cygunski @ece.wpi.edu

WILLIAM S. PAGE Daneliuk & Page, Kingston, Ontario

bill-pugel @sy~?~putico.cu

Abstract. Of several developments of unified field theories in the spirit of Einstein's original objective of a fully geometric description of all classical fields as well as quantum mechanics, two are particularly noteworthy. The works of Mendel Sachs and Gennady Shipov stand apart as major life works comprising tens ofpapers, several monographs and decades ofeffort. Direct comparison ofthese theories is hampered however by differences in notation and conceptual view-point. Despite these differences, there are many parallels between the fundamental mathematical structures appearing in each. In this paper we discuss the main tenets of the two approaches and demonstrate that they both give rise to a factofition ofthe invariant interval of general relativity. 1. Introduction The theories reviewed in this paper represent a return to the ideas initiated by Einstein after the developmentofgeneral relativity. After brie y introducing both theories we develop the representations used by each factorization of the general invariant space-time line element. [n his book, General Relativity and Matter [l] Mendel Sachs presents a unified Field theory incorporating gravitation, electromagnetism, nuclear interactions and the inertial properties of matter. In a later book, Quantum Mechanics from General Relativity [2] Sachs extends the formalism ofgeneral relativity in the munner originally envisioned by Einstein to obtain a general theory of matter including those properties of matter that are now usually described by quantum mechanics. To achieve this unification Sachs writes the field equations ofgeneral relativity in a factored form having a similar relationship to the usual field equations ofgeneral relativity that the Dirac equation has to the Klein-Gordon equation in relativistic quantum mechanics. Factoring the field equations involves introducing ageneralizationofRiemann geometry that admits coordinate transformations involving all 16 parameters of the Einstein group rather than the usual LO parameters of the Poincare group. These extra pameters represent spin degrees of freedom. 499 R.L Amoroso et a1 (eds.),Grcniitntinnand Cosmc~logy:From the Hubble M i u s to the Plmck Scnle, 499-506. 63 2002 KlinverAdernic P~Iblishers.P6ttecl in the Netherl~~uis.

500

D. CYGANSKI AND W. S. PAGE

Sachs expresses this in terms ofthe algebra of spinors and quaternions. Applying this same factorization to Maxwell's equations leads to an explanation the of Lamb shift without involving quantum field theory. Exact solution of the field equations corresponding to a ground state of bound particleantiparticle pairs suggests a picture of the physical vacuum quite different than the virtual particle sea of contemporary relativistic quantum mechanics. lnstead of annihilating. ground state particleantiparticle pairs constitute a ubiquitous very weakly interacting background which provides an alternate physical interpretation of phenomena such as anomalous scattering and magnetic moments that are well described numerically by contemporary relativistic quantum mechanics but lack an intuitive physical interpretation in that formalism. Gennady Shipov, in his book '-ATheory of Physical Vacuum" [3] also presents a unified field theory with conclusions very similar to those of Mendel Sachs. Shipov's program involves a completely geometric representation of the field equations of general relativity as equivalent to the structural equations of A4 geometry [8]. Gravitation and the inertial properties of matter in non-inertial frames ofreference are described in terms ofthe contorsion part ofthe general affine connection of A4while a generalization of electromagnetism is derived from the Christoffel part of this connection. Solutions to the structural equations for the situation corresponding to anti-particle particle pairs bound by the generalized electromagnetic interaction yield the same picture of the physical vacuum as proposed by Sachs. Shipov's theory achieves William Clifford's vision[4] that preceded Einstein's general relativity by more than 30 years, ofa representation of the material world entirely in terms of the curved and twisted geometry of space itself. It is remarkable that in addition to the description of gravitational, electromagnetic and nuclear interactions that are well known in physics, Shipov's theory also admits solutions involving only the torsion of space. Shipov proposes some novel interpretations and potential applications ofthis fact that are very controversial. From the surface resemblance seen in the above comparison one is led to consider the possibility that the two theories may be linked at a fundamental level However differences not only in notation but in the choice of affine connections and geometry act as barriers to direct comparison. We have undertaken a research p r o m directed towards construction ofa bridge between the formalisms and determination of their relationship to each other. The objectives of this paper which contains some early results from this effort include: familiarizing readers with the existence of the two theories; making available a readable derivation of the spinor affine connection used by Sachs and a parallel development for Shipovl s connection' identifying the number of spin degrees of freedom retained in each theory's metric factorization and as a result demonstrating another parallel between the approaches.

2. Spinors Fundamental to the development by Sachs is the application ofthe spinor representation of spacetime. Cartan[S] introduced the spinor as an irreducible representation of the proper Lorentz group of special relativity. The splitting of the four dimensional Riemannian space into a direct product oftwo spinor spaces was first introduced by Van Der Waerden and Infield [6] with the introduction of spinor analysis, This application of spinors was further developed by Bergmann[7] and many others, eventually taking a form that is today recognized as a theory ofconnections on a complex valued fibrebundle. Recall that the covariant derivative of a vector is given by

QUATERNIONS, TORSION AND THE VACUUM

where {

"

PP

501

) is the affine connection.

Id/ is represented by means of a set of

The covariant derivative of a two-component spinor

fundamental 2nd rank spinorfields called the spin-affine connection

a,. f i pplays the same role as

the tetrad field in the more well known tetrad tensor field formalism.

A mapping is needed between the space of spinors and tensors that allows us to represent tensors with full compatibility between actions carried out with tensor objects and the results one would obtain on first mapping into spinor space and then carrying out these same actions. shall find that such

-

a mapping can be found through a tetrad of fundamental fields objects that take a covariant tensor

bIr

into a 2nd rank spinor

q pwhich are

mixed tensor/spinor

P by means ofthe simple mapping

Our compatibility requirement when applied to the action ofthe covariant derivative then requires that we obtain the same value on differentiation followed by mapping as mapping followed by differentiation. Hence we need to find tetrad fields that obey the equality

P;, = (b,q");p-- b,,9"

(4)

Applying Liebnitz's rule we obtain the requirement

which can only be fulfilled if

4;

= 0.T h u s a necessary condition on the existence of a compatible

spinor representationof a tensor is the existence of a tetrad field with this property. Given such a tetrad field, we need now also obtain the spinor afine connection

a, that realizes this compatibility.

Before proceeding we will need to also introduce the covariant second rank fundamental spinor

(the contravariant form being given by 8 ) which plays a role for spinors similar to the fundamental metric tensor for the definition of an inner product and in the raising and lowering of spinor indices The covariant derivative of the fundamental spinor is obtained again by a correspondence principle. Given a spinor invariant formed via the metric property of this spinor, we require that the


502

covariant derivative of this new spinor valued object to behave appropriately in analogy with the covariant derivative of a scalar field:

Carrying out the covariant derivative we obtain

y;,Ey

* +ysy;;

=0

Thank to the antisymmetric nature of & it can be easily shown that all components of the first two terms cancel. leaving the requirement that E;, = 0 since

W

is arbitrary. It was shown by

Bergmann that the vanishing ofthe covariant derivatives ofthe tetrad quaternion field and ofthe metric spinor ere sufficient conditions to restrict the allowable solutions for the spin-affine connection to a unique solution in the case of a spinor space based upon the restriction of unimodular spinor transformations. By definition the fundamental quaternion fields q'transforms as a2nd rankcovariant spinor and as a tensor with respect to the tensor index U , . That is. we have that

q" = (ax" ldx')(b+)-lq'b-l. Applying the Leibnitz rule, the covariant derivative of these fields can be written as follows if we assume that the connection for tensor objects takes the form appropriate for a

v4space which

is given by the Christoffel symbols.

where

;{

are the Christoffel symbols.

q" denotes the time reversed quaternion field (so

named as the action that results is reversal of the sign of the xo component in x)

qp= e(q ')

S(

e

(1 1)

where * denotes the complex conjugate. The fundamental spinor plays the role of the fundamental metric tensor in the raising /lowering of spinor indices and in the construction of the inner product and magnitude of spinors. Bergmann requires that the covariant derivative ofthe fundamental quaternion fields vanish, i.e. that transport ofthese -ds and the fundamental spinor E from one point in space to an infinitesimally near point are both globally parallel.

As shown by Bergmann, on obtaining a solution of these equationsthe resulting spinor-affine connection is uniquely obtained as

QUATERNIONS, TORSION AND THE VACUUM

503

Sachs introduces the new notion of an algebraic structure for Bergmann's tetrad fields. He shows that the

can be interpreted as a quaternion valued four vector and as such admits the

manipulations of quaternion algebra.

2.1 CLIFFORD ALGEBRA The real-valued quaternion algebra is the even sub-algebra of the Clifford algebra of 3-dimensional space CZ(3). Cl(3) is isomorphic to the algebra of 2 x 2 complex matrices and has also been called complex-valued quaternion algebra. The matrix representation of the basis of Cl(3) consists of 8 matrices: the identity matrix (rank 0),

Pauli matrices (rank 1)

the (rank 2) products

and the pseudo-scalar (rank 3)

The even rank elements of CZ(3) constitute the basis for the quaternion


504

Since the second rank spinor fields have the form of 2 x 2 Hermitean matrices they may be represented as quaternions. As will be seen. Sachs exploits the associated algebraic structure to obtain his factorization of the metric.

3. Sachs' Factorization Sachs observes that the structure of the 10 parameter Poincare group. which includes translations, rotations and reflections, is represented in the Riemann geometry of conventional general relativity by the real-valued symmetric metric tensor g

PV

- But reflection symmetry is not required by any of the

postulates of general relativity. If operations of reflection are removed from the Poincare group, the result is a 16 parameter group that Sachs calls the Einsrein group.

g P"

does not provide a complete

representation of this group. But a faithful irreducible representation can be found in terms of the fundamental quaternion field. Hence the metric tensor can be written in the symmetric factored form

where the products ofthe field tensor components are understood as quaternion products. Now we can write the linear invariant infinitesimal line element as the quaternion differential

Thus ds is a quaternion-valued scalar invariant. In contrast to the conventional formulation this invariant no longer has any ambiguity of sign. It is invariant with respect to translations in space but has internal spin degrees of freedom. Sachs also defines the quaternion conjugate or time reversed quaternion field

q

from

qp

Their product is the ordinary quadratic real-valued line element ofRiemann space which is invariant with respect to changes in both spinor coordinates and translations. This factorization makes apparent "spin" degrees of freedom that are usually hidden. Sachs does not address the important questions of the number of degrees of freedom in the quatemion field that are preserved in the invariant interval differentials, By finding the rank of the

-

Jacobian of eight differential components in ds and ds with espect to the sixteen coefficients in the quatemion field we find that there are exactly four degrees of freedom. This is suggestive of the form ofthe intrinsic spin four vector.

4. Shipov's Tetrads Shipov concerns himself from rhe beginning of his development with associating angular reference frames to point-size entities. To accomplish t'lis he applies the concept of tetrads. We will briefly introduce this approach in this section. The method of tetrads or vierbien in the tensor analysis used in the early work on general relativity and unified field theories does not lead naturally to the fuli irreducible representation of the properties ofhigher order geometry. For example, in tensor notation, the Riemann metric is written as follows:

QUATERNIONS, TORSION AND THE VACUUM a b

g,v = ??abe,ev

505 (22)

2

where the tetrad e, consists offour linearly independent covariant vector fields which provide a local pseudo-Euclidean coordinate system at each point. We also have contravariant vector fields such that

and

,

We use Greek letters such as p V etc. to denote tensor indices and Latin letters such as a., b etc. to denote "tetrad" indices. Note that raising and lowering tetrad indices is done via the Minkowski metric qab= metric gpi

giv

ab

of the local coordinate system, while tensor indices involve the symmetric Riemann

= 6; .The invariant differential interval ds is written

ds2 = g,v&pdxv

(25)

The tetrad fields map a tensor into set of tetrad scalars

a is a "dead" index (Schouten) We may now corsider the covariant derivative of these tetrad scalars

i$ are Ricci rotation coefficients As in the discussion of the covariant derivative of spinors above, we may ask when is the mvariant derivative of the tetrad scalars compatible with tensor differentiation.

