Fluctuating Paths and Fields: Festschrift Dedicated to Hagen Kleinert on the Occasion of His 60th Birthday

Fluctuating Paths and Fields Festschrift Dedicated to Hagen Kleinert Editors

W. Janke, A. Pelster, H.-J. Schmidt & M. Bachmann

World Scientific

Fluctuating Paths and Fields Festschrift Dedicated to Hagen Kleinert

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Fluctuating Paths and Fields Festschrift Dedicated to Hagen Kleinert on the Occasion of His 60th Birthday

Editors

Wolfhard Janke Universitdt Leipzig, Germany

Axel Pelster Freie Universitdt Berlin, Germany

Hans-Jurgen Schmidt Universitdt Potsdam, Germany

Michael Bachmann Freie Universitdt Berlin, Germany

V

P

Q World Scientific

•

Sinaapore • New Jersey • L Singapore London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Fluctuating paths and fields : festschrift dedicated to Hagen Kleinert on the occasion of his 60th birthday / editors, Wolfhard Janke . . . [et al.]. p. cm. Includes bibliographical references and index. ISBN 981024648X(alk. paper) 1. Path integrals. 2. Quantum field theory. I. Kleinert, Hagen. II. Janke, W. (Wolfhard) QC174.52.P37F58 2001 530.14'3-dc21

2001026287

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover design by M. Bachmann

Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Printed in Singapore.

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Preface

The subject of this volume embodies a wide variety of research topics whose theoretical methods are based on path integrals, or more generally Fluctuating Paths and Fields ~ a heading which reflects best the numerous research areas of Hagen Kleinert to whom this Festschrift is devoted. The selection of authors and articles is guided by the difficult task of presenting a cross section of his research activities. With a few prominent exceptions, the contributors to this volume are his former and present students, postdocs, co-workers, or colleagues who have enjoyed working with him. With their articles they wish to celebrate his 60th birthday on June 15, 2001. We are especially grateful to the Physics Nobel Laureate Gerard 't Hooft for delivering the birthday colloquium at the Freie Universitat Berlin. a The first part of the book contains recent advances in the path-integral description of quantum physics, covering and supplementing a subset of the topics discussed in Hagen Kleinert's widely-known textbook on Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics. Among Kleinert's many contributions to this field, the best known is his path-integral solution of the three-dimensional hydrogen atom based on non-holonomic spacetime transformations in 1979. The clue to the successful treatment was his deep knowledge of the relation between three-dimensional Coulomb and fourdimensional oscillator systems which was established in the context of celestial mechanics by Kustaanheimo and Stiefel. In standard quantum mechanics, this relation is rooted in the existence of the dynamical group 0(4, 2) of the hydrogen atom which Kleinert discovered in collaboration with the late Asim O. Barut during the years 1965 to 1967, when working on his Ph.D. thesis at the University of Colorado at Boulder. The main ideas for the pathintegral solution are contained in a paper of 1979, written with his postdoc Ismail H. Duru. A fully consistent mathematical formulation, however, turned a

T h e abstract of this lecture is printed on p. xiii.

vii

out to be quite subtle. In particular, it required an understanding of the path integration measure in curved space-time with torsion. The complete solution of the problem was found by Kleinert in the late eighties. Generalizing the theory allowed him to calculate path integrals of all systems for which the Schrodinger equation can be solved analytically. A full account of the theory can be found in his textbook on path integrals. The impetus to tackle this fundamental problem of atomic physics goes back to Kleinert's discussions with Richard P. Feynman during a sabbatical at Caltech in 1973/74. As described in the Feynman biography by John and Mary Gribbin, "... Feynman had stopped teaching path integrals at a less advanced level, because he never derived a complete path integral description of the hydrogen atom, and was embarrassed by this failure. . . . Kleinert . . . not only solved the problem (much to Feynman's delight), but wrote a major textbook on the path integral approach, re-establishing path integrals as a research tool, not only conceptually useful but now capable of solving problems . . . " b Kleinert was invited to Pasadena by Murray Gell-Mann whom he had met before from 1968 to 1972 during various extended visits to CERN in Geneva, which was and continues to be one of the main centers for meeting important people (... such as his wife Annemarie). At that time, Gell-Mann was interested in a relation between current and constituent quarks derived by Kleinert and his colleagues at CERN, two of whom, Franco Buccella and Carlos A. Savoy, have written articles for this Festschrift. In Pasadena, Kleinert shared an office with Yuval Ne'eman whose pioneering work on SU(3) had paved the way to Gell-Mann's development of a quark field theory and his successful prediction of the fi~ particle. During that time Kleinert found a theoretical explanation for the algebra of Regge residues proposed by Ne'eman in collaboration with Cabibbo and Horwitz. This led to a deep friendship with Ne'eman, and even in his period as a Minister of Science and Technology in Israel, he found time to visit Kleinert in Berlin for discussing physics (see his contribution in this Festschrift). The articles in Part I cover topics ranging from rigorous definitions of functional integrals through applications to group spaces, quasi-classical approximations, and semi-classical dynamics to numerical Monte Carlo evaluations of path integrals, which Kleinert, while working mainly analytically, often enjoyed performing on his personal computer at home. One of the edib J . Gribbin and M. Gribbin, Richard Feynman: Books, London, 1997), pp. 215-216.

