Geometries of Nature, Living Systems And Human Cognition: New Interactions of Mathematics With Natural Sciences And Humanities

G e o m e t r i e s of N a t u r e , Living Systems and Human Cognition New Interactions of Mathematics with Natural Sciences and Humanities edited by

Luciano Boi

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Geometries of N a t u r e , Living Systems and Human Cognition New Interactions of Mathematics with Natural Sciences and Humanities

s i•

edited by

Luciano Boi Ecole der. Hautes Etudes en Sciences Sociales, Contra de Machematiques, and LUTH, Qbservatoire do Paris-Meudon, France

PP

G e o m e t r i e s of N a t u r e Living S y s t e m s and Human Cognition New Interactions of Mathematics with Natural Sciences and Humanities

\jjp World Scientific NEW JERSEY • LONDON

• SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

GEOMETRIES OF NATURE, LIVING SYSTEMS AND HUMAN COGNITION New Interactions of Mathematics with Natural Sciences and Humanities Copyright © 2005 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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-474-8

Printed in Singapore by Mainland Press

To my brother Piergiorgio Boi, whom I honour and still love, and my daughter Vega Boi who blossomed into a living being but she was never born.

To my son Aliocha Boi, who came to life and since then has inspired me in thinking and writing. I dedicate this book to him.

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"Geometry is a magic that works" RENE THOM

"Topology unlocks the secrets of the universe" HERMANN WEYL

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Contents Foreword

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About the Contributors PART I

MATHEMATICAL IDEAS AND TECHNIQUES IN CLASSICAL AND QUANTUM PHYSICS

1. T-Duality, Functional Equation, and Nonconunutative String Spacetime M. Lapidus {University of California at Riverside) 2. On the Soliton-Particle Dualities F. Helein & J. Kouneiher {University of Paris 7) PART II

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KNOTS IN MATHEMATICS AND EM NATURE

3. Knots L. H. Kaujfman {University of Illinois at Chicago)

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4. Topological Knots Models in Physics and Biology L. Boi {EHESS - CAMS, Paris)

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PART III

MATHEMATICAL AND LOGICAL MODELING IN THE NATURAL SCIENCES AND LIVING SYSTEMS

5. Beyond Modelling: A Challenge for Applied Mathematics P. T. Saunders {King's College, University of London)

281

6. Fondements Cognitifs de la Geometrie et Experience de L'espace A. Berthoz {College de France)

303

1. The Reasonable Effectiveness of Mathematics and Its Cognitive Roots G. Longo {CNRS and Ecole Normale Superieure)

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8. Pathways of Deductions A. Carbone {University of Paris 6 and IHES)

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9. Phenomenologie et Theorie des Categories F. Patras {Lab. J.A. Dieudonne, CNRS-University Nice)

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Foreword The collection of papers forming this book is intended to provide a deeper study of some mathematical and physical subjects, which are at the core of the recent developments and far-reaching researches in the natural and living sciences, as well as to highlight the ubiquitous character and the generative power of a number of mathematical concepts, mostly discovered recently, in some apparently disparate fields. The main goal is to show that a profound movement of geometrization of the most important fields of research is in course, and that this movement has enabled the discovery of a great variety of amazing structures and behaviors in the physical reality and in the living matter. This process seems to be closely related to the action of precise geometrical and topological objects and structures. In addition, some of those objects and structures are very likely scale-invariant and act at different levels of the organization of matter. Among others, the following central questions have been raised by the authors of the nine chapter composing this volume: 1) How to reach a coherent description of the behavior of the different fundamental forces acting in nature from the properties pertaining to space-time and from its geometrical and topological structures? 2) How to explain the kind of relationships existing between some principles of geometrical and topological ordering and stability and certain physical phenomena such as those of bifurcation and of symmetry breaking in dynamical systems, such as fractals? 3) How to robustly take into account the qualitative (and morphological) changes occurring during the molecular and cellular evolution of living beings? 4) How to explain the connection between the mathematical structures of substrate spaces where phenomena unfold and the more or less complex dynamics according to which those phenomena evolve in time? An answer to those questions would very likely demand that we be able to develop new conceptual tools and especially new mathematical methods, ideas and techniques suited to work out a dynamical theory of the spatial generation of physical phenomena, as well as a theory of the emergence of natural and living processes. For example, one particularly interesting task would consist of explaining to what extent the formal structures and spatial-temporal events that constitute the natural frame of living organisms may influence its bio-chemical, physiological and cognitive growth. Some of these new mathematical methods and ideas are suggested and sometimes formally developed in such works by various authors. xi

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The book is divided in three parts, each of which comprises several chapters written by some of the most qualified scholars directly involved in the research themes addressed here, from knot theory and topological quantum field theory, to symmetry breaking in dynamical systems and developmental biology and spatial perception. Instead of trying to review such a large and rich range of contributions and corresponding topics covered by the authors, it may seem more appropriate to focus on some introductory thoughts on three key topics addressed by the authors of this collective work, namely: 1) The seemingly central role of certain topological objects such as, in particular, knots in mathematics and nature, 2) The emergence in the last two decades of several unfamiliar ideas and techniques in classical and quantum physics, 3) The possibility of new modeling in natural phenomena and living systems, and the cognitive basis of spatial perception. 1. The role of topological knots in mathematical physics and in nature. In the nineteenth century knot theory has been studied by very few scientists, remaining thus a fairly isolated subject of interest. Moreover, the study of knots was related only to physics. The German mathematician Gauss used the properties of knots in the solution of an electromagnetic problem, and the British physicist Thomson planned to describe atomic physics and chemistry in terms of knots. However, the twentieth century has known a renaissance in the interaction between physics and knot theory, which was due essentially, on the one hand, to intensive research in topology, and on the other hand, to statistical mechanics studies. A very profound interrelationship between statistical mechanics and knot theory emerged when it became clear that the Jones algebra and knot and link polynomials are formally identical to the Temperley-Lieb algebra. The latter one was introduced in the study of the Potts model, which is a generalization of the Ising model in statistical mechanics. In other words, it became apparent that statistical mechanics can be done on a link diagram and produce new invariants of links in the process. That opened the way to the discovery of a number of new invariants of knots and links that generalized the original Jones polynomial. In particular, the Yang-Baxter equation is a statistical mechanics model that leads to new knot invariants. In the last three decades, it has been understood that there are three other different ways in which physics and knot theory are related. Not only topological quantum field theory, but also the theory of quantum invariants has proved to be closely related to conformal field theories. Witten, notably, has shown that the Jones polynomial and its generalizations are related to the topological

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Chern-Simons theory. More precisely, he attempted to understand the new Jones's invariants of knots and links in a systematic way by relating them to the vacuum expectation values of Wilson loops in Chern-Simons theory. Using the invariance of the Chern-Simons action under orientation-preserving diffeomorphisms, these vacuum expectations values should be isotopy invariants. In this connection, one of the most fruitful developments in the last decades in theoretical physics is the work relating quantum field theory and threedimensional topology. Thus, Atiyah, Witten, Turaev and Kauffman were led to construct link invariants for closed three-manifolds. The chapter of Boi provides a detailed mathematical account of those subjects. The chapter of Kauffman in the present book is very illuminating in this regard. Furthermore, Kauffman gives new mathematical results relating to the invariants in knot theory, and a very interesting and original interpretation of quantum mechanics in the light of the concept of skein relations of knots and links. Its contribution is a masterpiece of imaginative and creative mathematics and mathematical physics. Even more astonishing has been the finding that there exist several fundamental connections between topological knot theory and life sciences. In fact, it became more and more clear that the spatial conformation of DNA knots are phenomena that make up part of biophysical and living matter; in other words, knots and links are microscopic and nanoscopic objects carrying a tremendous amount of precious information on the emergence of new forms and the evolution of organisms. Moreover, one might say that knotting and unknotting are "universal" transformations underlying almost all living processes at the molecular and at the macromolecular and also maybe at the cellular levels. In fact, it became recently clearer and clearer that the topological conformation of some families of enzymes, called topoisomerases, and also the threedimensional structures of proteins that act on DNA inside the space of cell are essential in order for certain vital biological functions of organisms to be accomplished. This opened totally new perspectives for fruitful collaborative work between topologists and biologists, first of all because differential geometry and knot theory can be used to describe and explain the threedimensional structures of DNA and protein-DNA complexes. In his chapter, Boi gives a detailed analysis of that subject, and also suggests some ideas that could be helpful in order to address these questions. 2. Some open problems and perspectives in classical and quantum physics. The chapters of Lapidus, Helein and Kouneiher deal with the question of the role of geometrical and (algebraic)-topological concepts in the developments of theoretical physics, especially in gauge theory as well as string theory, and they