The analogous necessary condition is

e:. = 0

which leads to the definition of the Ricci rotation coefficients

where the inverse tetrad is defined by I.,= v

= {o,pV

(31)

Hence identical prescriptions are used by Shipov and Bachs in the derivation of the

D. CYGANSKlAND W. S. PAGE

506

connection for their respective geometries.

4.1 METRIC FACTORIZATION Using the tetrad bases we may form the four linear scalar invariants

ds" = e;dxp ds, = e:&, and from these usual quadratic Riemann inetds

Thus the tetrad fields aliow a factorization of the invariant interval without sign ambiguity but within the context of Shipoy's formalism and without introduction of the spinor/quaternion calcuius of Sachs. Now since ds,and

dsaare linearly related through a raising operation by the Minkowski

metric) by virtue of the construction given above, these represent only four degrees of freedom. Hence the Shipov differential ilivariants comprise the same number of spin degrees of freedom as the Sachs invariants In the Shipov construction, the raisingllowering operation in the internal Minkowski tangent space at each point in his A4 geometry is the parallel of the process o- quaternion conjugation in Sachs'

case. 5.

Conclusion

The resurrection of unified field theory, as originally envisioned by Einstein, Cartan arid many others following the development of general relativity, represents a clear alternative to the collection of phenomenological and mathematical procedures loosely referred to as the Standard Model. The approach to higher order geometry required to express absolute parallelism and exemplified by the spinor formulation obviate the need to appeal to physically unintuitive notions such as strings in 10 dimensional space. At this stage of our research program we conclude that there is a deep similarity between Sachs' spinor and quaternion development and Shipov's tetrad based formalism, not only in general perspectives but at the level of metric factorization On the other hand, there are such pronounced differences in notation and geometric formalism that further study will be required to determine whether or not the similarity extends to an isomorphism. It our intent to pursue this investigation to

such ends. References [I] General Relativity and Mattex; A Spinor Field Theory from Fermis to Light-Years, Mendel Sachs, D. Reidel Publishing Co., 1982. [2] Quantum Mechanics from General Relativity; An Approximation for a Theory of Inertia, Mendel Saclis, Reidel Publishing Co., 1986. [3] A Theory of Physical Vacuum. G. I. Shipov, English edition. Russian Academy of Natural Sciences. 1998. [4] Mathematical Papers, by William Clifford, London, 1882. Lectures and Essays, Vol. 1, London, 1879. [5] E. Cartan, Bull. Soc. France Math. 41,53, 1913. [6J I. Infield, B.L. Van Der Waerden, Sitzber. preuss. Akad. Wiss., Physik-math. Ki, 380, (1933). [7] Two-Component Spinors in General Relativity, Peter G. Bergmann, Physical Review, Vol. 107, No.2, p.624. [8] Tensor Analysis for Physicists, J. A. Schouten, 2nd edition, Dover Publications Inc., 1989.

HOMOLOIDAL WEBS, SPACE CREMONA TRANSFORMATIONS AND THE DIMENSIONALITY AND SIGNATURE OF MACRO-SPACETLME An Outline of the Theory

M . SANIGA Astronomical Institute of the Slovak Academy of Sciences SK-059 60 Tatranska' Lomnica, The Slovak Republic

1. Introduction

No phenomenon of natural sciences seems to be better grounded in our everyday experience than the fact that the world of macroscopic physical reality has three dimensions we caIl spatial and one dimension of a different character we call time. Although a tremendous amount of effort has been put so far towards achieving a plausible quantitative elucidation of and deep qualitative insight into the origin of these two puzzling numbers, the subject still remains one of the toughest and most challenging problems faced by contemporary physics (and by other related fields of human inquiry as well). Perhaps the most thought-provoking approach in this respect is the one based on the concept of a transfinite, hierarchical fractal set usually referred to as the Cantorian space, E ( ~ )In. its essence, &(w) is an infinite dimensional quasi-random geometrical object consisting of an infinite number of elementary (kernel) fractal sets; yet, the expectation values of its both topological and Hausdorff dimensions are finite. The latter fact motivated El Naschie [1,2] to speculate not only about the total dimensionality of spacetime, but also about its enigmatic signature. His reasoning goes, loosely speaking, as follows. It is assumed that the effective topological dimension of &(m), {la), grasps only spatial degrees of freedom, whereas its averaged Hausdorff dimension, (d), incorporates also the temporal part of the structure. These two dimensions are interconnected, as both depend on the Hausdorff dimension of the kernel set, And there exists a unique value of the latter, viz. d?) = 1/2,forwhich ( n ) = 3 (space) and (d) = 4 (spacetime)!

4'). 507

R.L. knoroso et a1 (eds..),Gravitutionarid Cosmclogy: From the Hubble Radius to the Plm1c.k Scale, 507-510. O 2092 Kl~nc*erAcdernic Publishers. Printed ivt the Netherluitds

508

M.SANIGA

2. Cremonian Pencil-Spacetimes

In our recent papers 13.41, we approached this issue from a qualitatively different, but conceptually similar to the latter, algebraic geometrical point of view. This approach is based on our theory of pencil-spacetimes 15-13]. The theory identifies spatial cmrdinates with pencils of lines and the time dimension with a specific pencil of conics. Already its primitive form, where all the pencils lie in one and the same projective plane. suggests a profound connection between the observed number of spatial coordinates and the internal structure of time dimension 15-7,9,11-131. A qualitatively new insight into the matter was acquired by relaxing the constraint of coplanarity and identifying the pencils in question with those of fundatttental elements of a Cremona transformation in a three-dimensional projective space [3,4]. The correct dimensionality of space (3) and time (I) was found to be uniquely tied to the so-called quadro-cubic Cremona transformations - the sitrtplest non-trivial, non-symmetrical Cremona transformations in a projective space of three dimensions. Moreover, these transformations were also found to fix the type of a pencil of fundamental conics, i.e. the global structure of the time dimension. A space Cremona transformation is a rational, one-to-one correspondence between two projective spaces 1141. It is determined in all essentials by a homaloidal web of rational surfaces, i-e. by a linear, triply-infinite family of surfaces of which any three members have only one free (variable) intersection. The character of a homaloidal web is completely specified by the structure of its base manifold, that is, by the configuration of elements which are common to every member of the web. A quadro-cubic Cremona transformation is the one associated with a homaloidal web of quadrics whose base manifold consists of a (real) line and three isolated points. In a generic case, discussed in detail in 131, these three base points (Bi7 i=1,2,3) are all real, distinct and none of them is incident with the base line (c*).In the subsequent paper 141, we considered a special 'degenerate' case when one of Bi lies on f B . It was demonstrated that the corresponding fundamental manifold still comprises. like that of a generic case, three distinct pencils of lines and a single pencil of conics; in the present case, however, one of the pencils of lines incorporates ,LB, and is thus of a different nature than the remaining two that do not. As a consequence, the associated pencil-space features a kind of intriguing anisotropy, with one of its three macro-dimensions standing on a slightly different footing that the other two. Being examined and handled in terms of the transfinite Cantorian space approach, this macrospatial anisotropy was offered a fascinating possibility of being related with the properties of spacetime at the microscopic Planck scale 141-

CREMONA TRANSFORMATIONS AND MACRO-SPACETIME

509

If this spatial anisotropy is a real characteristic of the Universe, then its possible manifestations, whatever bizarre and tantalizing they might eventually turn out to be, must obviously be of a very subtle nature as they have so far successfully evaded any experimental/observational evidence. Yet, conceptually, they deserve serious attention, especially in the light of recent progress in (super)string and related theories 1151. For alongside invoking (compactified) extra spatial dimensions to provide a sufficiently-extended setting for a possible unification of dl the known interactions, we should also have a fresh Look at and revise our understanding of the three classical macro-dimensions we have been familiar with since the time of Ptolemy. 3. Conclusion

The concept of Cremonian spacetimes represents a very interesting and fruitful generalization of the pencil concept of spacetime by simply raising the dimensionality of its projective setting from two to three. When compared with its two-dimensional sibling, this extended, three-dimensional framework brings much fresh air into old pressing issues concerning the structure of spacetime. and allows us to look at the latter in novel. in some cases completely unexpected ways, Firstly, and of greatest importance, this framework offers a natural qualitative elucidation of the observed dimensionality and signature of macro-spacetime, based on the sound algebro-geometrical principles. Secondly, it sheds substantial light at and provides us with a promising conceptual basis for the eventual reconciliation between the two extreme views of spacetime, namely physical and perceptual. Thirdly, it gives a significant boost to the idea already indicated by the planar model that the multiplicity of spatial dimensions and the generic structure of time are intimately linked to each other. Finally, being found to be formally on a similar philosophical track as the fractal Cantorian approach, it grants the latter further credibility.

Acknowledgeii~ent-Thiswork was partially supported by the NATO Collaborative Linkage Grant PST.CLG.976850. References 1. El Naschie, M.S.: Time symmetry breaking, duality and Cantorian space-time, Chaos, Solitons & Fractals 7 (1996), 499-5 18. 2. El Naschie, M.S.: Fractal gravity and symmetry breaking in a hierarchical Cantorian space, Chaos, Solitons & Fractals 8 (1997), 1865-1872. 3. Saniga, M.: Cremona transformations and the conundrum of dimensionality and signature of macro-spacetime, Chaos, Solitcms & Fractals 12 (2001), [in press].

510

M. SANIGA

4. Saniga, M.: On 'spatially-anisotropic' pencil-spacetimes associated with a quadro-cubic Cremona transformation, Chaos, Solitons & Fractals 12 (2001), [in press]. 5. Saniga, M.: Arrow of time & spatial dimensions, in K. Sato, T. Suginohara and N. Sugiyama (eds.), The Cosmological Constant and the Evolution of the Universe, Universal Academy Press, Tokyo, 1996, pp. 283-284. 6. Saniga, M.: On the transmutation and annihilation of pencil-generated spacetime dimensions, in W.G. Tifft and W.J. Cocke (eds.), Modenl Mathe?naticalModels of Time and their Applications to Phvsics and Cosnlology, Kluwer Academic Publishers. Dordrecht. 1996, pp. 283-290. 7. Saniga, M.: Pencils of conics: a means towards a deeper understanding of the arrow of time?, Chaos, Solitons & Fractals 9 (1998), 1071-1086. 8. Saniga, M.: Time arrows over ground fields of an uneven characteristic, Chaos, Solitons & Fractals 9 (1998), 1087-1093. 9. Saniga, M.: Temporal dimension over Galois fields of characteristic two, Chaos, Solitons & Fractals 9 (1998), 1095-1 104. 10. Saniga, M.: On a remarkable relation between future and past over quadratic Galois fields, Chaos, Solitons & Fractals 9 (1998), 1769-1771. 11. Saniga, M.: Unveiling the nature of time: altered states of consciousness and pencilgenerated space-times, Int. J. Transdisciplinary Studies 2 ( 1 998), 8-17. 12. Saniga, M.: Geometry of psycho(patho)logical space-times: a clue to resolving the enigma of time?, Noetic J. 2 (1 5)99), 265-274. 13. Saniga, M.: Algebraic geometry: a tool for resolving the enigma of time?, in R. Buccheri, V. Di Gesh and M. Saniga (eds.), Studies on the Structure of Tinle: From Physics to P~7lcho(patho)logy, Kluwer Academic/Plenum Publishers, New York, 2000, pp. 1 3 7 4 6 and pp. 301-6. 14. Hudson, H.P.: Crenlona Transfonllations in Plane and Space, Cambridge University Press, Cambridge, 1927. 15. Kaku, M.: Introduction to Superstrings and M-771eog: Springer Verlag, New York, 1999.