viii

A Life in Science (Viking Penguin

tors (W.J.) took part in a race of two good-old Atari computers against each other - one at the institute and the other at Kleinert's home, the latter PC, of course, turning out to be newer and thus faster . . . Part II of this volume collects articles on quantum field theory which has always been the backbone of Hagen Kleinert's research. In fact, this was one of the reasons why Werner Theis, then one of the senior professors of theoretical physics at the Freie Universitat Berlin, asked the 27 year-old Kleinert to join the faculty as an Associate Professor. After turning down two other offers for a full professorship elsewhere, he became a Full Professor in Berlin in 1976. Since then Kleinert, who grew up in Hannover and received his undergraduate education at the Technische Universitat, became a real "Berliner", refusing two further attractive offers from other universities in Germany and abroad. Kleinert's early applications of field theoretic methods include the introduction of composite fields via path-integral transformations in models of elementary particle physics which he called "hadronization of quark theories" and, by analogy, in models of superconductors and superfluid Helium. Field theoretic methods involving gauge fields play a crucial role in Kleinert's comprehensive work on defect-driven phase transitions which is collected in his two extensive monographs on Gauge Fields in Condensed Matter - Vol. I: Superflow and Vortex Lines, Vol. II: Stresses and Defects. In this context non-holonomic mappings are again of central importance. The articles in Part II discuss gauge theories, in particular quantum electrodynamics and quantum chromodynamics, and in some more detail 4-field theories, where Kleinert and his collaborators contributed most over the recent years. Variational perturbation theory and the resulting highly efficient scheme for resumming divergent perturbation series are discussed in Part III. This technique has its roots in a variational treatment of quantum-statistical partition functions in the path-integral representation developed by Kleinert in joint work with Feynman during his later sabbaticals at the Universities of California at Berkeley in 1979/80 and at Santa Barbara in 1982/83. After a considerable delay due to Feynman's illness, the manuscript was finally finished and submitted for publication in Physical Review during Kleinert's sabbatical stay at San Diego in 1985/86. This last period of intense scientific interactions with Feynman is also witnessed by the famous photographs of Feynman's last office blackboards. 0 The joint theory is an improvement of a 20 year-old less powerful approach found by Feynman himself. It is an amusc

Feynman's

Office: The Last Blackboards, in Physics Today 4 2 , 88 (February 1989). IX

ing fact of scientific history that, after such a long time, the same idea for an improved version was worked out independently also by Riccardo Giachetti and Valerio Tognetti in Florence in 1985/86. While subsequently the Italian group was mainly interested in applications to solid state physics, Kleinert pushed the method to higher orders and made it applicable to quantum field theory. Finally he arrived at an extremely powerful formulation of variational perturbation theory which may be viewed as a highly efficient resummation procedure for divergent, asymptotic perturbation series. Besides impressive applications in quantum mechanics and atomic physics, this has recently led to a successful treatment of 4-field theories in the strong-coupling limit. The results are collected in a monograph on Critical Properties of (f)4-Theories, written in collaboration with Verena Schulte-Frohlinde, who also contributes here. The articles in Part IV deal with phase transitions and critical phenomena, many of them centering around explicit calculations of critical exponents of 0 4 -theories with and without anisotropics or quenched, random disorder. They have important applications to phase transitions in magnets and the Atransition in liquid 4 He. Others are devoted to the related Ginzburg-Landau model and phase transitions in superconductors and superfluid 3 He where topological defects play a crucial role. Here, Kleinert's development of a disorder field theory dual to Ginzburg-Landau's order field theory has led to his discovery of the tricritical point in superconductors in 1982. By studying the limit of strongly bound electron pairs as the origin of high-Tc superconductivity he pointed out that pairing and phase decoherence may occur at different temperatures. In 3 He he found, together with Kazumi Maki who has written an article in this part, an interesting helical texture whose existence was subsequently confirmed experimentally. To explain the blue phase of cholesteric liquid crystals, Kleinert and Maki furthermore investigated icosahedral symmetries. Such structures were later discovered in sputtered aluminum and are now called quasicrystals. In between their joint calculations Maki especially enjoyed singing Italian operas with Kleinert, who knows a great number of them by heart. Part V comprises articles on topological defects and their role in phase transitions as well as on fluctuating strings and membranes. Defects play an important role in crystal melting, superfluidity, superconductivity, cosmology, and elementary particle physics as described in Kleinert's above-mentioned monographs. Several new developments are discussed there. As far as strings and membranes are concerned, the key point of Kleinert's research emphax