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show the great significance of those concepts for gaining a deeper understanding of the dynamics of physics. Though starting from different paths, they lead in some sense to the same conclusion, namely that physical phenomena very likely emerge from the geometrical and topological structure of space-time. The attempts to solve one of the central problems in twentieth theoretical physics, i.e. how to combine gravity and the other forces into a unitary theoretical explanation of the physical world, depends essentially on the possibility of building a new geometrical framework conceptually richer than Riemannian geometry. In fact, this modified differential geometry still plays a fundamental role in non-Abelian gauge theories, superstring theory and non-commutative geometry, thanks to which a great variety of new mathematical structures has emerged. A very interesting hypothesis is that the global topological properties of the manifold model of spacetime play a major role in quantum field theory and that, consequently, several physical quantum effects arise from the non-local metrical and topological structures of these manifolds. In his chapter, Lapidus offer an authoritative and deep study of these questions, and present some important new mathematical results. The unification of general relativity and quantum theory requires thus some fundamental breakthroughs in our understanding of the relationship between spacetime and quantum processes. In particular the superstring theories lead to guessing that the usual structure of space-time at the quantum scale must be dropped out from physical thought. Non-Abelian gauge theories satisfy the basic physical requirements pertaining to the symmetries of particle physics because they are geometric in character. They profoundly elucidate the fundamental role played by bundles, connections and curvature in explaining the essential laws of nature. Kaluza-Klein theories, and more remarkably superstring theory, showed that spacetime symmetries and internal (quantum) symmetries might be unified through the introduction of new structures of space with a different topology. That essentially means that "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain classes of (presumably) non-smooth manifolds. There are two key questions that are at the heart of the mathematical developments of quantum field theories, which include, among others, superstring theory, topological quantum field theory, noncommutative geometry, and loop representation (in quantum gravity). The first question concerns the meaning of quantum geometry. General relativity taught us that space-time geometry shares the same dynamical character of the electromagnetic field and the other physical fields; and we have learned with quantum theory that all

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dynamical objects exhibit quantum behavior: they can be in probabilistic superpositions of states and they manifest themselves in quanta at small scales. If we trust the general validity of these two physical discoveries, we are led to conclude that space-time geometry too must exhibit quantum behavior: it may admit probabilistic superpositions and it may manifest itself in quanta at small scales. What is then space-time geometry with such features? What kind of mathematics can describe it? What physical meaning can we give to it? We have a certain understanding of quantum field theory on a fixed geometrical background; can we understand how to describe a world in which the background geometry, space-time itself, is quantum mechanical? To illustrate the physical need of a diffeomorphism invariant quantum field theory, consider the following observations: Space-time geometry is dynamical, and thus rather similar to physical fields - indeed, space-time geometry is the gravitational field; but we can also turn this observation around and notice that the dynamics of the gravitational field is profoundly peculiar - indeed, the gravitational field is space-time geometry. In fact, this dynamics cannot be captured, as the other physical fields, by techniques that rely on the existence of a fixed background space-time. A diffeomorphism invariant quantum field theory very likely requires that we abandon local quantum field theory, in which physical operators are labeled by regions of a metric manifold, in favor of a diffeomorphism invariant quantum field theory, in which physical operators are diffeomorphism invariant objects. Thus, the definition and the construction of nontrivial diffeomorphism invariant quantum field theory is at present one of the most fascinating problems, and in our opinion it is very important in order to clarify the intimate relationship between physical intuition and mathematical conceptualization. 3. New modeling in natural phenomena and living systems, and in the cognitive basis of spatial perception. Most contributions in the third section of the collection are aimed at investigating relationships between the perceptual structures pertaining to our macroscopic world and the geometric properties which characterize the physical space and the beings that inhabit it. The main question raised by some authors, in particular by Berthoz and Longo and to some extent by Carbone is the following: What differentiates the topological and metrical structures of the 'physical' space, which is embodied into (or internalized by) the neurophysiological space in the brain, from the properties of the perceptive and representative space obtained very likely by deforming the first space according to certain general (mathematical and/or physical) laws? They try to answer this crucial question, which finds itself at the core of the present research in cognitive sciences, by raising some issues which concern: (i)

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The mechanisms by which our complex sensorial systems such as the visual, the sensorimotor and the vestibular systems drive our perception of the surrounding (phenomenal) space and of the movements required to be performed in order to reach it. (ii) The perception of movement and its neurophysiological basis, and the role of movement in the perception of the third dimension, (iii) The geometrical and dynamical grounds of human perception and cognition and, reciprocally, the biological and cognitive roots of our mathematical knowledge. In his very interesting chapter, Berthoz supports both theoretically and experimentally the thesis that perception of space, and of things and events that are part of it, is fundamentally grounded in movements and actions of the body. Hence, geometry is less an abstract set of formal rules, and much more a complex activity which has developed phylogenetically in the course of evolution in order that we be able to reconstruct mentally the world around us through a neural allocentric processing of space, instead of an egocentric one, as it is the case for animals including primates. The goal of Longo's stimulating contribution is to stress that we organize knowledge of space and numbers by formal structures distilled in a mathematical language. In addition, he points out that the meaning of space, numbers and language is primarily grounded in mathematical intuition, which stems from the biological and historical character of human cognition. The papers of Saunders, Carbone and Patras deal with different, although related, aspects of mathematical modeling of complex living systems and formal structures. Saunders proposes a new methodology for mathematical modeling in the living and human sciences; he derives examples of that methodology from catastrophe theory and the theory of pattern formation. He points out that in conventional modeling, and especially in the physical sciences, we often construct a model and then treat it largely as an exercise in mathematics, only translating back into science at the end, whereas in both examples mentioned above, arriving at the prediction required mathematics and biology to be used together. Carbone shows, first, that replication and folding are two basic operations of complex systems, such as the brain, and further that they play a crucial role in the structural complexity and geometry of proofs. Relying on the great mathematician Alexandre Grothendieck, Patras emphasizes that dealing with meaning, that does not rely only on formal or computational tools, is necessary for internal progress of mathematics. This is proven by considering modern mathematical theories, such as category theory. Let us point out that the new relationship, of a much more intrinsic and dynamic nature than in the past, between mathematics (in particular geometry and topology) and biology appears to be one of the most significant

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developments of science in the last three decades. Relying on previous works in the correlated fields of morphology, embryology, theoretical biology and differential topology, it has been proposed recently a new view of self-organization in which topological-biological models of evolution and development of living organisms play an essential role. These attempts aim at explaining the stability and the reproduction of the global spatial-temporal structures or patterns in terms of the organization of the structure itself. For example, the self-organizational properties of morphogenesis is its capacity to reduce the symmetry order (to break symmetry) without necessarily employing any discrete external "dyssymmetrization", instead only by means of certain internal geometrical transformations bound up closely with the growth of organisms. In fact, experimental evidence strongly suggests that embryos obey some general topological principles, such as preservation of the Euler characteristic of a developmental field on the surfaces over which these fields are defined. A new mathematical approach inspired from qualitative dynamics and combinatorial-differential topology seems to provide far reaching possibilities to attack the problem of the emergence and the stability of self-reproducing structures. In our opinion, which is well exemplified in some of the papers presented here, the validity of this type of dynamic mathematical description exceeds by far the biological realm, and may concern various domains of physics (especially the problem of phase transitions), a number of morphological spaces and processes (whether animate or inanimate), as well as certain mechanisms of perception and cognition. In conclusion, we would like to stress that the main purpose of the present collection is to explore some far-reaching interfaces where mathematics, theoretical physics and natural sciences seem to interact profoundly. The volume bring a great overlap of knowledge between mathematicians, physicists and natural scientists. I hope that the long time it took to prepare the edition of this book contributes to the originality and the interest of the work presented here. Finally, I take this opportunity to thank all the contributors for their efforts, collaboration and patience; in addition, I am indebted to some of them for the several interesting conversations about many mathematical and physical questions we have had over the years of this book. I should also like to mention the editorial help I received from my students Jerome Havenel and Michel Karma in preparing the present volume. Luciano Boi Paris, February 2004