PULSE INTERACTION IN NONLINEAR VACUUM ELECTRODYNAMICS A. M. IGNATOV General Physics Institute Moscow, Russia Email: [email protected]~ V.P. POPONIN International Space Sciences Organization San Erancisco, CA, USA Ernail: [email protected]

Abstract. The energy-momentum conservation law is used to investigate the interaction of pulses in the framework of nonlinear electrodynamics with Lorentz-invariant constitutive relations. It is shown that for the pulses of the arbitrary shape the interaction results in phase shift only.

1. Introduction Although classical electromagnetic theory deals with linear Maxwell equations, there have been numerous attempts to bring the nonlinear phenomena into the stage, All relativistic and gauge invariant versions ofelectromagnetism are based on the Lagrangian density, L, which depends on the invariants of the field tensor. Generally, in terms of the electric (E) and magnetic (B) fields the Maxwell equations in absence of external charges may be written in a standard form:

where we put c = 1 and

D = f $ , H = - g.The

Lagnngian L( I,J' )depends on

Poincare invariants I = E Z- B2and J= EB only. The distinctive feature ofEqs. (1.1) is that since the Poincare invariants are identically zero for the plane electromagnetic wave, the latter is insensitive to vacuum nonlinearity and propagates without distortion. Of particular interest are the nonlinear corrections to the linear electsodynamics arising due to vacuum polarization in the strong electromagnetic field. In the ultimate case of slowly varying fields this results in Heisenberg- Euler electrodynamics [I]. 511 R.L. Amomso et a1 (eds.),Gravitation and Cosmology: From the Hubble R~diusto the Pl~llrckScale, 511-514. O 2002 Kluwer Academic Publishers. Printed in the Nebherlanlls.

A. M.ZGNATOV AND V.P. POPONIN

512

The main point of this paper is to describe the simplest, nonlinear vacuum process: the interaction of two electromagnetic waveforms propagating in opposite directions.

2. Maxwell Equations We consider a linearly polarized wave propagating in the z direction of the E, = E(z, t), By = B(z, t ) with all other components being zero. In this situation. the second Poincare invariant vanishes, J

E 0, SO

the Maxwell equations are written as

where the subscript denotes the derivative with respect to the corresponding variable and I = E~ - BZ The Lagrangian in Eq.(1.2) is expanded in powers of I. Keeping the lowest-order nonlinear corrections we have

.

With the help ofthe appropriate scale transform, the coefficient a may be reduced to f 1. For the particular case ofthe Heisenberg-Eulerelectrodynamics, 0 = 1 [I]. Of interest also is to keep in mind the Born-Infeld electrodynamics [2]with the Lagrangian

3. Energy-Momentum Tensor The conservation laws for Eqs. (1 -2) are given by

where the components ofthe energy-momentum tensor, namely, the energy density, W, the momentum density, N, and the stress, P, may be obtained using standard variation procedure e.g. 91. Explicitly,

Usually Eqs. (1.5, 1.6) are thought of as a consequence of the Maxwell equations (1-2). However, we may consider the relations (1.6) as a constraint implied upon the components of the momentum-energy tensor, so there are two independent variables in Eqs. (1.5), for example, W and N. One can easily check that for the nontrivial solutions of Eqs. (1 -21, i.e.

NONLINEAR VACUUM ELECTRODYNAMICS

513

for I # 0 ,the Jacobian ofthe transform E. BW. N is non-zero. Thus. instead of looking for the solutions of Eqs. (1.2) we can solve Eqs. (1.5, 1.6) excluding the Poincare invariant I from Eqs. (1.6). 4. Solution

To exclude I it is convenient to introduce the invariants of the energy-momentum tensor, that is, its trace, S = P -W, and the determinant T = WP - N AS it follows from Eqs. (1.6)

.

The latter relations implicitly d e f i e the dependence T = T(S). Substituting the Lagrangian (1.3) into Eqs. (1.7) we find that the first nonvanishing term of the expansion of T in powers of S is linear and it is provided by the quadratic term of the expansion (1.3): T (S) = - as+... . It is noteworthy that the Born-Infeld Lagrangian (1.4) yields exactly the linear dependence T (S) = - S. The relations (1.5) are resolved introducing the potential y :W = Y ZZ ,. N =

- y,, P = y,. Restricting ourself with the linear relation between T and S, we obtain the Ampere-Monge type equation for

:

There are trivial solutions to this equation y /

+

(2,t ) = F(z t ) with an arbitrary function F,

which correspond to the plane electromagnetic waveforms described by Eqs. (1.2) with I = 0. Besides these, implementing the Legendre transform[4] one can easily obtain the general integral of Eq. (1.8) valid for T # 0 and, consequently,for I # 0 As aresult, we get the components of the energy-momentum tensor in a parametric form:

.

where

qP2 are arbitmy functions and A ( t , r ] ) = 1- &'(t) F;(1]).

Consider, for example, two localized pulses of the arbitrary shape propagating in opposite directions. This corresponds to the following initial conditions:

A. M. IGNATOV AND V.P. POPONIN

~ ( 2tll,,, ,

where of

=

w,(2+ t ) + 4 (2- fX

(1.10)

Fs2 ({)I +-+,0.This initial condition is provided by the following choice

4,in Eqs- (1-9):4r2(5) =

(t)alld 4 ( y )I

{ + ,

({)I

09F2

C+,

0. m e

asymptote of the solution (1.9) at t 4cn is then given by

W(z,t)=W,(z+t-~K~)+W,(z-t+a~,)~ where

is the net energy carried by the corresponding pulse.

5. Discussion Of interest is the geometrical sense of the obtained solution (1.9). The parameters { and 41 are, in fact, the light-cone coordinates disturbed by the electromagnetic field. One may say that the electromagnetic field alters the space-time metric due to the dependence of the speed of light on the field strength. In contrast with general relativity, the space-time remains flat. Another interesting point is that for a = 1 the increase in the pulse amplitude results in delay in energy (and information) exchange between distant points, that is, the solution described by (1-1 I) is subluminal. This takes place for both the Heisenberg-Euler electrodynamics, which is currently the only one ofphysical sence, and for the elegant Born-Infeld theory, for which our results are exact. However, for a = -1 the pulse propagation would be superluminal. From the viewpoint of nonlinear physics. the electromagnetic pulses in vacuum exhibit the soliton-like behavior: the collision results in a phase shift but the form of a pulse remains unchanged. The main interesting point with this respect is that unlike usual nonlinear equations, the shape ofthe soliton is arbitrary.

Bibliography [I] [2] [3] [4]

Heisenberg, W., Euler, H. (1936), Z. Phys, 38, 714 Born, M., Infeld, L. (1934), Proc. Roy. Soc. (London), A144], 425 Landau. L.D., Lifshitz E. M. (1971), The Classical Theory of Fields Oxford, New York. Pergamon Press, Courant, R. 119621, Partial Differntial Equations,N.-Y., London,

PROPOSAL FOR TELEPORTATION BY HELP OF VACUUM HOLES

CONSTANTIN LESHAN

S. OCTEABRISCOE & R-L SINGEREI Kiev University MD-6233 Moldova Email: con-phys @homail.com

Abstract. W e can teleport a body using geometrical properties of space. Teleportation would consist of sending a body outside the universe into zero-space in order to reappear at another point in the universe. Some teleportation properties we can see by a simple mechanical motion. Let body A have a linear and uniform motion in space. Body A passes distance ds during time dr without energy expenses. Consequently body A can be teleported without energy expenses. Energy expenses appear ifbody A moves between two points with different values of force field. Therefore teleportation is not possible if the startpoint and endpoint has a different value force field. How can time dt decrease? Unfortunately a body cannot move faster than light. There is a single solution. Superluminosity can occur in space where time properties do not exist. The same 0-space exists. 1. Zero-Space and Vacuum Holes As is well known the universe is curved and has a limited volume. After the Big Bang. the initial explosion giving birth to the universe, limited time would occupy a limited volume. This limited volume suggests boundary conditions, even if only in some points. What could be behind this border? Most authors assert "there is nothing outside the Hubble universe. neither galaxies, nor substance; absolutely nothing - neither space, nor time" genemlly [I]. In other words outside the universe could be a "point form space" where distance between two points is always equal to zero which we could name "zero-space" or "hole". Moreover this border can't exist in a single place because the cosmological principle, which states there are no privileged frames relative to another place or point in the universe, is violated The border ofthe universe must pass through every point of space. Virtual holes in spacetime must exist in every point of the universe. This physical object is so-called "vacuum with holes" or "hole vacuum" [2]. Vacuum holes exist as virtual particles. What would happen if we sent body A outside the Universe? Since zero-space is a point and where time as a property does not exist, therefore it cannot contain body A and consequently body A will appear in the real universe at that same moment in time. With the distances between zero-space and any other point of universe being equal to zero, these holes can potentially exist in every point of universe. Therefore body A could appear at random in at point of in the Universe. Let us do the following thought experiment. In order to send body A outside of the universe we must first create a closed surface which consist vacuum holes around body A, for but an instant in time. Then we ask, where was the body A during that time dt that it was surrounded by vacuum holes? Inside of hole sphere body cannot exist because outside of universe cannot exist any body, consequently body A was existing all the time dt in other point of universe already. If we destroy the hole surface around body A we destroy in this way the channel that connect this two points and 515 R.L. Amroso et a1 feds.).Gravitation and Cosmlugy: From the Hubble Radius to the Plunck Scale, 515-516. O 2002 Klutver Academic Publislieru. Piilrted in the Netlrerlunds.

5 16

LESHAN, OCTEABFUSCOE & SINGEREI

body A will remain in new place. The body was teleported from point 1 to point 2 through zerospace. Inside of hole surface exist the non-Euclidean geometry of A. Poincare. This place is completely isolated system toward to external universe. Not exist any ray or other radiation able to penetrate through the hole in space and time. For internal observer the distance between centre and hole surface is infinite, as the distances between every two points decrease from centre to border and all distances is equal to zero near hole surface. The same properties have the A Poincare model of non-Euclidean universe from 1882 for case of 3-dimencional space. He proposed a model of non-Euclidean universe that is a precise copy of our hole sphere. For imaginary people that live inside of Poincare circumference the distance to border is an infinite. All universe for they are inside of this circumference. The transmitter of matter can be with internal or external hole production. First present a station that have a spherical room where is introduced the material body for teleportation (the sphere has a minimal area). On the external surface of sphere is equipment that produces vacuum holes around the body A. Second method present a station that produce a hole surface around oneself, therefore this station can repeatedly teleport oneselfto at random points of universe. In this case the receiver of matter is not necessary. There is probability after a number of teleportations the station will appear on its planet. The equipment for hole production in this case is inside of sphere. The energy expenses are necessary for curvature ofthe space-time only but not for motion of body from point 1 to endpoint 2. First we must create the station with internal hole production. The station will be very durable as not have friction. During the exploitation term one will launch about thousand spacmfts into very deep space, on the distance of hundreds or millions of light years. For example spacecraft can appear in Cmb nebula or near Sirius. Unfortunately we don't know the endpoint after teleportation, but universe is uniformly and isotropic, therefore is all the same where look for a new planets for colonisation or extraterrestrial life. For hole teleportation we must create the equipment that able to produce the holes in space-time. For it is necessary to research interaction between vacuum holes and matter. Hole teleportation has a lot of advantages toward to quantum one. For QT we must collect all information about object therefore we must decompose one. Who agrees to be destroyed in transmitter in order to be rebuilt in the receiver? Would the soul be copied? So, quantum teleportation is not suitable for humans. The volume of information about a single human is enormous. Just how much information are we talking about anyway? (3) Well the visible human project by the American National Institute of Health requires about 10 Gigabytes (this is about ten CD ROMs) to give the full three dimensional details of a human down to one millimeter resolution in each direction. If we forget about recognizing atoms and measuring their velocities and just bits (a one scale that to a resolution of one-atomic length in each direction that's about followed by thirty two zeros). This is so much information that even with the best optical fibers conceivable it would take o v a one hundred million centuries to transmit all that information! It would be easier to walk! If we packed all that information into CD ROMs it would fit into a cube almost 1000 kilometres on a side! Enough said? The technology that permitted to build a human in receiver by help of information received from transmitter will appear only in a very far future. You see, QT is practically impossible. All this defects don't exist in hole teleportation. There is not necessary to decompose and rebuild a teleported bodies, therefore the HT is suitable for human teleportation. Objects are teleported faster that light at any distances, even to far stars. We must solve a single problem, how to create a closed hole surface.