sized the structural similarity between real membranes and the surfaces swept out by color-electric strings which hold quarks together. This led to the by now famous Polyakov-Kleinert action for curvature-dominated membranes and strings. Many field theoretic techniques could be successfully transferred between these physically quite different fields. In particular, a fundamental constant which rules the so-called Helfrich pressure exerted by fluctuating membranes was thus found analytically for the first time in good agreement with earlier Monte Carlo simulations. The final Part VI deals with questions related to gravitation, cosmology, and astrophysics. Here Kleinert's research interests focused mainly around extending Einstein's gravitational theory in such a way that also the torsion of space-time can be taken into account. In close analogy to forces between defects in real crystals he developed a replacement and extension of Einstein's equivalence principle in the form of a non-holonomic mapping principle which permits deriving laws of Nature in geometries with curvature and torsion from those in flat space-time. In using non-holonomic mappings, Kleinert closed the circle between gravity, the theory of dislocations and disclinations, and his early path-integral solution of the hydrogen atom. In gravity, he also obtained interesting consequences for the early universe by deriving a gravitational action which differs from Einstein's at high curvature as a result of quantum fluctuations of elementary fields. The comprehensive work of Hagen Kleinert is witnessed by his four books and more than 300 research papers, all accessible on his internet homepage http://www.physik.fu-berlin.de/~kleinert.

His work cannot be fully reflected by the selected contributions in this Festschrift. It contains 65 articles arranged in each part in topical order and further linked together through an extensive index. We can only hope that in this way at least a flavor of Kleinert's diverse research interests and achievements becomes visible. We refrain from reproducing his current list of publications as it continues to grow rapidly and would be outdated at the time of printing. We are grateful to all authors for their immediate agreement to contribute an article. Many of them emphasize Kleinert's stimulating influence on their research. His Italian-, Spanish-, French-, and certainly English-speaking visitors all have an easy time discussing with him since he converses with them in their own language. The number of former and present students, postdocs, collaborators, and friends of Hagen Kleinert is so large that we were unable xi

to ask all of them to contribute to this book. We apologize to all those we have not been able to contact. Financial support from the HSP III program of the German Ministry of Education and Research (BMBF) is appreciated. We thank Sieglinde Endrias for reading the manuscript and spotting many printing errors, Renate Schmidt and Lea Voigt for secretarial help as well as Ms. E.H. Chionh for the pleasant negotiations with the publishing company. Finally, we acknowledge the constant interest and valuable advice of Dr. Annemarie Kleinert. She was very helpful in the editing process and brought several former collaborators of her husband to our attention.

Leipzig, Berlin, and Potsdam, April 2001 Wolfhard Janke Axel Pelster Hans-Jiirgen Schmidt Michael Bachmann Editors

xii

Special Lecture

How Does God Throw Dice? Speculations A b o u t Q u a n t u m Mechanics at t h e Planck Scale. of Nobel Laureate

Gerard 't Hooft Spinoza Institute, Utrecht University, Leuvenlaan 4> Postbus 80.195, 3508 TD Utrecht, Netherlands E-mail: [email protected] in honor of

Hagen Kleinert on the occasion of his 60th birthday on June 15th, 2001 at the Freie Universitat Berlin

A t t e m p t s to arrive at consistent theories combining q u a n t u m mechanics with general relativity not only require new concepts of space, time, and matter, such as the ideas t h a t lead t o superstring theory, D-brane theory, and M-theory, but they may also require a reconsideration of what q u a n t u m mechanics itself really is about. Although completely deterministic scenarios appear to be ruled out by the Bell inequalities, it is nevertheless worth-while t o investigate a set-up where we s t a r t with a deterministic theory and add t o this the notion of information loss. Models proposed so-far all show deficiencies of some sort which make t h e m unrealistic for describing the real world, but they do show how chaotic phenomena in a deterministic theory might be found t o lie at t h e basis of t h e q u a n t u m n a t u r e of our world.

xiii

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Contents

Part I: Path Integrals and Quantum Mechanics Semiclassical Quantum Mechanics: A Path-Integral Approach Barry R. Holstein

3

Forward-Backward Semiclassical Dynamics Nancy Makri

15

The Feynman Path Goes Monte Carlo Tilman Sauer

29

Coordinate Independence of Path Integrals in One-Dimensional Target Space Alexander Chervyakov

43

Perturbatively Defined Path Integral in Phase Space Michael Bachmann

57

The Use of Non-Relativistic Path Integrals in Field Theories Ismail H. Duru

69

Path Integration and Coherent States for the 5D Hydrogen Atom Nuri Unal

73

The Aharonov-Bohm Effect in Four Dimensions De-Hone Lin

83

Path Integral for Separable Hamiltonians of Liouville-Type Kazuo Fujikawa

95

Conjecture on the Reality of Spectra of Non-Hermitian Hamiltonians Carl M. Bender, Stefan Boettcher, and Peter N. Meisinger xv

105

Time-Transformation Approach to g-Deformed Objects Akira Inomata

117

Feynman Integral on a Group Zbigniew Haba

129

Characterizing Volume Forms Pierre Cartier, Marcus Berg, Cecile DeWitt-Morette, Alexander Wurm

139 and

Vassiliev Invariants and Functional Integration Louis H. Kauffman

157

Part II: Quantum Field Theory Higher Algebraic Geometrization Emerging from Noncommutativity 173 Yuval Ne 'eman Dynamical Fermion Masses Under the Influence of Kaluza-Klein Fermions in Randall-Sundrum Background Hiroyuki Abe, Tomohiro Inagaki, and Taizo Muta