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About the Contributors Michel L. LAPIDUS has received his Ph.D., Doctorat es Sciences, and Habilitation to direct research (all in Mathematics) from the University of Paris VI (Pierre et Marie Curie) in France, in 1980, 1986, and 1987, respectively. After having taken a research position at the University of Paris VI and a postdoctoral position at the University of California at Berkeley, he has taught at the University of Southern California in Los Angeles, the University of Iowa, the University of Georgia in Athens, Yale University, and the University of California at Riverside where he is Professor of Mathematics since 1990. He has held a number of visiting positions at universities in the US and abroad, and has been a member of many research institutes, including most recently the Mathematical Sciences Research Institute in Berkeley, the Institut des Hautes Etudes Scientifiques in Paris (Bures-sur-Yvette), the Schroedinger Institute of Mathematical and Physics in Vienna, and the Newton Institute of the University of Cambridge. Since 1999, he is Deputy Governor and a honorary member of the Research Board of Advisors of the American Biographical Institute. In 2000, he was elected a Fellow of the American Association for the Advancement of Science (AAAS) for "Distinguished contributions to mathematical physics and fractal geometry". His research interests include mathematical physics, geometric analysis, partial differential equations, dynamical systems, number theory, as well as fractal and spectral geometry. Over the last year, he has published two books closely related to his research interests: the first one (joint with G. W. Johnson) entitled The Feyman Integral and Feynman's Operetional Calculus was published by Oxford University Press in May 2000. The second book (joint with M. van Frankenhuysen) was published by Birkhauser in January 2000 and is entitled Fractal Geometry and Number Theory, with the subtitle Complex dimensions offractal strings and zeros ofzeta functions. Frederic HELEIN is Professor of Mathematics at the Institut de Mathematiques Jussieu, Universite Denis Diderot Paris 7. He is the author and co-author of three books, among them, Harmonic Maps, Conservation Laws and Moving Frames (Cambridge University Press, 2002), and several research publications in differential geometry and mathematical physics. Frederic Helein is working on partial differential equations and their applications to differential geometry and to theoretical physics. His first contributions concerned the analysis of solutions XIX

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About the Contributors

of partial differential equations modeling static equilibria of liquid crystals in physics or of the harmonic map problem in geometry: he obtained several stability results by various geometrical methods (like for instance calibrations) and important regularity results on weak harmonic maps. He and F. Bethuel and H. Brezis developed a theory for the asymptotic analysis of two-dimensional solutions of the Ginzburg-Landau equation, used for modeling superconductivity and superfluidity. Later on, his interest turned to integrable systems arising in differential geometry and he proved that Willmore surfaces and (in collaboration with P. Romon) Hamiltonian stationary Lagrangian surfaces are integrable systems: this led to constructions and classifications results based on geometrical and algebraic methods. Lastly, in collaboration with J. Kouneiher, he is working on developing the multisymplectic approach to the calculus of variations, a framework suitable for a covariant Hamiltonian formalism for fields theory, and on its application to mathematical physics, for instance, the quantization of fields. Joseph KOUNEIHER is Researcher in the Laboratoire de l'Univers et de ses Theories (LUTH), CNRS and Observatoire de Paris, and is teaching in NiceSofia-Antipolis University. His research interests are in mathematical physics and in particular: quantum fields theories, quantum gravity and their topological aspects. At the same time he explore the conceptual development of the physics and the geometry of the twentieth century. His main contributions concern the cohomological aspect of quantum mechanics and the construction of the Khaler and symplectic structures underlying the foundations of the corresponding complexefied Hilbert space. In collaboration with Frederic Helein, he is now developing a covariant Hamiltonian formalism for fields and gauge theories. Louis H. KAUFFMAN is Professor of Mathematics at the University of Illinois at Chicago. He received his B.S. from MIT in 1966 and his Ph.D. from Princeton University in 1972. He has been on the faculty of the University of Illinois at Chicago since 1971, and has been a visiting professor at many universities and research institutions around the world. Professor Kauffman is the author of the book On Knots (Princeton University Press, 1987), the book Knots and Physics (World Scientific Publishing Co, 1991, 1994, 2001) and numerous books and research articles related to topology, knot theory, statistical mechanics and mathematical physics. He is the founding editor of the Journal of Knot Theory and Its Ramifications (World Scientific), and editor of the World Scientific Series on Knots and Everything.

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Luciano BOI is Associate Professor of Geometry, Scientific Theorization and Natural Philosophy at the Ecole des Hautes Etudes en Sciences Sociales, in the Centre de Mathematiques (Paris). He studied mathematics, physics and philosophy of science at the Universities of Bologna, Paris and Berlin, and received his Doctorate as well as his Habilitation from the EHESS (Paris). He has been a visiting professor and a member of several universities and research institutes, including the IHES (Bures-sur-Yvette), the IAS (Princeton), the UQAM and the University of Montreal. He has also been a visitor at the Universities of Cambridge (UK), Munich, Milan, Siena and Heidelberg, and invited lecturer in numerous international conferences and symposiums. He has received several awards, such as fellowships from the von Humboldt Foundation (1991, Berlin), from the Centre National des Lettres (Paris, 1992), and from the SSRCC (1996, Montreal), as well as a 1997 award from the Guggenheim Foundation (New York) and a 2000 award from the SingerPolignac Foundation (Paris) in recognition of the scope of his work. His main research interests include various aspects of mathematics, the geometrical foundations of theoretical physics, the interactions between topology and biology, the geometrical modeling of spatial perception, as well as the history and philosophy of sciences. He is the author of numerous books and articles on mathematical physics, theoretical biology, spatial perception, the epistemology of mathematical and physical sciences, and on natural philosophy and phenomenology. He is the editor of the "Philosophia Naturalis et Geometricalis" collection at Peter Lang (Bern), and co-director of the "Centra Internazionale di Studi e di Ricerche sulla Scienza, l'Arte e la Filosofia" Pharos at San Leo (Urbino) in Italy. Peter SAUNDERS was born in Canada and took a degree in mathematics at the University of Toronto. He came to London to do a Ph.D. in cosmology at King's College, London, where he is now Professor of Applied Mathematics. His present research interests are in mathematical and theoretical biology, including evolution theory, and in the study of complex systems. The aim of his research program is to explain the properties of organisms and other complex systems in terms of ordinary mathematics, physics and chemistry. More recently, his work is concentrated on two main areas: (a) physiological control, and (b) generic properties of complex nonlinear systems. He is the author of many research publications appeared in the Bulletin of Mathematical Biology, the Journal of Theoretical Biology, and the Journal of Physiology.

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Alain BERTHOZ is Professor of the chair Physiologie de la Perception et de l'Action at the College de France, and Director of the CNRS-College de France Center of Cognitive Neuroscience. He holds Ph.D. degrees in engineering and natural sciences at the University of Paris. He has been visiting professor at the McGill University, at the Universities of Montreal, Oxford, Tubingen, Pise and Pavia. He is a member of numerous Scientific Societies and Academies, notably the Academie Internationale d'Austronatique, the Academia Europea, and the Academie des Sciences (Paris). He organized numerous scientific and interdisciplinary conferences, and has been an invited lecturer in many international congress and symposia. He was awarded by several international prizes and distinctions for its outstanding contributions to the neurosciences. He is a leading researcher in the fields of neurology and physiology of perception and movement. Professor Berthoz is the author of several books, which include (with W. Graf and P. P. Vidal) The Head-Neck Sensori-Motor System (Oxford University Press, 1991), Multisensory Control of Movement (Oxford University Press, 1993), Le sens du mouvement (Odile Jacob, 1998), and of an impressive number of research articles appeared in the leading international journals in the fields of physiology of perception and action, neurosciences, psychophysics and cognitive sciences. Giuseppe LONGO has been former Associate Professor of Mathematical Logic and, later, Professor of Computer Science at the University of Pisa. Currently, he is Directeur de Recherche CNRS (DR1) at the Ecole Normale Superieure in Paris. He worked in Mathematical Logic and at various applications of Mathematical Logic to Computer Science. In particular, Type Theory, Higher Type Recursion, Category Theory and their applications to Functional Languages have been his main research interests. Longo is editor-inchief of Mathematical Structures in Computer Science, a leading scientific journal of the Cambridge University Press. He is presently extending his research interests to Epistemology of Mathematics and Cognitive Sciences. He is the author of several books, among which, Categories, Types and Structures, MIT Press, 1991 (with A. Asperti), and published extensively, over the last years, on mathematical logic, foundations and philosophy of mathematics, and cognitive sciences. Alessandra CARBONE is Professor of Computer Science at the University Pierre et Marie Curie Paris 6 and a visitor at the Institut des Hautes Etudes Scientifiques (Bures-sur-Yvette). Her research touches on various aspects of