References 1 LD. Novicov, Evolution of universe, Science, Moscow, 1990 2. Conference proceedings, ICPS 94, S. Peterburg, 1994 3. http://www.sees.bangor.ac.uk/-schrnueYtpo

COSMOLOGY, THE QUANTUM UNIVERSE, AND ELECTRON SPIN MILO WOLFF Technotran Press 11 24 Third Street, Munhattan Beach, CA 90266 [email protected]

Abstract. Clifford, Mach, Einstein, Wyle, Schrodinger all pointed out that only a wave structure ofparticles (nlatter)can conform with experimental data and fulfill the logic of reality and cosmology. 1 describe a quantum Wave Structure of Matter (WSM) that satisfies their requirements and predicts the origins of natural laws. This structure is a simple pair of spherical outward and inward quantum waves convergent to a center. The wave pair is the physical origin of the electron quantum spin, which results when the inward quantum wave undergoes spherical rotation to become the outward wave. These two waves are a Dirac spinor, thus this physical quantum wave structure satisfies the theoretical Dirac Equation. But it also forms the structure ofthe universe! 1. Introduction - A True Science Odyssey Discovering quantum wave structure is a beautiful adventure in which you find the origin of the natural laws, a powerful tool of science. and an exciting window on the quantum wave universe. Every electron. proton or neutron are quantum wave structures. Understanding their reality demands learning the quantum wave rules. It is easy but one has to think anew and discard the false notion of a material point particle, and recognize that location, charge and mass are properties of the wave structure. Since the quantum wave universe is not directly useful to our personal survival. nature has not equipped us to observe it as easily as apples and tigers. Lacking personal experience people imagine that the electron is a "particle" like a baseball. Laboratoq evidence does not support this idea. Clearly, scientists must change their belief from particle to wave. The difficulty of that change is shown in the story below from Omni Magazine: Imagine that are the commander of the fifth inter-galaxy survey unit and the survey captain reports to you, 'They're made of meat." "You mean, mea!?" 'There's no doubt about it. We picked several from different parts of the planet. took them aboard our vessels, probed them all the way through. They're completely meat." 'That's impossible. What about the radio signals? The messages to the stars?" 'They use radio waves to talk, but the signals come from machine..." "So who made the machines? That's who we want to contact." 'They made the machines. That's what I'm explaining. Meat made the machines-" "Ridiculous! You're asking me to believe in sentient me*"

517 R.L. Arnoroso et a1 (eds.).Gravitation and Cosmology: From the Hubble Ruclius to the Plunck Scale, 517-524. O 2002 Klutver Acaclemic Publishers. Printed in the Netherluncls.

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"I'm not asking you, I'm telling you the results ofour research." "Okay, maybe they're part meat, like the Weddilei, a meat head with an electron plasma brain inside." "Nope. They do have meat heads but we probed them all the way through." "No brain?" "Oh. there is a brain all right. It's just that the brain is made out ofmeat!" 'Oh? What does the thinking?" "You're not understanding, are you? The meat brain does the thinking." "Thinking meat? You're asking me to believe in thinking meat'" "Yes, thinking meat! Conscious meat! Loving meat! Dreaming meat" "Really? You're serious then. They're made out ofmeat." "Finally! Yes. And they've been trying to contact us for a hundred of their years." "So what does the meat have in mind?" "It wants to talk to us. Then I imagine it wants to explore the universe-contact other sentients, swap ideas and information. The usual." "They actually do talk. then. They use words. ideas. concepts?" "Oh, yes. Except they do it with meat." "I thought youjust told me they used radio." "They do. but what do you think is on the radio? Mear sounds. Singing meat" "Omigosh! Singing meat! This is too much. Any true sentients in the galaxy?" 'Yes, a rather shy hydrogen core cluster intelligince in a class nine star i n - ~ 4 4 5 zone. Was in contact two galactic rotations ago, wants to be friendly again." "And why not? How unbearably cold the universe would be if one were all alone!"

New truths of science are often unwelcome. Emotional rejection occurs if the new truth conflicts with established belief. Max Planck once said, "New scientific truth does not triumph by convincing its opponents, but because the opponents die and a new generation grows up unopposed to the new idea."

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2. The New Structure of Matter Spherical Space Resonances Below is the new truth of the structure of matter that agrees with experimental facts. It is overwhelmingly simple because it uses only three principles to establish the Wave Structure ofMatter as the basis of all scientific laws. The proposal that mass and charge were properties of a wave structure in space was consistent with quantum theory since quantum mathematics does not depend on a belief in particle substance or charge substance. The reality is that space waves are real while mass and charge points are mere appearances, "Schaumkommen" in the words of Schroedinger. The famous English geometer, William Clifford (1876), wrote "All matter is simply undulations in the fabric of space." Einstein and Emst Mach reasoned that particles must be "spherically spatially extended in space." Einstein wrote, "..hence the nraterial particle has noplace as a fundairrental concept in afield theory. " Paul Dirac was never satisfied with the point particle because the Coulomb force law had to be corrected by bbrenormalization." He wrote (1929) "This is just not sensible nrathenlatics. Sensible inathenlatics involves neglecting a quantity because it turns out to be snrall, not neglecting it because it is infinitely large andyou do not wanl it!" Wheeler and Feynman (1945) modeled the electron as spherical inward and outward electronragnetic waves. seeking to explain radiation forces, but encountered difficulties because there are no spherical solutions of vector e-m wave equations. Nevertheless their

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particle sends yuanturlz waves outward and

Milo Wolff (1990-93-97), using a scalar wave equation with spherical wave solutions, found the Wave Structure of Matter (WSM) described here. It successfully predicted the natural laws and all properties of the electron, except its spin. This paper provides a physical origin of spin which accords with the Dirac Equation. 2.1 THE NATURAL LAWS ARE FOUND IN THE WAVE STRUCTURE Our knowledge of science and the universe is based on natural laws, the rules for calculating electricity, gravity, relativity, quantum mechanics, and conservation of energy and momentum. The origin of these laws was unknown. Now their the origin is found to be a quantitative result of the WSM. The wave-snuctured panicle, Figure 1, is termed a space resonance (SR). The medium of the waves and the leading player in the new scenario is space. Space resonances and the laws they produce are derived from propaties of the wave medium. Thus, this single entity, space, described by the three principles, underlies everything. Out F i p I. 'The Dynamic Waves oTa Space Resonanw The resonance 9 composed of a spherical IN wave, which converges to the center, ~otatesto become an OUT wave, and diverges from the center. These two waves combine to form a standing waw whose peaks and nodes are like layers of an onion. The wave amplitude is a scalar number, not an electromagnetic vector. At the center, the wave amplitude is finite, not infinite, in n o r ~ ~ m with ~ n t nh~ervatinlr

3. Principle I - A Wave Equation This equation describes how quantum waves are formed and travel in the space medium. If the medium is uniform, typical nearly everywhere, only spherical waves occur. If the medium is locally dense, as in the central region of a proton, waves can circulate like sound waves in a drum or sphere. If observed in relative motion, Doppler modulation and elliptical waves appear. F'rinciple I is:

Quantum matter waves exist in space and are solutions of a scalar wave equation. The wave equation is:

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( g r a d ) 2 ( ~ ~ (l/c ~ ) 2) d 2 ( ~ ~ p ) / d=t02

Where AMP is a scalar amplitude, c is the velocity of light, and t is the time. A pair of spherical idout waves forms the simple structure of the electron or positron. The mathematical properties of this combination display the laws of: mechanics, quantum mechanics, special relativity, and e-m. The waves decrease in intensity with increasing radius, like the force of charge. The inward and outward waves move oppositely, thus

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forming a resonant standing wave. Arriving at the center. the IN-wave rotates. producing 'spin' and becoming the OUT-wave. All properties of the waves depend on their medium, space, as is true for all oscillators. There are &o combinations of the resonances, electrons and positrons. Thus matter is constituted of two binary elements inward and outward waves. It is curious that the universe, like computer hardware. is binary. Does this have a profound meaning? The rules of quantum mechanics (QM) and special relativity (SRT) are the result of the motion, with a velocity b = vlc, of one SR relative to another, which produces a Doppler shift in ! &I the IN- and OUT- waves. All parameters of QM and SRT for a moving particle appear as algebraic factors in the Doppler-shifted waves; that is, the deBroglie wavelength of QM, and the relativistic mass and momentum increases, exactly as experimentally measured. This can be shown by writing the amplitude received at either SR - both are alike: Received amplitude = llr (2 AMP-max) exp[ikg (ct + br)] sin &g (bct + r)]. This is an exponential carrier oscillator modulated by a sinusoid. In the carrier: Wavelength = hlgmv = deBronlie wavelength with relativistic momentum. Frequency = kgc /2Tw gmc2/h = mass frequency with relativistic energy. And in the sinusoid: Wavelength = h/gmv = Compton wavelength with relativistic momentum. Frequency = b gmc2/h = b x (mass frequency) = relativistic momentum frequency. There are solutions of Principle 1 and two combinations which correspond to electrons and positrons. Charge properties depend on whether there is a + or - amplitude of the lN wave at the center. If a resonance is superimposed upon an anti-resonance they annihilate. The amplitude at the center is finite as observed, not infinite as in the Coulomb rule. They obey Feynman's Rule: "A positron is an electron going backward in time." See this by replacing the variable t with a -t in an electron resonance; a positron resonance is obtained. The change o f t has exchanged the roles of the IN and OUT waves.

4. Energy Transfer And The Action-At-A-Distance Paradox tt is essential to recopize that communication or acquisition of knowledge of Q kind occurs only with an energy transfer. Storage of information, whether in a computer disk or in our brain. always requires an energy transfer. Energy moves a needle. magnetizes a tape. and stimulates a neuron. This rule of nature is embedded in biology and our instruments. The energy transfer mechanism is found in Principle 11below. One major failure of the classical force laws is they have no physical mechanism for energy transfer. This is the fault ofthe point particle model, which contrasts with the dynamic, spatially extended Space Resonance, Ernst Mach observed positive evidence of cosmological energy transfer in 1883, noticing that the inertia of a body depended on the presence of the visible stars. He asserted: "Every local inertial frame is determined by the conlposite matter of the universe. " His

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concept arises from two different methods of measuring rotation. First, without looking at the sky, one can measure the centrifugal force on a rotating mass m and use the inertia law f=ma, to find circumferential speed v. The second method is to compare the object's angular position with the fixed (distant) stars. Both methods give exactly the same result! Mach's Principle was criticized because it appears to predict instantaneous action-at-adistance. How can information travel from here to the stars and back again in an instant? The solution lies in the space resonance. Space is not empty because it is filled with the waves of every particle in the universe (Principle I1 below). Inertia is an interaction with the space medium. There is no need to travel across the universe.