185

The Ultraviolet Fixed-Point in Quantum Electrodynamics Adler Conjecture: Is QED Trivial? Raghunath Acharya

197

From Z Operator to SO(10), Neutrino Oscillations, and Fermi-Dirac Functions for Quark Parton Distributions Franco Buccella

201

A Tilt at Constituent Quarks Carlos A. Savoy

207

The Breaking of Isospin and the p-u> System Harold Fritzsch

219

Analytic Confinement and Regge Spectrum Garij V. Efimov

225

xvi

Recursive Graphical Construction of Tadpole-Free Feynman Diagrams and Their Weights in 4-Theory 235 Axel Pelster and Konstantin Glaum Recursive Construction of Feynman Graphs in Spontaneously Broken 0 (AT)-Symmetric 0 4 -Theory Boris Kastening

247

Critical Behavior of Correlation Functions and Asymptotic Expansions of Feynman Amplitudes 259 Adolfo P.C. Malbouisson Gauge Symmetry and Neural Networks Tetsuo Matsui

271

Part I I I : Variational Perturbation Theory Note on the Path-Integral Variational Approach in Many-Body Theory Jozef T. Devreese

283

Variational Perturbation Theory: A Powerful Method for Deriving Strong-Coupling Expansions Wolfhard Janke

301

Variational Perturbation Theory for the Ground-State Wave Function Axel Pelster and Florian Weifibach

315

Uniformly Suitable Estimation for Thermodynamic Values Ilya D. Feranchuk and Alexei Ivanov Path-Integral and Perturbation Methods for Debye-Waller Factors Observed by Extended X-Ray-Absorption Fine Structure Spectroscopy Toshihiko Yokoyama

xvii

327

337

The Highest-Derivative Version of Variational Perturbation Theory Bodo Hamprecht and Axel Pelster

347

Fast-Convergent Resummation Algorithm and Critical Exponents of 4-Theory in Three Dimensions 365 Florian Jasch Critical Exponent a of Superfluid Helium from Variational StrongCoupling Theory Bruno van den Bossche

377

Five-Loop Expansion of the 4-Theory and Critical Exponents from Strong-Coupling Theory 387 Verena Schulte-Prohlinde

Part IV: Phase Transitions and Critical Phenomena Nonasymptotic Critical Behavior from Field Theory Claude Bagnuls and Claude Bervillier

401

Nonanalyticity of the Beta-Function and Systematic Errors in Field-Theoretic Calculations of Critical Quantities Michele Caselle, Andrea Pelissetto, and Ettore Vicari

413

A Remark on the Numerical Validation of Triviality for Scalar Field Theories by High-Temperature Expansions Paolo Butera and Marcattilio Comi

425

Multiloop 0 4 -Theory at Criticality John A. Gracey

433

Phase Ordering Dynamics of 4-Theory with Hamiltonian Equations of Motion Bo Zheng, Volkard Linke, and Steffen Trimper Phase Transition in the Random Anisotropy Model Maksym Dudka, Rheinhard Folk, and Yurij Holovatch xviii

445

457

Gaps and Magnetization Plateaus in Low-Dimensional Quantum Spin Systems 469 Karl-Heinz Mutter Effective Free Energy of Ginzburg-Landau Model Adriaan M.J. Schakel

485

Scaling and Duality in the Superconducting Phase Transition Flavio S. Nogueira

497

Thermal Fluctuations in the Gross-Neveu Model with [/(l)-Symmetry at Small N 507 Egor Babaev FLEX-Theory for High-T c Superconductivity Due to Spin Fluctuations Dirk Manske and Karl H. Bennemann From Superfluid 3 H e to Triplet Superconductor Sr 2 Ru04 Kazumi Maki and Hyekyung Won

517

533

Part V: Topological Defects, Strings, and Membranes Description of Vorticity by Grassmann Variables and an Extension to Supersymmetry 553 Roman Jackiw Non-Equilibrium Worldline Duality in Condensed Matter Ray J. Rivers Field Theories and the Problem of Topological Entanglement in Polymer Physics Franco Ferrari The Role of Topological Excitations at Second-Order Transitions Luis M.A. Bettencourt

xix

565

577

589

Topological Singularities, Defect Formation, and Phase Transitions in Quantum Field Theory 601 Giuseppe Vitiello Confinement in the Ensembles of Monopoles Dmitri Antonov

613

Nambu-Goto String without Tachyons Between a Heavy and a Light Quark 625 Gaetano Lambiase and Vladimir V. Nesterenko Deconfinement of Quarks in the Nambu-Goto String with Massive Ends Vladimir V. Nesterenko and Gaetano Lambiase

635

Principles of Non-Local Field Theories and Their Application to Polymerized Membranes Kay J. Wiese

645

Random Paths and Surfaces with Rigidity Bergfinnur Durhuus

663

Part V I : Gravitation, Cosmology, and Astrophysics String, Scalar Field, and Torsion Interactions Richard T. Hammond Metric and Connection: Kinematic and Dynamic Solutions of the Space Problem Horst-Heino von Borzeszkowski and Hans-Jilrgen Treder