About the Contributors

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logic, symbolic dynamics, combinatoric and the interface between mathematics and molecular biology. She is co-author with S. Semmes of A Graphic Apology for Symmetry and Implicitness (published by Oxford University Press, 2000) and co-editor of several collection, including Pattern formation in Biology, Vision and Dynamics with M. Gromov and P. Pruzinkiewicz (published by World Scientific, 2000). Frederic PATRAS received his Ph.D. in Mathematics from the Universite Paris 7 in 1992. Currently, he is Senior Research Fellow of CNRS and Head of the "Algebra and Topology" group at the Laboratoire J.-A. Dieudonnee, Universite de Nice-Sophia Antipolis. His research interests include various aspects of algebraic topology, such as Hopf algebra, polytopes groups and homological algebra, as well as philosophy of mathematics and phenomenology. He is the author of the book La pensee mathematique contemporaine (Paris, 2001), co-editor (with L. Boi and P. Kerszberg) of the collection Rediscovering Phenomenology. Phenomenological essays concerning mathematical beings, physical reality and perception (Kluwer, 2005), and of several articles in mathematical journals, including Topology (1999), Algebraic K-Theory and its Applications (1999), Journal of Algebra (2000).

Parti

Mathematical Ideas and Techniques in Classical and Quantum Physics

"It seems reasonable to think that general relativity and quantum theory are intrinsically incompatible and that, rather than merely developing technique, what is required is some fundamental breakthrough in our understanding of the relationship between spacetime structure and quantum process." Edward Witten, Physics and Geometry, 1987.

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T-DUALITY, FUNCTIONAL EQUATION, AND NONCOMMUTATIVE STRING SPACETIME Michel L. LAPIDUS* University of California Department of Mathematics Riverside, CA 92521-0135, USA lapidus @math. ucr. edu

To the memory of Moshe FLATO, long-time friend and mentor, a brilliant scholar as well as an outstanding, compassionate, and generous human being

God made the integers, and all the rest is the work of man. Leopold Kronecker. (Quoted in [Bel, p. 477] and [Boy, p. 617].) String theory carries the seeds of a basic change in our ideas about spacetime and in other fundamental notions of physics. Edward Witten, 1996 [Wit 15, p. 24]

•Research partiallly supported by the U.S. National Science Foundation under Grants DMS-9623002 and DMS-0070497. 3

4

M. L. Lapidus

Abstract. We first give a broad (but brief) introduction to several of the main themes encountered in this work: arithmetic geometry, noncommutative geometry, quantum physics and string theory, prime number theory and the Riemann zeta function, along with fractal and spectral geometry. We then explain how string theory in a circle (or in a finite-dimensional torus)—considered from the point of view of noncommutative geometry—can be used as the starting point for a geometric and physical model for the Riemann zeta function C, = C,{s) (and other arithmetic L-series). In particular, we contend that the classic functional equation satisfied by C, corresponds to T-duality in string theory. The latter, a key symmetry that is not present in ordinary quantum mechanics, allows one to identify physically and mathematically two circular spacetimes with reciprocal radii. Furthermore, we suggest that the Riemann Hypothesis may be related to the existence of a fundamental length in string theory. This essay may serve as a relatively nontechnical introduction to the beginning of a theory presented in a book in preparation by the author and entitled In Search of the Riemann Zeros. In the latter monograph is developed, in particular, a theory of fractal membranes—viewed as suitable quantum or multiplicative analogues of the author's fractal strings. The notion of fractal membrane—interpreted heuristically as a noncommutative (adelic) infinite dimensional torus, in the spirit of the present discussion—may help in the long-term provide a unified geometric, algebraic and physical framework within which to understand number-theoretic zeta functions and their associated arithmetic geometries.

1

Introduction The recipes for quantization are a primitive manifestation of the fact that the space of internal degrees of freedom "at a single point" in vacuo is already infinite dimensional because of the virtual generation of particles. Further understanding is blocked until we relinquish the idea of space-time as the basis for all of physics. Yuri I. Manin, 1979 [Manil, p. 94]

One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation. Bernhard Riemann, 1859 [Riel], introducing his famous "Riemann Hypothesis". (Translated in [Edw, p. 301].)

T-Duality, Functional Equation

1.1

5

Arithmetic and Spacetime Geometry

I believe that at its deepest level, the geometry underlying the integers—in the old language, the 'geometry of numbers', and in modern terminology, 'arithmetic geometry', including the twin mystical notions of the 'arithmetic site' [Den3,6;Har2] and of the 'field of one element' [Mani4, So]— would have to reflect the physical and geometrical properties of what we traditionally call 'spacetime', for lack of a better word. I have held this belief, at least consciously, ever since reading in the mid-eighties the beautiful paper by Yuri I. Manin [Mani2], entitled New Dimensions in Geometry1. It was later comforted and turned into an intimate conviction by my own reflections and research experiences in developing since the late 1980's the theory of 'fractal strings' 2 and exploring its relationships with aspects of number theory, particularly the Riemann zeta function and the Riemann Hypothesis [Lapl-4,LapPol-3,LapMal-2,HeLapl-2, Lap-vFl-5]. Several years ago, I was startled to hear Alain Connes express a similar belief during a debate held at UNESCO in Paris on the occasion of the International Congress of Mathematical Physicists in July 1994. From our ensuing conversations about this subject—and from our ongoing dialogue (since the summer of 1993) about our respective approaches to the Riemann zeta-function ([BosConl-2], surveyed in [Con6,§V.ll], and [LapPol-2,LapMal-2,Lap2-4,HeLapl-2], now pursued in [Lap-vFl-4])—it appears, however, that his vision then (and probably still now in his new approach [Con9,10]) is quite different from the model I am about to propose, although key aspects of noncommutative geometry play an important role in both cases.3

1.2

Riemannian, Quantum and Noncommutative Geometry

During the course of the 20th century, and ever since the resounding success of the application of Riemannian geometry to the study of gravity in

' I am grateful to Christophe Soule for sharing with me his enthusiasm for this paper and for Arakelov theory [SoABKJ when I first met him in Berkeley in August 1984. 2 or 'fractal harps', as referred to in [Lap-vFl,2], not to be mistaken with the strings encountered in the classical string theory [Gre,GreeSWit,Kak,Mani3,Polc3-4,Schl], although part of the point of the present essay is that the two theories can be related, albeit in unexpected ways. 3 See, however, the relevant discussion towards the end of [Lap7] for some possible connections.

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Einstein's theory of general relativity, geometry has been a focal point for many mathematicians and physicists interested in apprehending aspects of physical reality. We now know that symplectic geometry is well suited to (and in fact, largely motivated by) the study of phase space in classical mechanics. As mentioned above, Riemannian geometry—in its Lorentzian version—is adopted in most models of classical physics concerned with gravitational fields. More recently, the geometry (and topology) of principal bundles over differentiable manifolds has been found to be an ideal tool to explore gauge field theories. It is to be noted, however, that most of the mathematically rigorous investigations of gauge theories have been limited so far to the classical rather than to the quantum aspects of such theories. It is much less clear, at the moment, how to determine what is "the" geometry underlying quantum mechanics, let alone quantum field theory. More generally, we do not understand what are the true mathematical foundations of quantum field theory [Witl7]. Of course, this question has been the object of many speculations and controversies. In recent years, noncommutative geometry has arisen in large part as a possible answer to such a question, although it is fair to say that we still seem to be far from having resolved this crucial problem. Beginning with the algebraic and functional analytic work of Murray and von Neumann [Mu-vN,vN], as well as of Gelfand and Naimark [GelNai], noncommutative geometry has truly emerged and flourished as an independent subject with the deep work of Alain Connes. (See, for example, the books [Con5] and [Con6]; also see [GraVarFi].) In a nutshell, the central objects of noncommutative geometry are no longer spaces of points, as in ordinary geometry, but (typically noncommutative) operator algebras, the elements of which can be thought of heuristically as representing quantum fields on the underlying 'noncommutative (or quantum) spaces'. In recent years, Connes [Con7,8] has proposed a set of axioms for noncommutative geometry that requires a much richer structure for a noncommutative space. It involves, in particular, the existence of a suitable Dirac-type operator acting on the Hilbert space on which the operator algebra is represented. (Intuitively, the noncommutative algebra itself can be thought of as the 'algebra of coordinate functions' on the associated quantum space.) This enables one, for example, to measure distances within a noncommutative space much as in a Riemannian manifold, via a formula in some sense dual to the geodesic formula. (See [Con4] and [Con6,Chapter VI].) Under appropriate assumptions, this also provides a noncommutative analogue of the de Rham complex and of aspects of differential topology and geometry (see [Conl-8] and [GraVarFi]). It is good to keep in mind, as is often stressed by Daniel Kastler [Kas2,3], that the abovementioned axioms are largely motivated by models from quantum physics, particularly the so-called 'standard model for elementary particles'

T-Duality, Functional Equation

1

(see [DV-K-M], [ConLo], [Con5,6]), as well by the long-standing problem of quantum gravity (see, e.g.,, [ChaConl,2]).