5. Principle I1 - Space Density Principle (SDP) This principle defines the medium of quantum waves in space. It is very important because the natural laws depend on the waves of the electron-positron which in turn depend on the medium. ~ h u sthe medium is the actual origin of the natural laws. Principle I1 is:

Waves from all particles in the universe combine their intensities to form the wave medium of space.

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The medium = space density mc2 = hf = k'{SUM OF:{(AMP,,)* x (llr,,?) Specifically, the frequency f, or mass m, of a particle depends on the sum of squares of all wave amplitudes, AMP,, from the N particles in the Hubble universe, which decrease inversely with range r, squared. This universe exists inside a radius R = c/H, where H is the Hubble constant. This principle contains a quantitative version of Mach's Principle because the space medium is the inertial frame of the law F=ma. Energy exchange takes place between the mass m and the surrounding space medium. Because N = loa0 particles in the Hubble universe, the medium is nearly constant everywhere and we observe a nearly constant speed of light. But near a large body, a larger space density produces a tiny curvature of the paths of quantum waves (and thus of light). But note that the self-waves of a resonance are counted too. Thus space becomes dense near the resonance center due to its own amplitude. Space is non-linear at the central region, which provides the coupling between two resonances needed for energy transfer. We observe this and call it "charge." Can this principle be tested? Yes. [fa resonance's self waves affect space density, then the intensity of self-waves at some radius, ro, must equal the total intensity of waves from the other N particles in the Universe. Evaluating this equality yields

This is called the Equation of the Cosmos, a relation between the size ro of the electron and the size R of the Universe. Astonishingly, it describes how all the N particles of the Hubble Universe create the space medium and the 'kharge" of each electron as a property of space.

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meters. N = Continue the test. Insert values into the equation above. R = particles. Then ro = 6 x meters. This should be near the classical radius, e2/mc2'of an electron, which is 2.8 x 10-l5meters. The test is satisfied. Let's discuss Energy Transfer and the Conservation of Energy. Typically energy transfer occurs between two atomic or molecular quantum states: a source and a receiver. In the source, an electron's energy shifts downward; in the receiver, there is an equal shift upward. Only oscillators with similar frequencies 'tuned' to each other can couple and shift frequency. Accordingly, the frequency (energy) changes must be equal and opposite. This is exactly the content of the Conservation of Energy law.

6. Principle I11 - Minimum Amplitude Principle (MAP) The third principle is a powerful law of the universe that determines how interactions take place and how wave structures will move:

The total amplitude of all particle waves in space always seeks a minimum. Thus energy transfers take place and wavecenters move in order to minimize their total wave amplitude. This principle is the disciplinarian of the universe. Amplitudes are additive, so moving two opposite resonances closer together will minimize amplitude. Thus, this principle dictates "Like charges repel and unlike charges attract." because those rules minimize total amplitude. The MAP produces the Heisenberg Exclusion Principle, which prevents two identical resonances (fermions) from occupying the same state. This is disallowed because total amplitude would be maximum, not a minimum.

7. Spin And The Dirac Equation The physical nature and cause of electron spin was unknown before the WSM. However, a successful mathematical theory of spin had been developed by Nobel laureate Paul Dirac (1926, see Eisele, 1960). It predicted thepositron (Anderson, 1931) and a spin of h14pi angular momentum units. Dirac was seekiig a connection between Schroediger's quantum equation and the conservation of energy given by

Unfortunately, Eqn (1) uses squared terms and Schriidinger equation cannot. Dirac had a crazy idea: 'Try replacing Eqn (1) with a 4-matrix equation."

Where [Identity], [alpha],and [beta] are new 4-matrix operators. This worked and his Equation (2) became famous. As a result Dirac realized that only wave functions were needed for the electron. So Dirac simplified the matrix algebra by introducing number pairs, termed spinors, creating a two-number algebra instead of our common single number algebra. His spinor algebra, gave no hint of the physical structure of the

COSMOLOGY, QUANTUM, AND SPlN electron. But now, we see that the idout wave pair are counter-part of Dirac's theoretical spinor functions.

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spinor waves, the physical

Spin occurs when the IN wave rotates to become an OUT wave. It uses a 3D property known as 'spherical rotation' (Gravitation, Misner et al, 1973) in which space returns to its initial state after turns. It is necessary that space return, otherwise it would twist up without limit. Two turns produces an angular momentum of & h/4pi, exactly what is observed. There are only two ways to rotate, CW and CCW. One is the electron; the other, the positron. This is why each charged particle has an anti-particle. Analysis of spherical rotation by Batty-Pratt & Racey (1980) showed that exponential oscillators ei"' were spinors, spinors. Wolff (1990) realized that the in-out waves of the WSM were satisfying the Dirac Equation. The SR also displays other physical properties of an electron including CPT and conversion to a positron. To see this start with the two solutions of the wave equation in spherical coordinates which are: m-wave = (l/r){AMP - max) exp (iwt + ikr) OUT-wave = (Ih){AMP - max) exp (iwt ikr)

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The electron SR = AMPThe positron SR = AMP'

= =

{IN-wave + OUT-wave) Rc" (-IN-wave OUT-wave) Rm

(c)

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(d)

You can experiment with the CPT inversions. To perform a Time inversion. change t to t, which converts the positron into an electron. To perform a mirror inversion (Parity), imagine that the waves are viewed in a mirror: a positron is a mirror image of the electron. To change a particle to an anti-particle (Charge inversion). switch the in-waves and the out-waves, and the spin . The successive C, P, and T inversions return an electron to its initial state proving the CPT rule, as a property of the wave structure. Another physical property ofthe quantum wave electron is that inverting its spin axis is equivalent to reversing its spin, in contrast with cylindrical rotation. Test this. The electron spinor is [ei"', 01. To reverse the spin, change t to -t. The spin reversed spinor becomes fe-jwl,0] which exchanges the outgoing wave with the incoming wave. Compare with inverting the spin axis of the electron spinor using the inversion matrix: 0 -1 1 0

ei"' 0

yields

ei"

0

Contrary to our human intuition, inversion and spin reversal are the quantum wave structure of charged particles.

the same, verifying

8. The Origin Of The IN Waves And The Response Of The Universe At first thought it is a puzzle where the IN waves come from. But we have ignored the waves of all other particles in space. To find reality and a rational origin of the inward waves, we must deal with the wave-filled universe. Christian Huygens, a Dutch

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mathematician, found that if a surface containing many separate wave sources was examined at a distance. the combined wavelets of the sources appeared as a single wave front with the shape of the surface. This is termed a 'Huygens Combination' of the separate wavelets. Figure 2. IN Waves. The o u t - w e of every particles interacts with other rnaner in the universe. The response to the outgoing wave is Huygens wavelets from other matter, which converge back to the cenm of the initial, out-wave. When the wavelets approach the center their combined ampIitude become larger forming the 1N wave. Each contribution to the in-wave is a tiny part of the entire fabric of the universe. Thus every particle depends on all other particles for its existence. We are prt ofrh universe and the universe is part of us.

Figure 2. Formation of

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When the outgoing wave encounters other particles in the universe, their out-waves are mixed (^reflected') with a component of the initial out-wave. These components return, in phase, to the initial center. Thus waves from all the other particles combine to form a Huygens Combination wave front that is the in-wave of the initial particle. We should not imagine each particle as one pair of IN and OUT waves, because one pair cannot exist alone. Each particle depends on all others particles in the universe to create its IN wave. We have to think of ourselves as inextricably joined with other matter of the universe. In conclusion, the value of studying the WSM is the insight it provides to deeply analyze quantum wave structure, the cosmos, the natural laws and their application, especially to ICs, computer memory devices, and energy.

References E. Battey-F'ratt and 7. Racey (1980), lntl. J. Theor. Phys. 19, pp. 437-475. Louis Duc de Broglie (1924), PhD Thesis "RecherchC sur la Theorie des Quanta," U. of Paris. W i a m Clifford (1876), in The World of Mathematics. Simon & Schuster, NY. 1956. Paul Dirac (1929), Proc. Roy. Soc. A 117, p. 610. P a d Dirac (1937), Nature, London. 174, p. 321. Albea Einstein (1950), Generalized Theory of Gravitation. John A. Eisele (1%0), Modem Quantum Mechanics with - Elementary Particle Physics, John Wiley, NY. C.W. Misner, K- Thome, and J.A. Wheeler (1973, Gravitation, W.H. Freeman Co., p. 1149. Walter Moore, (19891, Schroediwer - Life and Thought. Cambridge U. Press, p. 327. J. Wheeler and R Feynman (1945), "Interaction with the Absorber.." Rev. Mod. Phys. 17, p. 157. Milo Wolff (1990). Exploring Physics of Unknown Universe, ISBN 0-9627787-0-2.. Technotran Press. CA. Milo Wolff(199 11, Invited paper at 1" Sakharov Conf. Phys, Moscow, May 21-31 p. 1131 Nova Publ., IVY. Milo Wolff (1993), "Fundamental Laws, Microphysics and Cosmology," Physics Essays 6, pp. 181-203. Milo Wolff (19%), "A Wave Structure for the Electron," Galilean Electrodynamics 6, No. 5, pp. 83-91. Milo Wolff (1997) "Exploring the Universe.." Temple University Frontier Perspectives 6, No 2, pp. 44-56. Milo Wolff(1997) 'The Eight Fold Way ofthe Universe," Apeiron 4, No. 4. Milo Wolff (1997) "Mass Change and Doppler Shift .." Galilean Electrodynamics 8, No. 4. Milo Wolff, " Quantum Science Comer" The website: http://members.tripod.comlmwolff

ON SOME IMPLICATIONS OF THE LOCAL THEORY Th(fi AND OF POPPER'S EXPERIMENT

THOMAS D. ANGELIDIS Centre for Mathematical Physics 19 Cheval Place, Suite 5014 London SW7 1 EW, United Kingdorn. E-mail: [email protected]

A propos

Writing this paper in honour of Jean-Pierre Vigier's 80th birthday. brings to mind the time when I was first introduced to him by my late friend Karl Popper in 1979 at a seminar. Since then I found in Vigier not only a most captivating person, but also a thinker utterly devoted to his problems. We both, I trust, belong to the fraternity of critical rationalists, the fraternity of those who are eager to argue, to learn from one another, and who have the intellectual courage in changing their mind, under the influence of criticism, even on points of fundamental importance to their cherished conjectures and beliefs.

1. Locality Versus Nonlocality: A Verdict On "Nonlocal Interactions" The central theme in Vigier's diverse work [l] has been his battle against the Copenhagen interpretation of the theory of the quantum formalism TQFand, partly following the steps of de Broglie, Einstein, Schradinger et al., his valiant quest to construct a stochastic interpretation of TQ,. This is a realistic interpretation of Tw, where particles are presumed to have definite timelike trajectories in space-time (in contrast to the Copenhagen interpretation which denies this). based on Dirac's notion of a (covariant) aether. presumed to be a real physical (stochastically fluctuating) field. mediating "real interactions among particles" induced by the so-called "quantum potential" Q presumed to act at any distance. Vigier's interpretation (like Bohm's [2]) fails to provide a local explanation of the quantum statistical correlrtions exhibited in the Einstein-Podolsky-Rosen (EPR) [3]Bohm (B) [4] ideal experiment. In place of this hiatus, and deviating from the steps of de Broglie, Einstein, Schrodinger et al., Vigier (like Bohm) advances the notion of interactions among particles at rrny distance, now referred to as "nonlocal interactions" induced by the "quantum potential" Q, as a necessary assumption for a "nonlocal yet causal" explanation of the EPRB ideal experiment. 525 R.L. Amoroso et a1 (eds.),Grmitationand Cosmc~logy:From the Hubble Radius to the Planck Scale, 525-536. O 2002 Kllnver Acc~cienzicPublislrers. Printed in the Netlrerland~.