677

685

Asymptotic Freedom in Curvature-Saturated Gravity 697 Salvatore Capozziello, Gaetano Lambiase, and Hans-Jiirgen Schmidt What Can Ising Spins Teach Us about Quantum Gravity? Christian Holm

xx

707

Gravitational Excitons - Fluctuating Particles from Extra Dimensions Uwe Giinther and Alexander Zhuk

721

Path Integrals in Quantum Cosmology Claus Kiefer

729

Varying Light Velocity as a Solution to the Problems in Cosmology John W. Moffat

741

Renormalization Group Method and Inhomogeneous Universe Yasuada Nambu

759

Analogies, New Paradigms, and Observational Data as Growing Factors of Relativistic Astrophysics 771 Remo J. Ruffini Index

799

XXI

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Part I

Path Integrals and Quantum Mechanics

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SEMICLASSICAL Q U A N T U M M E C H A N I C S : A PATH-INTEGRAL A P P R O A C H

B.R. HOLSTEIN Department

of Physics,

University E-mail:

of Massachusetts,

[email protected].

Amherst,

MA 01003,

USA

edu

It is well known that one can approach problems in electrodynamics at either the macroscopic or microscopic level. However, we have less knowledge about the same approach in the case of semiclassical quantum mechanics. This case will be discussed here.

1 Introduction When dealing with simple optical phenomena such as transmission and reflection, one can proceed in two quite different but totally equivalent ways. One is macroscopic and utilizes the solution of the Maxwell equations subject to appropriate boundary conditions. The second is microscopic. It is based on the multiple scattering series and follows a simple ray through all possible paths to its final destination. This procedure offers a very visual and a much clearer intuitive picture for what is taking place. The equivalence of macroscopic and microscopic techniques in this case is well known. Less familiar to many physicists is that an equivalent microscopic approach can be used in the case of semiclassical quantum mechanics. This will be discussed here. Our results are implicit in the work of other authors [l], so the presentation will be more of a didactic nature. 2 Semiclassical Methods It is possible to treat quantum mechanics via macroscopic and/or microscopic methods, at least in the situation that the action divided by h is large - the

3

4

B R . Holstein

so-called semiclassical limit. The macroscopic procedure is well known as the WKB technique, the basic idea of which is that for a slowly varying potential V(x), a stationary state solution can be written as [2] i>WKB(x,t) ~ J~—expU(

k(x')dx'-

Et\\

,

(1)

with the local wave number k(x) = y/2m(E - V{x)).

(2)

This approximation is valid when refraction dominates over reflection, or equivalently when the change in potential energy over a distance of the de Broglie wavelength is much smaller than the local kinetic energy, i.e.

JdV/dxl X

E-V(x)^1

•

(3)

This approximation clearly breaks down at a classical turning point x = a, where E = V(a). The usual procedure to deal with this breakdown is to use the WKB form of the wave function except in the immediate vicinity of the turning point, wherein a linear approximation to the potential is used in order to match the form of the wave function. This technique is a standard one and a prototypical problem treated via WKB is that of barrier penetration, i.e. the case of a particle of mass m and energy E incident on a potential barrier V{x) with maximum height such that VmaK > E. The results of this procedure are well known and yield reflection and transmission formulae [3]

"-'-^(MW

via t(n)

I/ ++ Ir Jxi

Jb

dx dx

iIm

m

+D + l) f dx ' ^ '+i(2n + i(2n (E-V(x)) Ja \ 2(E-V(x)) Ja

"r p(v(x)-E)

(27) Here the turning points a and b are functions of E(x2,x\,t^), but A is independent of energy (and time). The Fourier transform is performed via the stationary phase approximation. Writing D(x2,t;x\,0) = p(t) e1^ (where we have suppressed the dependence upon X2, xi) we have />oo

/ dtei(-Et+^^p(t), Jo and the stationary phase point i is determined via D(x2,xi;E)=

d d ( fX2 0 = — {Et + <j>(t)) = — lEt+ / „ ~, . dE = E-E{t)-t—+ dt

(28)

k{x) dx - Et

X2

f , dk dE dx-^r — . JX1 dE dt

29)

However, t — JX2 dxdk/dE = 0, since this is the equation which defined E(t) in the first place. Thus the stationary phase point t is defined via E = E(t). Finally, using dE

^{Et+m)

-—J-—-= Jxi

ax

~d&

(m2/ v

dxk

Jxi

\x)\ J

(30) the stationary phase approximation yields 1/2

D(x2,xi; E) = I ————- ) ^k(x2)k(x1)/

exp \i I

k(x) dx > ,

(31)

where the integration is understood in the sense of Eq. (26). As the ft") are complex, the stationary phase procedure requires that the path from £ = 0 t o £ = +oois deformed into the complex plane in such a way that it passes through each i^ with n — 0 , 1 , 2 , . . . . We then find oo

D(x2,xl;E)

=

Y,D(nHx2,x1;E)

Semiclassical Quantum Mechanics: A Path-Integral Approach ,2

\ 1/2

x ^ A 2 7 l e x p l - ( 2 n + l) / /<x)ete

n=o

11

Ja

\

.