1.3

String Theory and Spacetime Geometry

Over the last twenty years, string theory—which originated as a theory of strong interactions in the early 1970's, soon to be superseded by quantum chromodynamics (QCD)—has emerged as the best candidate for unifying the four known fundamental forces (or interactions) of nature; namely, the electromagnetic force, the weak force and the strong force—all described by Yang-Mills gauge field theories—along with the gravitational force, described by Einstein's theory of general relativity. In this sense, it may eventually provide a means of reconciling quantum mechanics (or quantum field theory) with general relativity, and thereby to resolve the riddle posed by quantum gravity. Caution must be exerted, however, because despite its great beauty and mathematical power, string theory is still far from being a complete physical or mathematical theory. Moreover, due to the extremely high energies (or equivalently, the minuscule scales) involved, it has been notoriously difficult in string theory to make predictions that can be verified experimentally with our present-day technologies or even in the foreseeable future. We note, however, that although experiments involving high-energy accelerators seem to be out of the question—except to verify some of the most basic assumptions of (super)string theory, such as the existence of supersymmetry [Kan,Freu,Wein4]—interesting large-scale astronomical experiments currently under way may provide useful clues within the next ten to fifteen years. Roughly speaking, in string theory, point-particles are replaced with tiny strings (i.e., one-dimensional open strings or else closed loops) vibrating in a (target) space-time, which is assumed to be ten-dimensional in superstring theory. As time evolves, a given string sweeps out a two-dimensional world-sheet, viewed mathematically as a Riemann surface. Hence, the Feynman path integral approach to quantum mechanics ([Feyl], [FeyHi], see also [JohLap]) extends naturally to this setting, with the path integral being replaced with an integral over all possible world-sheets, or more precisely, with integrals over suitable moduli spaces of Riemann surfaces (with a given finite genus and a given number of marked points). The resulting heuristic Feynman-type integral is often referred to as a 'Polyakov integral' [Polyl-3] in the literature. (See, for example, [GreeG,GreeSWit,Kak,Polc3-4,Wit4], along with [JohLap], Chapter 20, especially Section 20.2.B.) The associated Feynman (or string) diagrams take a much simpler form than in quantum

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M. L. Lapidus

field theory and their detailed analysis provides a good understanding of perturbative string theory, at least at the physical level of rigor. The miracle is t h a t the divergences caused by the'coincidence of points in space-time (and hence the vanishing size of point-particles) in standard quantum field theory now disappear because of the extended size of the strings. In physical terms, superstring theory is said to be renormahzable or, more precisely, "finite to all orders in perturbation theory". In this respect, in concluding his plenary lecture at the International Congress of Mathematicians delivered in Berkeley in 1986, Edward Witten made the following statement ([Wit4, p. 302], 1987): / have tried to make it plausible that path integrals on Riemann surfaces can be used to formulate a generalization of general relativity. What is more, the resulting generalization is (especially in its supersymmetric forms) free of the ailments that plague quantum general relativity. If the logic has seemed a bit thin, it is at least in part because almost all we know in string theory is a trial and error construction of a perturbative expansion. [The Feynman-Polyakov path integrals over moduli spaces of Riemann surfaces] are probably the most beautiful formulas that we now know of in string theory, yet these formulas are merely a perturbative expansion ... of some underlying structure. Uncovering that structure is a vital problem if ever there was one. Such was the situation up to the late 1980's. However, during the 1990's, progress has been made towards developing a nonperturbative string theory, called M-theory, in which (one-dimensional) strings are replaced with higher-dimensional geometric objects, called 'membranes' or 'D-branes'. The associated 'dualities' (including the so-called 'S-duality' and 'T-duality') enable one to relate the five basic types of string theories, and thereby to obtain a more unified picture of string theory. (See, for example, [Witl5-17] and [GivePR,Gre, Polcl-4, Sch2-4,Val-2].) These recent developments are sometimes referred to as the "second superstring revolution" [Sch2]. Edward Witten often begins his conferences on string theory—especially when addressing himself to a mathematical audience—by stressing the following striking contrast between the historical development of string theory and general relativity (see also, for example, the introduction of [Wit4]). In ([Witl3, pp. 205-206], 1994), he writes: More fundamentally, I believe that the main obstacle [to further progress] is that the core geometrical ideas—which must underlie string theory the way Riemannian geometry underlies general relativity— have not yet been unearthed.

T-Duality, Functional Equation

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Whatever the true underlying geometric foundations of string theory (or of M-theory), there seems to be an emerging consensus among theoretical and mathematical physicists that one needs to significantly revisit the notion of spacetime, both from a geometrical and physical point of view. In particular, at extremely small scales (typically, below the Planck scale4), the classical model of spacetime as a smooth Riemannian (or Lorentzian) manifold is probably no longer valid. For example, the small-scale structure of spacetime may be discrete, or partly discrete and partly smooth. Alternately, it may be of a fractal nature. In fact, in early work on quantum gravity by Wheeler [Whe,WheFo], Hawking and others (see, for example, [GibHaw,Haw,HawIs]), there have been intriguing references to the existence of some kind of 'fractal foam' (sometimes also called 'quantum foam'). (More recently, see also [Not] in another context.) More radically, it has even been suggested to do away with the notion of spacetime altogether, at least as a primary concept. (See, for instance, Witten's article [Wit 15] entitled Reflections on the Fate of Spacetime and from which the second quote heading this essay is excerpted. Also, for a different perspective on a similar theme, see Manin's quote from [Manil] heading the present introduction.) Perhaps an appropriate modification or extension of Connes' noncommutative geometry [Con5,6] will provide clues as to how to proceed in suitably altering or replacing the concept of spacetime. Indeed, there has already been a number of attempts in this direction, several of which will be key to aspects of our present work. (See, for example, [Wit3], and more recently, [FroGa,ChaFro,Chal-2,LiSzl2,FroGrRel-2] along with [ConDouSc.LanLiSz].) Whatever the answer to these fundamental questions will turn out to be ultimately, the relationship between physics and geometry (in a broad sense) will continue to be at the center of the ongoing dialogue between physicists and mathematicians during the next few decades of the 21st century. 4 The Planck length (or scale) is the fundamental scale of quantum gravity. It is approximately equal to 1.6 x 10~ 3 3 cm (in international units) and is expressed in terms of the following three universal constants, h (Planck's constant or quantum of action), c (the speed of light), and G (Newton's gravitational constant). It is also equal to the reciprocal of the Planck mass, about 1.22 x 10 19 GeV, the natural mass (or energy) scale of quantum gravity. (It may be useful to note—as is frequently stressed by physicists—that Planck's length is about 20 orders of magnitude smaller than the size of a proton.) In much of this work, we will choose units so that Planck's length (or rather, the string length, see §2.3) is equal to one.