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Four questions arise here: (1) Do particles have timelike trajectories in space-time?; (2) Are "nonlocal interdctions" a necessary assumption?; (3) Are "nonlocal interactions", if they exist, compatible with special relativity?; (4) Do such "nonlocal interactions" exist ?. Our arguments in support of our replies to the four questions are as follows. Our reply to question (1) is: Yes. We have shown [5] that the demonstration of conservation of momentum as a theorem of TQFrequires that particles do have timelike trajectories in space-time. Furthermore, the recent results of Popper's proposed experiment [68] speak for particles having trajectories in space-time as Tw itself requires and against the Copenhagen interpretation of TQpwhich denies this (we will not elaborate further here). Our reply to question (2) is: No. The consistent theory Th(4i) gives a local and causal ("common cause") interpretation of the EPRB ideal experiment [9,10]. The theory Th(g shows that local action suffices to explain all that TQFpredicts for the EPRB ideal experiment and its experimental realization by Aspect et. al. [ll]. And so there is no need to introduce "nonlocal interactions" in order to explain the quantum statistical correlations exhibited in the EPRB ideal experiment. Whence, the assumption of "nonlocal interactions" is not necessary. Our reply to question (3) is: No. If assumed to exist. "nonlocal interactions" are not compatible with special relativity. In the Minkowski space-time W of special relativity. the "quantum potential" Q, (the subscript m in Q, stands for "Minkowski") is interpreted by Vigier [I] and Bohm et al. [12] as inducing influences ("nonlocal interactions") connecting spacelike-separated events, events which lie outside each other's light cones. As we have alluded in 110, p. 16361, it is not difficult to show that, if it exists, any influence connecting spacelike-separated events clashes with the causal structure (order) of events in W. The formal demonstration is as follows: According to Vigier et al. (Ref.1, pp.95-loo), and Vigier has recently confirmed this to us, Q, is a symmetric function on w. Since a function is a one-to-one relation, Q, is a symmetric relation on W. Let '(x)' and '(y)' stand for '(VXEW ) ' and ' ( V ~ E W)' respectively. The definition of a sym?rzetric relation is, in symbols, (x)(y)(Q,,,xy 3 Q,yx). In words, for any event x and any event y, if Q, holds between x and y, then Q, holds between y and x. By Zeeman's theorem [13], relativistic causality R is a partial ordering on wand, therefore, all (elements) events in %t must satisfy, inter alia, the antisy?rzr~zetric relation (x)(y)(x#y & Rxy S- ~ R y x ) ,where "1" is the negation sign. In words, for any pair of distinct events x and y, if x and y are causally connected (R holds between x and y), then y and x are not causally connected ( R does not hold between y and x). Now let Q,xy hold between a pair of distinct events x and y ( x ~ y ) Then, . by modus ponens, from Q,xy and the conditional (x)(y)(Q,xy* Gyx), we deduce Q,yx. Assume that the &-"causal connection" is consistent with the R-causal connection, that is, whenever Q, holds between x and y, R holds between the same x and y, in symbols, (x)(y)(Q,xy 3 Rxy) or, equivalently, (x)(y)(lRyx+ lQ,yx), where this equivalence is deduced by contraposition and by an alphabetic change of the Rxy), bound occurrences of x and y. By modus ponens. from Q,xy and (x)(y)(Q,xy we deduce Rxy. By modus ponens, from xicy and Rxy and (x)(y)(x#y & Rxy a ~ R y x ) ,

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. modus ponens, from 1Ryx and (x)(y)(lRyx a ~ Q , y x ) , we deduce we deduce ~ R y xBy 1Qmyx.Whence, from the assumption that Q,,,xy holds between a pair of distinct events x and y and from the assumption that the a-"causal connection" is consistent with the Rcausal connection, we deduced both Q j x and lQ,,,yx, an obvious contradiction. 1 As a consequence, if the "quantum potential" Qmis interpreted in Vigier's sense as inducing "nonlocal interactions" connecting spacelike-separated events and, if as Vigier [I] and Bohmet. al. [12] assume such "nonlocal interactions" exist, then such "nonlocal interactions are not be compatible with the causal structure (order) of events in W. We are aware of the claim by Vigier et. al. (Ref.1, pp.95-100) that such "nonlocal interactions" satisfy Einstein's causality. However, their attempt to establish their claim nowhere addresses, nor considers, nor brings in explicitly the (underlying) causal structure of events in W ,And so their attempt apparently fails to establish their claim. We are also aware of suggestions in the literature [12] that such "nonlocal interactions" do not manifest themselves at the statistical level in the shape of a signal being exchanged faster than light and thus no relativistic prohibitions are violated. However, such suggestions do not address the real issue. In reply, we note that our (coordinate-free) demonstration above is not only independent of any signalling assumption, but also shows that the conflict with relativistic causality lies deeper than the statistical level. It lies at the level of individual events and of their particular outcomes in Wwhere, if it exists, the bbnonlocalinteractions" are presumably at work by exerting changes at any distance in the physical properties attributed to individual particles located in spacelike-separated regions. There is no 'peaceful coexistence" [14] between special relativity and such "nonlocal interactions", if they exist. And, as we have alluded (Ref.10, p.1637), for any realistic interpretation of TQF, such "nonlocal interactions", if they exist, lead to an iirlpasse which can only be resolved by experiment. Fortunately, such an experiment is now at hand: Popper's experiment shows that such "nonlocal interactions" do not exist. Our reply to question (4) is: No. Popper's experiment, which is essentially Aspect's experiment with slits rather than with polarizers, speaks against Vigier's prediction that the scatters of individual particles would be correlated by "nonlocal interactions" induced by the "quantum potential" Q. That is, when the particle moving to the left scatters upwards, the particle moving to the right scatters downwards since, according to Vigier, the measurement by the left slit on the left particle has a "nonlocal" influence on the right particle, even with the right slit removed. In a discussion with Popper (in May 1983), Vigier argued that his prediction would correspond to the situation in the Bohm version (measurement of polarization or spin rather than position) of the EPR experiment. In Vigier's own words [15]: Now the question is: Are these rizeasurerrlents correlated Therqore, also in your [Popper's] experiment there will be nonlocal or not ?.. correlations between the two photons. Ifone slit twists the spin of one photon, there will be an irrlirlediate action on the otherphoton going in the opposite direction. This irrzplies that we have nonlocality... we have shown that the quantuin potential gives rise to an action at a distance between the two photons. 7

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THOMAS D. ANGELIDIS

In sharp contrast to Vigier's prediction. our prediction - based on TQPand Th(@ -was that Popper's proposed experiment would show a "null result" - no deflection of particle trajectories [16,17]. Nothing would happen. The right particle (photon) would go on undisturbed in the absence of the right slit B, precisely as locality and TQFrequire. The recent results [18] of Popper's experiment clearly show that nothing happens: the right particle (photon) goes on undisturbed in the absence of the right slit B. This "null result" shows that there are no "nonlocal actions" between the left particle (photon) and the right particle (photon) causing the right particle (photon) to scatter (to "twist") in the absence of the right slit B. Popper's experiment gives a striking demonstration of locality and of our prediction of a "null result" (we believe a long-standing bet of $5 with Vigier as to whose prediction would be falsified by Popper's experiment is now due!). Sudbery [19] (with Redhead [20] following) claims that "there must be some flaw" in Popper's deduction from the Copenhagen interpretation, namely, "narrowing slit A increases the spread of momentum at B". and that "this would be a strikiig nonlocal effect...shared by any interpretation that incorporates the projection postulate...". Sudbery's own deduction - "in order to implement the projection postulate" - is that "narrowing slit A does not increase the spread of momentum- at B because it is already ivlfinite ".So the "flaw" Sudbery attributes to Popper is that the "spread of momentum at B" is infinite rather than finite, and presumably the "spread of momentum at B" cannot possibly be further increased. But if so, then the actual "null result" obtained, almost "no spread of momentum at B" (cf. Fig.5, Ref.18). decisively refutes Sudbery's own deduction of an infinite "spread of momentum at B". Popper's "own conjecture" [7,8] was the same as our prediction of a "null result". It was essentially based on prel nary work in the long quest to realize Einstein's hope of strengthening TQFintoa realistic and local theory (like Th(@). Popper was aware of our work in its most intimate details since its inception. and he steadfastly supported it to his last days. At the same time. he invited others to apply to it the k i d of earnest critical scrutiny Popper himself had applied to it in all sincerity. Popper's invitation has so far been met with an (uneasy, perhaps) official silence, broken only by a few idle rumours and fleeting whispers which, as far as we know, have not even begun to take the shape of an argument [21]. The need to carry out Popper's proposed experiment and its crucial importance, inter alia, for the issue of locality vs. nonlocality was for almost 20 years summarily dismissed by most physicists nurtured in the Copenhagen doctrine as being part of "rubbish of a most stimulating k i i [22]. Setting aside such disrespectful remarks, it turned out that what had been relegated to "rubbish" was an unremgnised nugget of wisdom waiting to be retrieved. It was our unwavering promise to Popper to see that his proposed experiment was carried out (knowing how close it was to his heart). It required a lot of dedication and perseverance to overcome the stagnant, if not hostile, initial situation. Our arguments persuaded some experimental physicists to suspend their disbelief (albeit not for long), weaned them from some mistaken criticisms of the proposed experiment, steered them clear of what was mistakenly considered to be Popper's proposed experiment. and tutored them to understand and focus on the real issues involved in Popper's experiment [23].

LOCAL THEORY Th(g AND POPPER'S EXPERIMENT

529

This eventually led to the realization of the proposed experiment almost as intended. Yet. even after the experiment was done, a lingering disbelief has remained as is evident from the statement: "Indeed, il is astonishing to see that the experirnenhl results agree with Popper'sprediction. " [18]. Why the astonishment? In a real local world, as envisaged (say) by the theory Th(9, this is as expected. Rather it is the Copenhagen doctrine that cannot come to terms with our prediction of a "null result" which gives rise to the exclaimed astonishment at, and the lingering disbelief in, the experimental results actually agreeing with Popper's prediction. Notwithstanding the results, and without any explanation, later the authors suddenly reverted to the trail of mistaken attributions to Popper, and asserted that [18]: "Popper and EPR made the same error...". What "error" ?. The "error" of allegedly confusing the "twoparticle entangled state" with the "state of two individual particles". We find nowhere an argument in support of this allegation except some rather dubious assertions that "the conditional behaviour" of a particle "in an entangled two-particle system is different" and that "the uncertainty principle is not for 'conditional' behaviour". We find it difficult to decipher such assertions, let alone argue either for or against them (incidentally, the authors denied our request for a note in their paper [18] disclaiming any implied agreement with their assertions). Instead of their attempt to explain the results away (2 la Copenhagen with nothing more on the menu than "click-click" measurements), we invite the authors 1181 to make the earnest effort (like Sudbery [19]) to enlighten us with a clearly formulated argument that could at least be sufficiently understandable to enable an assessment.