(32)

J

The contribution from paths involving n > 1 are exponentially suppressed and of questionable accuracy, as the corrections to the semiclassical approximation to the path integral could well be larger. Nevertheless, it is important to include the effects of these "interior bounce" solutions since only then we have a unitary and fully consistent picture of the transmission and reflection process which conserves probability. Thus calculating the propagator for transmission, as given above, we can perform the summation over n, yielding D(x2,x1 ;E)

( m2 \k(x2)k(xi);

N 1/2

i

X2

exp

k(x) dx Jx!

Jb

\2e-2o> (33)

where a =\f*K(x)dx. We see that this path-integral approach to the semiclassical approximation allows a very appealing and graphical picture of the transmission process. Instead of the sum over all possible paths required in calculating the complete path integral, the semiclassical approximation utilizes a sum over all "classical" paths connecting the initial and final points. (Here "classical" is used in the sense defined above, wherein analytic continuation into the complex time plane is permitted.) The form of the propagator can be found by using the simple rules: b i) Propagation from x\ to x2 in a classically allowed, forbidden region produces a factor exp \i

k(x) dx > ,

exp < — /

K(X)

dx > ,

(34)

respectively; ii) Reflection from a classical turning point within a classically forbidden region yields a factor A = i/2, as discussed in Ref. [7]. b

For completeness, it should be noted that there exist corresponding phases exp(±i7r/4) which arise upon transmission through a barrier. However, these do not play a role when the transmission or reflection probabilities are determined and therefore, for simplicity, will be omitted here.

B.R. Holstein

12

In order to deal with the corresponding reflection coefficient, we require one additional rule iii) Reflection from a classical turning point within a classically allowed region yields a factor rj = —i, according to Ref. [7]. The propagator for the reflection process may then be constructed by taking x\,X2 both to the left of the barrier and including all possible "classical" trajectories: i) Propagation from x\ to the left-hand wall at x = a followed by a reflection and propagation back to x^\ ii) Propagation from x\ to the left-hand wall at x = a followed by transmission into the barrier, reflection from the right-hand wall, transmission back to x — a and then propagation from x = a to x2, etc. The total contribution to the propagator is then D{X2 XliE)=

'

(k{xfk(Xl))

^{ijx+ijjk{x)dx^f

(35)

with

f = v+T,x2n+1 e~{2n+2)a =»»+ i - k - » "

(36)

n=0

where a is defined in Eq. (5). The connection with the transmission and reflection coefficients R=\r(E)\2

and

T =\t{E)\2

(37)

can now be made via the identifications

m2

\1/2

( m2 D(x„Xl;E)={k{x2)k{xi))

\1/2

f

D{x2,x1;E)=[k{x2)Kxi))

t(E),

x1«a,

r(E),

xux2«a,

and we find, using Eqs. (32) and (35), T=\t(E)\2

h_A2e-2a|2

( 1 +

le-2a)2'

b«x2, (38)

Semiclassical Quantum Mechanics: A Path-Integral Approach

R=\r(E)\2

=

11 _

_ A e _-2a 2

1-A
-00

2L

C

„^4_ ^L

(11)

derivable by making use of the cyclic property of the trace, is convergent of order 0(1/L4) and amounts to simply replacing the potential eV in (4) by an effective potential [32]

v

+

—" S(^'-

0 with j3 fixed or, equivalently, of L —> oo for local, importance sampling update algorithms, like the standard Metropolis algorithm, a slowing down occurs because paths generated in the Monte Carlo process become highly correlated. Since autocorrelation times diverge in simulations using the Metropolis algorithm as [35] TQ1 OC L Z with z s=s 2, the computational effort (CPU time) to achieve comparable numerical accuracy in the continuum limit L —> oo diverges as L x Lz = Lz+1. To overcome this drawback, ad hoc algorithmic modifications like introducing collective moves of the path as a whole between local Metropolis updates were introduced then and again. One of the earliest more systematic and successful attempts to reduce autocorrelations between successive path configurations was introduced in 1984 by Pollock and Ceperly [36]. Rewriting the discretized path integral, their method essentially amounts to a recursive transformation of the variables x, in such a way that the kinetic part of the energy can be taken care of by sampling direct Gaussian random variables and a Metropolis choice is made for the potential part. The recursive transformation can be done between some fixed points of the discretized paths, and the method has been applied in such a way that successively finer discretizations of the path were introduced between neighbouring points. Invoking the poly-