M. L. Lapidus

The Riemann Hypothesis and the Geometry of the Primes The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found in it a mysterious attraction impossible to resist. ... Very different results are revealed when we turn to the second principal branch of the modern theory, the theory of the average or asymptotic distribution of primes. This theory (though one of its most famous problems is still unsolved) is in some ways almost complete, and certainly represents one of the most remarkable triumphs of modern analysis. The theory centres round one theorem, the Primzahlsatz or Prime Number Theorem; and it is to the history of this theorem, which may almost be said to embody the history of the whole subject, that I shall devote the remainder of this lecture. ... The next great step was taken by Riemann in 1859, and it is in Riemann's famous memoir Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse that we first find the ideas upon which the theory has now been shown really to rest. Riemann did not prove the Prime Number Theorem: it is remarkable, indeed, that he never mentions it. His object was a different one, that of finding an explicit expression for n(x) [the number of primes not exceeding a;], or rather for another closely associated function, as a sum of an infinite series. But it was Riemann who first recognized that, if we are to solve any of these problems, we must study the Zeta-function as a function of the complex variable s = a + it, and in particular study the distribution of its zeros. ... To these propositions [Riemann] added certain others of which he could produce no satisfactory proof. In particular he asserted that there is in fact an infinity of complex zeros, all naturally situated in the 'critical strip' 0 < a < 1; an assertion now known to be correct. Finally he asserted that it was 'sehr wahrscheinlich' that all these zeros have the real part \: the notorious 'Riemann hypothesis', unsettled to this day.

T-Duality, Functional Equation

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We come now to the time when, a hundred years after the conjectures of Gauss and Legendre [about the asymptotic distribution of the primes], the theorem was finally proved. The way was opened by the work of Hadamard on integral transcendental functions. In 1893 Hadamard proved that the complex zeros of Riemann actually exist; and in 1896 he and de la Vallee—Poussin proved independently that none of them have the real part 1, and deduced a proof of the Prime Number Theorem. It is not possible for me now to give an adequate account of the intricate and difficult reasoning by which these theorems are established. But the general ideas which underlie the proofs are, I think, such as should be intelligible to any mathematician. ... The arguments which I have advanced are not exact: I have merely put forward a chain of reasoning which seems likely to lead to the desired result. The achievement of Hadamard and de la ValleePoussin was to replace these plausibilities by rigorous proofs. It might be difficult for me to make clear to you how great this achievement was. Some branches of pure mathematics have the pleasant characteristic that what seems plausible at first sight is generally true. In this theory anyone can make plausible conjectures, and they are almost always false. Nothing short of absolute rigour counts; and it is for this reason that the Analytic Theory of Numbers, while hardly a subject for an amateur, provides the finest possible discipline in accurate reasoning for anyone who will make a real effort to understand its results. Godfrey H. Hardy, 1915 [Hard2, pp. 350-354], in his lecture on Prime Numbers The zeta-function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity. ... The main interest comes from trying to improve the Prime Number Theorem, i.e., getting better estimates for the distribution of the prime numbers. The secret to the success is assumed to lie in proving a conjecture which Riemann stated in 1859 without much fanfare, and whose proof has since then become the single most desirable achievement for a mathematician. Martin C. Gutwiller, 1990 [Gut2, p. 308] It is perhaps fitting t h a t the same mathematician who brought Riemannian geometry to the world (with such an impact on physics, especially general relativity, half a century later) also proposed what later came to be known as the most famous open problem of mathematics, the so-called Riemann Hypothesis. In developing his geometry, Georg Friedrich Bernhard Riemann (1826-1866) was motivated by the work of his predecessors—including

12

M. L. Lapidus

Karl Friedrich Gauss (1777-1855) and the co-discoverers of non-Euclidean geometry, Nikolai Ivanovich Lobachevsky (1793-1856) and Johann (or Janos) Bolyai (1802-1860)—as well as by philosophical and physical considerations.5 On the other hand, Riemann's Conjecture (or Hypothesis) concerning the location of the critical zeros of the Riemann zeta function £ = £(s) (namely, C(s) = 0 with 0 < Re s < 1 implies that Re s = | ) seems to have had entirely different and purely 'internal' motivations. The Riemann Hypothesis has fascinated mathematicians since its introduction by Riemann in his famous inaugural lecture to the Berlin Academy of Sciences in 1859 (see [Riel]). Curiously, it was presented almost as a passing remark or conjecture within [Riel], the only paper by Riemann devoted to number theory. (See the second quote heading this introduction.) One of the main goals of Riemann in [Riel] (admittedly never stated overtly) seems to have been to provide the tools needed to establish the then still unproved 'Prime Number Theorem' (conjectured by Gauss and Legendre) according to which, in particular, (i.4.i)

n(x) = ^ ( i

+ 0 (i))

as x —• +oo, where o(l) denotes a function tending to zero as x —* +oo and n(:r) = E p ^ l denotes the 'prime number counting function', equal to the number of primes p not exceeding x > 0. The Prime Number Theorem6 was eventually proved almost forty years later in 1896, simultaneously and independently by Jacques Hadamard [Had2] and Charles-Jean de la Vallee Poussin [dVl]. (See also the earlier key papers [vMl,2] and [Hadl], along with the later and more precise error estimate obtained in [dV2].) We refer the interested reader to Edwards' book [Edw] or to W. Schwarz's recent survey article [Schw] for a detailed history of the Prime Number Theorem. As is wellknown (see, for example, [Edw], [In], or [Pat,§1.8]), the Riemann Hypothesis is equivalent to the statement that the prime numbers are asymptotically distributed as harmoniously as possible or, more precisely, that the error 5 Referring, in particular, to Riemann's groundbreaking Habilitationschrift—titled On the Hypotheses at the Foundations of Geometry and presented in 1854 to the University of Gottingen—Sir Arthur S. Eddington—the British astronomer whose observation of the 1919 total eclipse of the Sun first confirmed the bending of light rays grazing a massive body (like the Sun), as predicted by Einstein's theory of general relativity—made the following statement (quoted in [Ac,p.l9]): "A geometer like Riemann might almost have foreseen the more important features of the actual world." 6 either in the form (1.4.1) or in the following stronger form conjectured by Gauss, II(x) = Li(x)(l + o(l)) as x -> +oo, where Li(x) = l i m £ ^ 0 + ( / 0 '" + fi+Jukldt denotes the integral logarithm.

T-Duality, Functional Equation

13

term in the statement of the Prime Number Theorem (in the form given in the last footnote) is best possible.7 Arguably, the most beautiful and useful result obtained by Riemann in [Riel] is the so-called Riemann 'explicit formula', connecting U.(x) (or related counting functions) and the critical (or nontrivial) zeros of the Riemann zeta function ((s). (See, e.g., [Edw,Chapter3], [In], [Pat,Chapter3], [ParShal,§2.5], and [Lap-vF2,p.4 and pp.75-76].) Riemann's formula is sometimes referred to as the Riemann-von Mangoldt formula (see, e.g., [LapvF2,§4.5]) because a suitable version of it was later proved in full rigor by von Mangoldt [vMl,2] in the mid-1890's. (See Equation (2.4.20) in Section 2.4.1 below for a classic version of Riemann's formula.) We note that such an explicit formula—along with its later generalizations to other parts of number theory—has recently been extended to the setting of fractal geometry in [Lap-vF 1,2] in order to develop the theory of complex dimensions of fractal strings and to precisely describe in terms of the underlying complex dimensions the oscillations intrinsic to the geometry or the spectrum of fractals. (See [Lap-vF2], Chapter 4 and the relevant applications discussed in Chapters 5-9.) Earlier, in [LapMal,2], a geometric reformulation of the Riemann Hypothesis had been obtained in terms of a natural inverse spectral problem for the vibrations of fractal strings. Rephrased in a more pictorial language, the work of [LapMal,2] can be stated as saying that the question (a la Mark Kac [Kacl]) Can one hear the shape of a fractal drum?—suitably interpreted as the abovementioned inverse problem, connecting the geometric and spectral oscillations of a fractal string—is intimately connected with and, in fact, equivalent to the Riemann Hypothesis. This characterization of Riemann's Conjecture was extended and placed in a broader context in [Lap-vF2], especially Chapter 7. In particular, the intuitive picture of the critical strip 0 < Re s < 1 for £(s)—suggested by the work in [LapPol,2] (see especially [LapPo2,§4.4b], along with [Lap2,Figure 3.1 and §5] and [Lap3,§2.1,§2.2 and p.150]) and corroborated by the results of [LapMal,2]—has been justified rigorously in [Lap-vFl,2]. (See [Lap-vF2,Figure 7.1,p.165] and the discussion surrounding it.) In my opinion, the importance of the Riemann Hypothesis does not lie solely in the incredible multiplicity of its equivalent forms, but also in the cryptic message which it carries with it: one about the geometry of a landscape so far inaccessible to us, that underlying the prime numbers (or the integers). Once we will have found the clues needed to decode this message, 7 Namely, for every 5 > 0, Il(a;) = Li(x) + 0(x2 +<s ) as x —> +oo; see, e.g., [Pat,§1.8 and §5.8].