2. A Local Extension of Tap: A Verdict on "Impossibility Proofs" The EPR argument is that either we must give up locality or we must admit the incompleteness of the theory of the quantum formalism TQF[3]. We strengthened TQFby adding four postulates to it (Section 5). And by making manifest that the extended theory Th(9 is local we thereby showed [9,10] that the (unextended) theory TQFis incomplete as the EPR argument would have it. The strengthening of To, affects only the universe of discourse of TQFby extending the range of its variables in a demonstrably consistent way. The consistency proof proceeds h(9in the model-theoretic sense [lo]. The by giving a model g o f the extended theory T theory Th(@ is a proper extension of TQF,in symbols. TQFc T h(9(proper inclusion). This strengthening of TQFseems to be rninimcrl. The suggested "impossibility proofs" of a consistent local extension of To, not only fall short of their stated goal, but are apparently refuted in the presence of the consistent local theory Th(fl. In the class of "impossibility proofs" belong the Bell "impossibility proof' 1241, the Kochen-Specker (KS) "impossibility proof' [25], and the GreenbergerHome-Zeilinger (GHZ) "impossibility proof" [26], and their v a r i i t s [27]. We know that the KS argument adds to TQFcertain formal constrcrints, which not only cannot be deduced from TQF(and, therefore, do not belong to TQF),but also apparently go

530

THOMAS D. ANGELIDIS

well beyond the three formal postulates of locality (or locality constraints) enunciated by Bell et. al. (Section 3). Setting aside the issue of the unclear physical significance of the KS constraints, we know that the KS theory, obtained by adding the KS constraints to TQF,is inconsistent. Now an inconsistent set of statements, say, CD are just those which have ever?. statement A as a consequence (Q, is inconsistent w 0 1 A, for every statement A). So from the inconsistent KS theory one could deduce locality and nonlocality too (and perhaps, if one likes, that the Moon is made of peen cheese). It is almost trivial that the inconsisten! KS theory is stronger than the weaker consistent theory T h ( 9 (obtained by adding the locality constraints to TQF).Ln other words, the KS "contradiction" cannot be attributed to locality. And locality. as incorporated in the weaker consistent theory Th(fl, is safe and well. The theory T h ( 9 and locality cannot possibly be the target of the KS "impossibility proof, let alone be rehted by it. In the same vein, T h ( 9 and locality cannot possibly be the target of the GHZ "impossibility proof, let alone be refuted by it. We therefore question the validity of the claims [27] that these "impossibility proofs" refute locality. It is not difficult to show (we shall not do so here) that there is a common flaw in the arguments leading to the KS and the GHZ "impossibility proofs". The flaw lies in a certain misconception of the link between probability functions and semantic notions. Bell's argument is based on the socalled "violated Bell inequality", call it X, and purports to show that TQFitselfis nonlocal. In Sections 7 and 8. we show that g T h ( 9 and, in the presence of TQFc 'I%(@, that e TQpWith this, Bell's argument fails and so does his "impossibility proof". And with it fails Bell's conjecture of nonlocality. As a consequence, all experiments based on do not establish that TQFitself is nonlocal since does not belong to TQF, g TQ, Apparently, all such experiments have altogether missed their intended target, namely, locality. On the other hand, there is a proper Bell inequality z deducible in Th(41). In other words, Th(41)satisfies the inequality Z.We have called [28,29], and here we call again, for an experiment to test the validity of the proper Bell inequality z E Th(41), and submit it as a crucial test against the apparently false yet spellbinding conjecture of nonlocality. Before Bell the most one could assert was that the theory TQ, itself was silent on the issue of locality. In fact, Einstein was right never to have claimed that TQFwas nonlocal. But Einstein did claim that the Copenhagen interpretation was nonlocal. Einstein clearly did not identify To, with its Copenhagen interpretation. Einstein drew this distinction in order to use his Principle ofLocal Action against the Copenhagen interpretation and not against TQF.Einstein's distinction is now affirmed by the fact that z E Th(41)and z E TQF. What the Copenhagen interpretation adds, inter alia, to TQFis von Neumann's famous projection postulate asserting an 'acausal' state transition ('acausal' in the sense of not falling under the remit of a state-evolution equation, say, the Schrodinger equation), also referred to as "collapse of the wave packet" or as "state reduction". It is the addition of this postulate that renders the Copenhagen interpretation nonlocal in the sense of action-at-adistance and (historically) gave rise to the spellbinding conjecture of nonlocality.

x

x

x x

x

LOCAL THEORY Th(g AND POPPER'S EXPERlMENT

531

In the presence of TQFc Th(@, not only the consistent local theory Th(@, but also TQFitself is apparently incompatible with the addition of von Neurnann's projection postulate and its Copenhagen interpretation. And, as hinted in our reply to Sudbery (Section 1). the "null result" of Popper's experiment not only gives a striking demonstration of Einstein's principle of locality, but also refutes von Neurnann's projection postulate and with it the nonlocal Copenhagen interpretation. For a non "null result" (deflection of particle trajectories) would be the prediction of any interpretation that adds to TQFthe projection postulate (as Sudbery apparently admits).

3. The Formal Postulates of Locality Bell et al. [24,30] enunciated three conditions for locality. We re-state them here in the shape of three formal postulates of locality which any theory T, with some of its postulates expressed by the quadruple < bl,6,,p,A> of specified functions, must satisfy if T is to qualify as a local theory in the sense of Bell et al. ( L l ) Any joint probability function p,, must be defined as a specified instance of the syntactical form p,,(a,P):= Lpfi) bl(h,a)6, (h,P)dh. where any specified function 6, must not depend upon the variable P, and where any specified function 6, must not (h,a)C,(h,P) is known as depend upon the variable a.The product form b12(h,a,P):=6, the "factorixibilitycondition ". (L2) Any specified function p must not depend either upon the variable a or upon the variable P. (L3) Any specified range A of the variable h must not depend either upon the variable a or upon the variable P. Note well that (Ll) stipulates that the values of the functions 6, and 6, must be bounded by 0 and 1, as probabilities should be. And note well that 6 2 ) does not exclude the possibility that the function p may be chosen to depend upon some other variable, say 1.1provided , p (like 3,) is a variable distinct from both variables a and P.

4. Bell's Conjecture of Nonlocality Bell's "impossibility proof', which we shall here call Bell's conjecture of nonlocality, purports to show "the incompatibility of any local hidden variables theory with certain quantuln illechanical predictions" [3 11. We shall slightly sharpen Bell's own formulation here. Let T be a theory with some its postulates expressed by the quadruple < bl,c2,p,12>,where A is the range of the

532

THOMAS D. ANGELIDIS

c2,

are specified functions variable h, p is a specified function defined on &and 6,, defined on A x D, and D is the range of the variables aand P. Let (Ll)r\(L2)r\(L3) be the logical conjunction of the three formal postulates of locality where the symbol "A" stands for the (truth functional) c:onjunction.Then, Bell's conjecture of nonlocality asserts that:

There exists NO consistent theory T whose postulates < bl, ,p,A> satisfy (Ll)r\(L2)~(L3)and such that:

c2

holds. Or, in Bell's [24] own words. the QF probability function P,:

defined by

"cannot be represented, either accurately or arbitraiily closely, in the forrrl of Eq(1)".

5. The Postulates Added to the Theory TQF By a theo~p(in some formal language) we mean a set of sentences (well-formed formulae, abbreviated to "wwf") which is closed under deducibility, that is. such that for each sentence (wff) 4 , if T 1 0, then 0 E T. A subset I'of a theory T is called a set of postulcrtes for T if I' 10 for every (P E T. be the set of postulates of the quantum formalism (QF). Then the theory of Let rQF the quantum formalism TQFis the set defined by: TQF= { (P I (P is a sentence of QF and rQF 1(P ). We have added to TQFthe four following postulates n , ,

n,, n,, IJ, and gave a model

9thereby establishing the consistency of the extended (or strengthened) theory T h ( 9 [lo]. We have shown that the theory T h ( g is a proper extension of TQFsince there are sentences, say, z which belong to T h ( g but not to TQF,in symbols, TQFc T h ( g (proper inclusion).

The added postulates n,, ll,, ll,, 114manifestly satisfy (Ll)n(L2)r\(L3). Their physical interpretation has been given elsewhere [9,10]. By the rules of substitution, the range of the variable p must be M:= {N --< p < +-). The values of b, and 6, are

LOCAL THEORY Th(g AND POPPER'S EXPERIMENT

533

bounded by 0 and 1. as probabilities should be. This answers Feynman's "jundamental problem" [32]. The theory Th(@ is consistent, as the theory of the model 4 is always a consistent theory (Ref. 10, p.1652). And from Th(@ one deduces [9,10] a family of functions {pr21piM) which converges uniformlj to a unique limit function identical with the QF function for Va,P E D, given by Eq(2), as the syntactical form of Eq(1) precisely requires. If we understand Bell's own words correctly, the possible existence of a family of functions endowed with this property was denied by him. Whence, the consistent theory Th(@ refutes Bell's conjecture of nonlocality. In the presence of n3, the sum of products of probability amplitudes in QF, corresponding to the two mutually exclusive alternatives involved in the entangled state lyl,y2>, can be transcribed into the sum of products of conditional probabilities in n(fl for the alternatives in question with the predictions of QF preserved. This is seen from:

pE

deduced from (Ll) using the postulated distribution ppgiven by I&. Eq(3) shows that each pr2 belonging to {pr21pczM) can be written as the sum of two real-weighted products 6,of conditional probabilities corresponding to the two mutually exclusive alternatives [9,10]. Furthermore, Eq(3) shows how Bell's hypothesis of "local causalitl." or of "no action at a distance" [33], formulated as the 'yactorizahilify condition " (Ll), is satisfied in Th(@. Incidentally, from Popper's [34] formal theory of (conditional or relative) probability, all the theorems of Boolean algebra can be deduced (Kolmogorov's theory of probability fails to do this). And Boolean algebra can of course be interpreted in many ways including, if one likes. "classical" ways. As a consequence, we have questioned [35] the validity of a 'general proof' by Deutsch and Ekert purporting to have established that entangled quantum states "genemlly have no chssical analogue", that is, without exception. For, if one likes, Q(3) can be interpreted as the "classical analogue" or counterpart in Th(fl of the entangled quantum state lyl,y,r.

cl

6. The Proper Bell Inequality z

E

Th(9)

Proposition: Th(@ 1z (w g 1z ) Proof . - . .I The proper Bell inequality (first-order sentence) z is a theorem of Th(@ or, equivalently, 9 is a model of z. The proper Bell inequality z is deduced from the conjunction of n=KI,A&^~,AII, together with an arithmetical lemma [30] and reads:

534

THOMAS D. ANGELIDIS

where the symbols a,, a,, PI, P, occurring in z are individucrl constants which the structure $?maps to designated individuals objects (numbers) of the domain G of 4 (i.e. a,gis a real number etc.) [36]. This mapping under gmust be distinguished from a value assignment 8 in$? which does not affect the denotation of individual constants [%I. The designated individual objects (numbers) are determined by 4and are here interpreted as the distinct directions of the settings of the polarisers. We have shown [28,29] that in T h ( g all the values of the QF probability function satisfy the proper Bell inequality z E Th(9,

pz

7. The "Violated Bell Inequality"

x

tic

Th(g)

Now a more pressing question may be the following: Does the so-called "violated Bell inequality", call it X,belong to Th(@ ? Reply: No. does not belong to Th(4i). Proposition: tic Th(@. Proof: Let 9, be the structure that maps the individual constants a,,a,, PI, P, to the designated individual objects (numbers) of gm called [371 the "mxirnurn violation vcilues". Then, in $ ,, the false propositional sentence ("violated Bell inequality"):

x

x

x

would be deducible from the true Bell sentence (inequality) T (z E Th($,,,)i ) there existed a value assignment 8 in $? such that the variable p is assigned the value pl= a, in the first two terms occurring in z AND p is assigned the value pl= a, inthe last two terms occurring in z with a, + ol, (due to the distinct settings of each polariser), that is: 8

:

p, = a,

AND p, = a,

with a, # a,.