38

T. Sauer

mer analog of the discretized path this method was christened the "staging" algorithm by Sprik, Klein, and Chandler in 1985 [37]. The staging algorithm decorrelates successive paths very effectively because the whole staging section of the path is essentially sampled independently. In 1993, another explicitly non-local update was applied to PIMC simulations [35,38] by transferring the so-called multigrid method known from the simulation of spin systems. Originating in the theory of numerical solutions of partial differential equations, the idea of the multigrid method is to introduce a hierarchy of successively coarser grids in order to take into account long wavelength fluctuations more effectively. Moving variables of the coarser grids then amounts to a collective move of neighbouring variables of the finer grids, and the formulation allows to give a recursive description of how to cycle most effectively through the various levels of the multigrid. Particularly successful is the so-called W-cycle. Both the staging algorithm and the multigrid W-cycle have been shown to beat the slowing down problem in the continuum limit completely by reducing the exponent z to z « 0 [39]. Another cause of severe correlations between paths arises if the probability density of configurations is sharply peaked with high maxima separated by regions of very low probability density. In the statistical mechanics of spin systems this is the case at a first-order phase transition. In PIMC simulations the problem arises for tunneling situations like, e.g. for a double-well potential with a high potential barrier between the two wells. In these cases, an unbiased probing of the configuration space becomes difficult because the system tends to get stuck around one of the probability maxima. A remedy to this problem is to simulate an auxiliary distribution that is flat between the maxima and to recover the correct Boltzmann distribution by an appropriate reweighting of the sample. The procedure is known under the name of umbrella sampling or multicanonical sampling. It was shown to reduce autocorrelations for PIMC simulations of a single particle in a one-dimensional double well, and it can also be combined with multigrid acceleration [40]. The statistical error associated with a Monte Carlo estimate of an observable O cannot only be minimized by reducing autocorrelation times TQ1. If the observable can be measured with two different estimators £/, that yield the same mean [/> ' = (Ui) with O = l i m / , - ^ Jj\ ', the estimator with the smaller variance afj. is to be preferred. Straightforward differentiation of the discretized path integral (4) leads to an estimator of the energy that explicitly

The Feynman Path Goes Monte Carlo

39

measures the kinetic and potential parts of the energy by

^-IE^HE^,).

(.3,

The variance of this so-called "kinetic" energy estimator diverges with L. Another estimator can be derived by invoking the path-integral analog of the virial theorem

J._^(^)y> (1 , ntj)) ,

(14)

and the variance of the "virial" estimator

j=l

2= 1

does not depend on L. In the early eighties, investigations of the "kinetic" and the "virial" estimators focussed on their variances [32,41,42]. Some years later, it was pointed out [43] that a correct assessment of the accuracy also has to take into account the autocorrelations, and it was demonstrated that, for a standard Metropolis simulation of the harmonic oscillator, the allegedly less successful "kinetic" estimator gave smaller errors than the "virial" estimator. In 1989 it was shown [44] that conclusions about the accuracy also depend on the particular Monte Carlo update algorithm at hand, since modifications of the update scheme such as inclusion of collective moves of the whole path affect the autocorrelations of the two estimators in a different way. A careful comparison of the two estimators which disentangles the various factors involved was given only quite recently [45]. Here it was also shown that a further reduction of the error may be achieved by a proper combination of both estimators without extra cost. 5 Concluding Remarks The application of the Monte Carlo method to quantum systems is not restricted to direct sampling of Feynman paths, but this method has attractive features. It is not only conceptually suggestive but also allows for algorithmic improvements that help to make the method useful even when the problems at hand require considerable numerical accuracy. However, algorithmic improvements like the ones alluded to above have been proposed and tested

40

T. Sauer

mainly for simple one-particle systems. On the other hand, the power of the Monte Carlo method is, of course, most welcome in those cases where analytical methods fail. For more complicated systems, however, evaluating the algorithms and controlling the numerical accuracy is also more difficult. Only recently, a comparison of the efficiency of Fourier- and discrete time-path integral Monte Carlo for a cluster of 22 hydrogen molecules was presented [46] and debated [47,48]. Nevertheless, path-integral Monte Carlo simulations have become an essential tool for the treatment of strongly interacting quantum systems, like, e.g. the theory of condensed helium [49]. Acknowledgments I wish to thank Wolfhard Janke for an instructive and enjoyable collaboration. References [1] R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948). [2] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics (World Scientific, Singapore, 1990). [3] W. Aspray, in The History of Modern Mathematics. Vol. II: Institutions and Applications, Eds. D.E. Rowe and J. McCleary (Academic Press, Boston, 1989), p. 312. [4] N. Metropolis and S. Ulam, J. Am. Stat. Ass. 44, 1949 (335). [5] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953). [6] L.D. Fosdick, J. Math. Phys. 3, 1251 (1962). [7] L.D. Fosdick and H.F. Jordan, Phys. Rev. 143, 58 (1966). [8] H.F. Jordan and L.D. Fosdick, Phys. Rev. 171, 129 (1968). [9] L.D. Fosdick, SI AM Review 10, 315 (1968). [10] S.V. Lawande, C.A. Jensen, and H.L. Sahlin, J. Comp. Phys. 3, 416 (1969). [11] S.V. Lawande, C.A. Jensen, and H.L. Sahlin, J. Comp. Phys. 4, 451 (1969). [12] I.H. Duru and H. Kleinert Phys. Lett. B 84, 185 (1979). [13] T. Morita, J. Phys. Soc. Jpn. 35, 980 (1973). [14] J.A. Barker, J. Chem. Phys. 70, 2914 (1979). [15] W. Janke and H. Kleinert, Lett. Nuovo Cim. 25, 297 (1979). 16 M. Creutz and B. Freedman, Ann. Phys. (N.Y.) 132, 427 (1981).