14

M. L. Lapidus

we should be able to discover and unify large new parts of mathematics, lying at the confluence of arithmetic and geometry. Along similar lines, one could perhaps consider the Riemann Hypothesis— together with the related information concerning the statistical distribution of the prime numbers (for example, that connecting the critical zeros of the Riemann zeta function and aspects of random matrix theory [Mon,Ber3-4, Odl-2,Gut2,KatSarl-2,BerKe])—as a mathematical analogue of the recent COBE observations regarding the extraordinary uniformity (and the tiny fluctuations) of the Penzias-Wilson cosmic background radiation [Tri,PuGis].

1.5

Motivations and Objectives of This Essay

At least from the physical point of view, our goal in the present essay is more modest than suggested earlier. Indeed, we do not plan to discuss a model of geometry that is supposed to fit physical reality at scales at which quantum gravity plays an essential role. Instead, we would like to propose a geometric and physical model that may help us to better understand aspects of number theory, particularly the set of prime numbers (or of integers) and the associated Riemann zeta function, along with their various generalizations for algebraic number fields and curves over finite fields arising naturally in arithmetic geometry (see, for example, [ParShal,2]). This new model (along with its broad extension in [Lap7]) is motivated in part by several mathematical and physical sources, including the following ones: (i) The theory of fractal strings [Lapl-4,LapPol-3,LapMal-2,HeLapl2,Lap-vFl-5,HamLap] (to be viewed here as 'fractal membranes') and the corresponding theory of (fractal) complex dimensions recently developed in the author's research monograph joint with Machiel van Frankenhuysen and entitled Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions [Lap-vF2]. (ii) More generally, the study of the vibrations of fractal drums, associated with Laplacians (or more general elliptic differential operators) on open sets with fractal boundary or on suitable (self-similar) fractals themselves. (See, for instance, [Lapl-6,LapFl,LapPol-3,LapMal-2,HeLapl-2,KiLapl2,LapPan,LapNRG,GriLap,Lap-vFl-5] and the relevant references therein related, in particular, to the "Weyl-Berry Conjecture" [Weyl-2,Berl-2].) We note that fractal strings correspond to the one-dimensional case of 'drums with fractal boundary' but also share certain features with the latter situation.

T-Duatity, Functional Equation

15

(iii) String theory from theoretical physics and its striking dualities, especially the so-called T-duality, a key symmetry not present in ordinary quantum mechanics and which enables us, for example, to identify physically two circular spacetimes with reciprocal radii. (See, e.g., [Asp,EvaGia, GivePR,Gre,Polcl-4,Sch2-4,Val-2,Witl4,16-17].) (iv) Noncommutative geometry and the recent attempts to connect it with conformal field theory and string theory. (See especially [FroGa, ChaFro,Chal-2,LiSzl-2,FroGrRel-2].) (v) Recent attempts at connecting aspects of noncommutative geometry and fractal geometry. (See especially [ConSul], [Con6,§IV.3]—particularly [Con6,§IV.3(e)], motivated in part by [LapPol-2]—as well as [Lap3,Part II], [Lap5], [Lap6] and [KiLap2].) (vi) The intriguing work of Deninger [Denl-7] on a possible cohomological interpretation of analytic number theory and on the Extended Weil Conjectures and in particular, on the (Extended) Riemann Hypothesis.8 Next, we present the contents of the rest of this work: In the main part of this paper (Chapter 2), we discuss the simple but important model of (closed) string theory on a circle (or more generally, on a finite-dimensional torus). We do so both from the standard physical point of view (in Section 2.2) and—following the work of Frohlich and Gawgdzki [FroGa], pursued by Lizzi and Sabo in [LiSzl,2]—from the point of view of noncommutative geometry (in Section 2.3). T-duality—which identifies the physics associated with string theory on toroidal spacetimes with dual lattices and in particular, on circles of reciprocal radii—is presented from each perspective in Sections 2.2 and 2.3, respectively. In Section 2.4, we suggest that in this context, the functional equation for the Riemann zeta function C, = C(s) is a natural counterpart of T-duality for string theory on a circle (or more generally, on a fractal membrane, in the sense of [Lap7]), while the Riemann Hypothesis may be connected, in particular, to the existence of a fundamental (or minimum) length in string theory, itself a consequence of T-duality. (We point out for the interested reader that in the first part of Section 2.4, we review some of the basic properties of £(s)—and of other number-theoretic zeta functions—that are used throughout much of this work; see Section 2.4.1 .) 8

See, e.g., Appendix B to [Lap7] for a brief discussion of the classic Weil Conjectures [Wei5] (and Theorem [Wei 1-4], in the case of curves over finite fields), along with some of their motivations.

M. L. Lapidus

16

This essay may serve as a relatively nontechnical introduction t o the beginning of a theory presented in a book in preparation by the author [Lap7], entitled In Search of the Riemann Zeros (and subtitled Strings, fractal membranes, and noncommutative spacetimes). In the latter research monograph is developed, in particular, a theory of fractal membranes—viewed as suitable quantum or multiplicative analogues of the author's fractal strings. The notion of fractal membrane—interpreted heuristically as a noncommutative (adelic) infinite dimensional torus 9 , in the spirit of the model presented in Chapter 2—may help in the long-term provide a unified geometric, algebraic and physical framework within which to understand number-theoretic zeta functions and their associated arithmetic geometries. More concretely, the well-known Euler product representation of the Riemann and other arithmetic zeta functions is naturally interpreted in this context in terms of the quantum partition function of the corresponding fractal (or prime) membrane. We close this introduction by pointing out t h a t in an epilogue to the present paper, entitled Fractal Membranes and Arithmetic Site, we will provide a somewhat more detailed description of the part of [Lap7] dealing with the theory of fractal membranes and its connections with zeta functions. 10 It should be made clear to the reader that the research program outlined in this essay (as well as in the forthcoming book [Lap7]) is still in a preliminary form and t h a t some of the mathematics involved or implied is rather formidable or even not yet formulated in a precise manner. We hope, nevertheless, t h a t some of the ideas presented here—once suitably modified or extended—may serve as a motivation for future research at the interface of noncommutative geometry, fractal geometry and aspects of arithmetic geometry and number theory.

2

String Theory on a Circle and T-Duality: Analogy with the Riemann Zeta Function Nature is simple if we look at it the right way. For example, I believe that God created just two dimensions—one of space and one of time. 9

or equivalently, as an adelic Hilbert cube, with opposite faces identified. We note that because in [Lap7], we consider the vibrations of fractal membranes rather than of fractal strings, the role played by the Riemann zeta function and the Riemann Hypothesis (or their generalization) is rather different than in the previous work of the author and his collaborators, Carl Pomerance, Helmut Maier, Christina He and Machiel van Prankenhuysen [Lapl-4,LapPol-3,LapMal-2,HeLapl-2,Lap-vFl-3]. 10

T-Duality, Functional Equation

What could be simpler? And then at a later epoch, there was a phase transition to four dimensions, plus six internal ones. Two is the heart of the matter in this theory and this number two will not be changed to three. Abdus Salam, 1988 [DaviBr, pp. 175-176] Recall that ... a particle's quantum wavelength is inversely proportional to its momentum, which, roughly speaking, is its energy. And so, by increasing a point particle's energy, its quantum wavelength can be made shorter and shorter—quantum smearing can be decreased further and further—and hence we can use it to probe ever finer physical structures. Intuitively, higher-energies particles have greater penetrating power and are therefore able to probe more minute features. In this regard, the distinction between point particles and strands of string becomes manifest. Just as [is] the case for plastic pellets probing the surface features of a peach pit, the string's inherent spatial extend prevents it from probing the structure of anything substantially smaller than its own size—in this case structures arising on length scales shorter than the Planck length. Somewhat more precisely, in 1988 David Gross of Princeton University and his student Paul Mende showed that when quantum mechanics is taken into account, continually increasing the energy of a string does not continually increase its ability to probe finer structures, in direct contrast with what happens for a point particle. They found that when the energy of a string is increased, it is at first able to probe shorter-scale structures, just like an energetic point particle. But when its energy is increased beyond the value required for probing structures on the scale of the Planck length, the additional energy does not sharpen the string probe. Rather, the energy causes the string to grow in size, thereby diminishing its short distance sensitivity. In fact, although the size of a typical string is the Planck length, if we pumped enough energy into a string—an amount of energy beyond our wildest imaginings but one that would likely have been attained by the big bang—we could cause it to grow to a macroscopic size, a clumsy probe of the microcosmos indeed! It's as if a string, unlike a point particle, has two sources of smearing: quantum jitter, as for a point particle, and also its own inherent spatial extent. Increasing a string's energy decreases the smearing from the first source but ultimately increases the smearing from the second. The upshot is that no matter how hard you try, the extended nature of a string prevents you from using it to probe phenomena on sub-Planck-length distances. But the whole conflict between general relativity and quantum mechanics arises from the sub-Planck-length properties of the spatial fabric. / / the elementary constituent of the universe cannot probe subPlanck-length distances, then neither it nor anything made from it