(4)

Going from right to left in Eq(4), we would then have p., # p, under one and the same value assignment 4 in $? which is impossible as no number is different from itself, an obvious contradiction. Thus, no such value assignment 8 in $? exists. And since the consistent theory Th(gm)is closed under deducibility, it follows that Th(qJ.1 What went wrong ? Bell's "substitution" leading to the false sentence ("violated Bell inequality" 4 Th(q,)),distinct from the true Bell sentence proper Bell inequality z E Th(%)), would amount to the existence of a value assignment 8 in $? under which a number is different from itself. We would respectfully submit that such an absurdity as Bell's "substitution" has nothing to do with locality or with the quantum formalism. For arbitrary substitutions are not always admissible in derivations (in sharp contrast to proofs, where the prernisses are logically true): A derivation cannot admit a step which depends on an arbitrary substitution because an initial formula generally does not logically imply a substitution variant of itself [38] (we will not elaborate on this point further in this paper).

x

x

x

LOCAL THEORY Th(g AND POPPER'S EXPERLMENT 8. Bell's Argument Based on

x Fails

What does the proposition X Th(g',) tell us about Bell's argument purporting to show that the theory of the quantumformalism TQFitselfis nonlocal ? Proposition: e TQF. Proof: Since TQFc Th(gand X, e Th(g, it follows that x 4 TQp( Since x tE TQF,Bell's argument fails to show that To, itselfis nonlocal. And with it fails Bell's 'impossibility proof' and his conjecture of nonlocality - this conjecture having been already refuted by the theory Th(fl. As a consequence. all experiments based on x do not establish that Tw itself is nonlocal since # does not belong to TQF, e TQp Apparently, all such experiments have totally missed their intended target, namely, locality. Fortunately, not all is lost. Although the "null result" of Popper's experiment gives a striking demonstration of Einstein's locality. we call for an experiment to test the validity of the proper Bell inequality z E Th(fl and submit it as a crucial test against the apparently false conjecture of nonlocality, should there still be a lingering disbelief in locality.

x

x

Keferences 1. 2. 3. 4. 5. 6. 7.

8 9. 10. 11. 12. 13. 14. 15. 16.

17.

18. 19.

Vigier, J.P.. in Jean-Piewe Vigierarrd the Stochastic Interpretation of Quantum Mechanics, eds., Jeffers, S., Lehnert. B., Abramson, N., and Chebotarev, L., Apeiron, Montreal, 2W. An anthology of Vigier's papers. Bohm, D.J., Phys. Rev. 85, 166, 180 (1952). Einstein, A., Podolsky, B., and Rosen, N., Phys. Rev. 47, 777 (1935) and Einstein, A., in Albert Einstein: Philosopher-Scientist, ed. Schlipp, P.A., La Salle (3rd edn.), Open Court, 1970, pp. 81-87. Bohm, D.J., Quantum T h e o ~Prentice-Hall, , Englewmd Cliffs, NJ, 1951, p1, 611-623. Angelidis, Th. D., Fouild. Phys. 7, 431 (1977'). Popper, K.R., Quantuin Theor?.and the Schism in Physics, Hutchinson, London, 1982, pp. 27-30. Popper, K.R., in Open Questions in Quantuin Phy.sics, DReidel Publishing Co., Dordrecht, 1985, pp. 5-11. Popper, K. R,in Determinism in Phyics, eds., Bitsakis, E. et.al., Gutenberg, Athens, 1985, pp. 13-17. Angelidis, Th D., Proc. Athens Acad. 66, 292 (1991). Angelidis, Th. D., J. Math. Phys.34, 1635 (1993). Aspect. A., Dalibard J., and Roger, G.. Phys. Rev. Lerr. 49. 1804 11982'). Bohm. D.J., and Hiley, B.J., The Undivided Universe, Routledge, London. 1993. Zeeman, EC., J. Math. Phys 5, 490 (1964). Shimony, A., Search for a Naturalistic World View, Vol. 2., Carnb. Univ. Press, 1993, pp. 151-1.54. Vigier, J.P., in Open Questions in Quantum Physics, D-Reidel Publishing Co., Dordrecht, 1985, pp. 26-27. Popper's dedications (Ref.17) to me were, interalia, in recognition of my contributions in our discussions towards a sharper formulation of his proposed experiment: (a) my proposal for the removal of one of the two slitted screens, which I considered to be a crucial test for a "null result" (no deflection of particle trajectories), and (bj that a "point source" was not necessary to carry out the experiment (only the individual emission events need be localized, as they actually are, within a somewhat extended real source j. In his book, Quanhim Theor?.and the Schism in Physics (Ref.6), Popper wrote by hand (as always): "With all good wishes for Thornas Angelidis from Karl Popper, 7-12-83, a dare at which this book wus super.seded". In his copy of the Proceedings of the Bari Workshop (May 19831, Open Que.stions in Quantum Physics (Ref.7). Popper wrote: "To Thoinas from Kurl infriedship. March 15, 1985". And on one of his papers, Popper wrote: "For Thornas, with adinlrarion and love, from Karl, 4-2-1987". Kim, Y.H., and Shih, Y.H., quant-ph/9905039 v2, 19 October 1999, available from the Web site: http:/lxxx.lanl.gov, to be published in Foundations of Physics. Sudbery, A., in Microphysical Realit?. and Quantum Fonnalisrn, Vol. 1, eds., van der Merwe et. al., Kluwer Academic Publishers, Dordrecht, 1988, pp. 267-277. In the quoted text, we interchanged 'A' and 'B' for agreement with Refs. 6-8,18. Also see references therein to other similarly mistaken arguments.

536

THOMAS D.ANGELlDIS

20. Redhead, MLG., in K. Popper: Philosophy ard Problenzs, ed. O'Hear, A., Cambridge University Press, Cambridge, 1995. pp. 163-176. Here Redhead claims (p.168) "the flaw in Popper's argument is that he misunderstands the nature of the EPR correlations" and refers to Sudbery's (Ref. 19) "critique..making broadly similar points". Redhead also writes (p.163) that "my p a t regret is that he cannot respond to this paper with criticisms of r~zyarguments!". The results (Ref.18) of Popper's experiment decisivel?. rejiute Redhead's own prediction, and we believe this would have been Popper's reply to Redhead 21. In Ref. 20, Redhead refers (pp. 175-176) to Popper's "suppolt to the work of Thomas Angelidis". The rest of Redhead's story calls for some Frank clarifications. (A) In a rather extensive correspondence (from February 94 to April 94), Popper did reply in detail to Redhead's comments on my work (Ref.10). (B) Redhead does not mention Popper's letter of 16 April 1994 where Popper wrote "both Thomas and I should very much like" Redhead to write up "a short critical papet" and submit it to J. MatlzPhys. together with my reply. That was more than six years ago and Redhead has yet to do it. (C) As for Redhead's comments: (i) he multiplied my postulate n,- with an ad hoc expression, which he had not realized its value was zero, and he called it a "generalization"!; and (ii) he asserted that a substitution of one variable for another amounts to an identity. Presumably, this may explain Redhead's reluctance to publish his comments on my work (Ref.10). Also see Note Added in 2002 below (Ref. 39). 22. Mermin, N.D., Boojurns All the Way Through, Cambridge University Press, Cambridge, 1990, p. 195. 23. Shi,Y.H., private cor~zr~tunication(e-mail, 20 October 1997), "Dear Thomas,...I remember these discussions and thank you for teaching me about Professor Popper's experiment. One of my student(s) started that experiment two weeks ago. It is not an easy experiment...I will keep asking you questions about Popper's experiment..". Yet my essential contributions (described as "important suggestions" [I 8]), which played a key role in bringing about the realization of Popper's experiment, remain unacknowledged. 24. Bell, J.S., Physics (N.Y.) 1, 195(1964). 25. Kochen, S., and Specker, E.P., J. Muth. Mech. 17,59-87 (1967). , Conceptions of 26. Greenberger, D.M., Home, M.A., Zeilinger, A., in Bell's Theorei~z,Quantur~tT h e o ~urul the Univer,se,ed., Kafatos, M., Kluwer Academic Publishers, Dordrecht, 1989, pp. 73-76. 27. Mermin, N.D., Php. Rev. Len. 65, 3373 (1990). 28. Angelidis, Th. D., in New Develop~zents on Fundcirtzental Problerns in Quuntur~tPhysics, eds., Ferrero, M., and van der Merwe A., Kluwer Academic Publishers, Dordrecht, 1997, pp. 17-29. 29. Angelidis, T k D., in Causalig ardlocalig in Modern Physics, eds., Hunter, G., Jeffers, S., and Vigier, J.P., Kluwer Academic Publishem, Dordrecht, 1998, pp. 451-462. 30. Clauser, JF-, and Home, M.A., Phys. Rev. D 10, 536 (1974). 31. Shimony, A., Found Phys 19, 1426 (1989). 32 Feynman, R.P., Intl. J. Theor. Phys 21,467 (1982). 33. Bell, J.S., J. de Phys. (Paris) 42, C2, 41 (1981). 34. Popper, K-R., The Logic of Scientific Discovery, Hutchinson. London. 1972 (6th impression). p. 329 and Appendices *iv and *v. 35. Angelidis, Th D., in Mysteries, Puzzles and Partlcloxe.s in Quantur~zMeclzanics, ed., Bonifacio, R, American Institute of Physics (Conference Proceedings 461), New York, 1999, pp. 255-259. 36. Bell, J.L., and Machover, M., A Course in Mather~mticalLogic, North-Holland, Amsterdam, 1977, pp. 10, 49-50, 162-163. 37. Clauser, J.F., and Shimony, A., Rep. Prog. Php. 41, 1881 (1978). 38. Camap, R., Introduction to *r~zbolic Logic, Dover, New York, p. 49. 39. Note Added in 2002: I take Redhead to have laid to rest his two stillborn comments (Ref. 21), since he nowhere upholds them in his later joint paper [J. Math-Phys. 40. 4290 (1999)I.The thesis of the third comment is also stillborn since the alleged counterexample is not a counterexample. The proof of uniform convergence rests on the inequality I py2(a, @)- p E ( a p) l S %Ip-a I valid for Vp E Mand Va,P E D. (Ref. 10, p. 1651, line 3). Substitute the given "counterexample" values, namely, "for any p let CX= pW4 and let p= p7d4". Then I H2(a,@)- p K ( a P) I = I 1/44)( = 114 and 1/2 1 p-011=1/2 1 -d4 1 = d B . Since 114 5 d8, their alleged "counterexample" values clearly satisf the inequality and thereby satisfy the sentence Z (the formal definition of uniform convergence). d e n c e , the sentence Z is true. What has gone wrong? Redhead and Melia would have established their presumed failure of uniform convergence had they succeeded to show (but did not) that (the negation of Z) is true, where is (3~>0KVrl>OK3p~ W(39p~ DX [ ( I p-a I c rl) v (I P-P I c rlM A I #2(aJ P)- pZ(a, P) I 2 €1 . And the irony is that their alleged "counterexample" values show that 1 Z is false! Upon instantiation, from 4one deduces (Vq>O)[ (d4

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