The Feynman Path Goes Monte Carlo

[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

[36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]

41

E.V. Shuryak and O.V. Zhirov, Nucl. Phys. B 242, 393 (1984). E.V. Shuryak, Sov. Phys. Usp. 27, 448 (1984). P.K. MacKeown, Am. J. Phys. 53, 880 (1985). J.D. Doll and D.L. Freeman, J. Chem. Phys. 80, 2239 (1984). D.L. Freeman and J.D. Doll, J. Chem. Phys. 80, 5709 (1984). J.D. Doll, R.D. Coalson, and D.L. Freeman, Phys. Rev. Lett. 55, 1 (1985). R.D. Coalson, D.L. Freeman, and J.D. Doll, J. Chem. Phys. 85, 4567 (1986). N. Makri and W.H. Miller, Chem. Phys. Lett. 151, 1 (1988). N. Makri and W.H. Miller, J. Chem. Phys. 90, 904 (1989). I. Bender, D. Gromes, and U. Marquard, Nucl. Phys. 5 346, 593 (1990). M. Suzuki, J. Math. Phys. 26, 601 (1985). M. Suzuki, Phys. Lett. A 146, 319 (1990). M. Suzuki, J. Math. Phys. 32, 400 (1991). W. Janke and T. Sauer, Phys. Lett. A 165, 199 (1992). H. De Raedt and B. De Raedt, Phys. Rev. A 28, 3575 (1983). M. Takahashi and M. Imada, J. Phys. Soc. Jpn. 53, 963, 3765 (1984). X.-P. Li and J.Q. Broughton, J. Chem. Phys. 86, 5094 (1987). H. Kono, A. Takasaka, and S.H. Lin, J. Chem. Phys. 88, 6390 (1988). W. Janke and T. Sauer in Path Integrals from meV to MeV: Tutzing '92, Eds. H. Grabert, A. Inomata, L.S. Schulman, and U. Weiss (World Scientific, Singapore, 1993). E.L. Pollock and D.M. Ceperley, Phys. Rev. B 30, 2555 (1984). M. Sprik, M.L. Klein, and D. Chandler, Phys. Rev. B 31, 4234 (1985). W. Janke and T. Sauer, Chem. Phys. Lett. 201, 499 (1993). W. Janke and T. Sauer, Chem. Phys. Lett. 263, 488 (1996). W. Janke and T. Sauer, Phys. Rev. E 49, 3475 (1994). M.F. Herman, E.J. Bruskin, and B.J. Berne, J. Chem. Phys. 76, 5150 (1982). M. Parrinello and A. Rahman, J. Chem. Phys. 80, 860 (1984). A. Giansanti and G. Jacucci, J. Chem. Phys. 89, 7454 (1988). J.S. Cao and B.J. Berne, J. Chem. Phys. 91, 6359 (1989). W. Janke and T. Sauer, J. Chem. Phys. 107, 5821 (1997). C. Chakravarty, M.C. Gordillo, and D.M. Ceperley, J. Chem. Phys. 109, 2123 (1998). J.D. Doll and D.L. Freeman, J. Chem. Phys. I l l , 7685 (1999).

42

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[48] C. Chakravarty, M.C. Gordillo, and D.M. Ceperley, J. Chem. I l l , 7687 (1999). [49] D.M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).

Phys.

C O O R D I N A T E I N D E P E N D E N C E OF PATH INTEGRALS IN O N E - D I M E N S I O N A L T A R G E T SPACE

A. CHERVYAKOV Institut

fur Theoretische Physik, Freie Universitat Arnimallee 14, D-14195 Berlin, Germany E-mail:

[email protected].

Berlin,

de

The recent perturbative formulation of the quantum mechanical path integral in a curved space allows one to calculate the Feynman diagrams involving multiple temporal integrals over products of distributions. To test the consistency of this formulation the exactly solvable path integral for the particle in a flat space is transformed into the curvilinear form of a nonlinear sigma model where the only solution is the perturbation expansion. By an explicit three-loop calculation we show that the perturbatively defined path integral in new curvilinear coordinates reproduces the ground-state energy of the original system thus ensuring the coordinate independence.

1 Introduction In honor of Professor Kleinert's birthday I would like to review some of the most interesting results we found in a series of recent papers written in a common collaboration [1-3]. By an arbitrary coordinate transformation from flat to curved space the exactly solvable path integral for the particle in a flat space can be turned into the curvilinear form of a nonlinear sigma model where the only solution is the perturbation expansion. The formal perturbative definition of quantum mechanical path integrals in curvilinear coordinates, however, poses problems. To exhibit the difficulties, consider the associated partition function calculated for periodic paths on the imaginary-time axis r:

Z = Jvq{T)yfge-^"\ 43

(1)

A. Chervyakov

44

where A[q] is the Euclidean action with the general form

AQ\

= J dr

l9^{g{TW{T)qv{T)

+ V{q(T))

(2)

The dots denote r-derivatives, g^(q) is a metric, and g = detg^ its determinant. The path integral may formally be defined perturbatively as follows: The metric g^(q) is expanded around some point q$ in powers of the deviation 8q^ = qt* — qft. The same thing is done with the potential V(q). After this, the action A[q\ is separated into a free part -4o[