18

M. L. Lapidus can be affected by the supposedly disastrous short-distance quantum undulations. ... An electron microscope has the ability to resolve surface features to less than a millionth of a centimeter; this is sufficiently small to reveal the numerous surface imperfections [of a well-polished block of granite, for example]. By contrast, in string theory, there is no way to expose the sub-Planck-scale "imperfections" in the fabric of space. In a universe governed by the laws of string theory, the conventional notion that we can always dissect nature on ever smaller distances, without limit, is not true. There is a limit, and it comes into play before we encounter the devastating [effects of] quantum foam. Brian R. Greene, 1999 [Gre, pp. 155-156]

2.1

Quantum-Mechanical Point-Particle on a Circle

It is well-known t h a t the space of states of a quantum-mechanical pointparticle moving in a circle of radius R, S(R) = R/2irRZ, is described by the complex Hilbert space Ti = L2(S(R),ds) consisting of all square-integrable complex-valued functions on S(R), where ds denotes (normalized) Haar measure on S{R).U Equivalents, H = L2(S\ ^ ) , where S1 = 5(1) = M/27rZ denotes the unit circle equipped with the Riemannian metric ds2 = R2dx2. Note that the radius R (or equivalently 2wR, the length of the circle) is the only metric invariant in this situation. Hence the positive half-line {r : r > 0} parametrizes the space of all Riemannian circles (in the plane). The circle S1 is equipped with its natural Dirac operator D = -hsR-1-^, the square of which yields the (negative) Laplacian or (free) Hamiltonian on the circle, H = D2 = — \R~2^Moreover, from the point of view of quantum mechanics, we can recover the geometry of the circle—viewed as a set of points, or more precisely, as a smooth Riemanian manifold—from the spectral triple associated with the Dirac operator [Con5]

(2X1)

R(uj) and Zst,R{-\/u) = Zst,R(uj)), we now apply a special case of the Poisson Summation Formula (2.4.11), namely (for t > 0 and tiel), oo

(2.2.29)

oo

exp(-7rtv2 + 2niuv) = t~1/2

^ V= — OO

7r(m — u)

J ^ exp

t

T71= —OO

Writing the double sum, ^2V w as J^ w ^Z m the ^as^ e Q u a hty of (2.2.25) and letting, for each fixed w € Z, t = u^/R2 and u = wiw in formula (2.2.29), we obtain after some elementary manipulations (and noting that \m — UJW\2 = m2 +UJ\W2 + w2w2 — 2mwuJi):

(2.2.30)

ZstM(uj) = RZ,(u)

TTR

J2 exp

\m — wu>\ UJ2

m,w=—oo

where Z» = (2.2.31)

Z„{UJ)

is independent of Z? and is defined by Z.(u>) :=

\r,(w)\-W/2,

with 77 = r/(a;) given by (2.2.26). Since u 2 = I m w = I m ( u + l), we have Z*(UJ) = Zt(uj+1) by (2.2.28a), and since Im(—1/w) = LU2/\U}\2, we also have Zt(—l/w) = Z»(<j) by (2.2.28b). Hence, Z* is modular invariant. Further, the double sum on the right-hand side of (2.2.30) is easily seen to be invariant under u> u> + 1 (by making the change of variables m' = m + w, w1 = w). Moreover, it is also seen to be invariant under w —1/ui (by making the change of variables m' = —w, w' = m).2i Therefore, in light of (2.2.30), we conclude that the partition function, Zst = ZstR, associated with string theory on a circle of radius R, is modular invariant, as desired: (2.2.32a) 24

ZsttR(uj + 1) = Zst,R(u)

which, not coincidentally, is identical to the transformation v — w and w «-» v (exchanging vibration and winding numbers) in our earlier discussion of circle duality preceding Equation (2.2.24).

34

M. L. Lapidus

and (2.2.326)

= ZsttR(u;).25

Zst,R(-l/cj)

Since the translation w *—• w + 1 together with the inversion u — i > — \/uj generate SL(2, Z), 26 this means that Zst = Zst,R is invariant under the action of the modular group SL(2, Z). Finally, we close this part of our discussion of circle duality by noting that, in light of the last equality of (2.2.25), the partition function Zst = ZstR is clearly invariant under the transformation R 1dxv{= | Sui/=i g^dx^dx", according to Einstein's summation convention), where {x^}d=1 are the angular coordinates on Td. As in [FroGa,§2.2] and [LiSzl,2], we allow for the possibility of torsion, due to the presence of an anti-symmetric (topological) instanton, namely a 2-form (3 = (3lil/dxfldx1' on Td satisfying appropriate integrality conditions. Then the (nonsingular) matrices d± = {rf* }d „ =1 are defined by (2.2.34)

d% =

9liv

± /V

At the quantum level, the center-of-mass coordinates {X^i}dl and the left-right momenta operators {pt}t=\ satisfy the following commutation relations (a higher-dimensional analogue of (2.2.13)): (2-2.35) 25

[X¥,fi] = -i&i,

At least at the physical level of rigor, the expression (2.2.30) for Zst—along with the modular invariance (2.2.32) of Zst—can be obtained from a Feynman-Polyakov path integral representation for Zst (and a standard application of the method of stationaryphase involving infinite dimensional Gaussian integrals), along with a suitable change of variables. See, for example, [Polc3,§8.2,pp. 237-238]. 26 Recall that 5L(2,Z) is the group of 2 x 2 matrices (la) with determinant one and coefficients in Z, acting on the Poincare upper-half plane {u> = U\ + iu>2 6 C : W2 > 0} via the Mobius transformations u — i > ?"fS.

T-Duality, Functional Equation

35

with all the remaining commutators vanishing. Further, the oscillatory modes { a * " : n G Z}£ =1 , with (a±-")* = Q^M, a j ' " = g»up±, and acting as creation (n < 0) and annihilation (n > 0) operators on a commuting pair of Fock spaces, satisfy the following commutation relations (the counterpart of (2.2.14)): (2-2.36)

[a^,a^]=ng^Sn+mfi,

with all other commutators vanishing and where {g,iVY^,v=\ denotes the inverse matrix of ( v l ^ i Moreover, the left-right momenta, also denoted by {pt}, are given by (2.2.37)

p± = - ^ ( ^ ± d > " ) ,

where {fcM}^=1 £ L* and {w M }^ =1 € L stand for the space-time (or uniformvibrations) momenta and the winding modes, respectively. (Compare with Equation (2.2.10) above, where the notation are somewhat different.) Here, following standard notation in Riemannian geometry, we have k^ = g^k"', for example. Further, L* is the dual lattice of L, relative to the inner product induced by {g^}- [Physically, the dual lattice L* represents the set of allowed (spacetime) momentum modes while the lattice L itself represents the set of allowed winding modes. Also, w^ corresponds to the number of times the closed string wraps around the /i-th circle (or cycle) of the torus Td = {Sl)d.} Hence, the Narain lattice is the set of left-right momenta A = {(p+,p~)}, with p± := {pt}t=i> equipped with the following quadratic form (an analogue of (2.2.11)): (2.2.38)

Q((P+,P-))

= p+gTPt - P»9TPZ = 2 * ^ " € 2Z.

It is an integral, even self-dual Lorentzian lattice, of rank 2d and of signature (d, d), given as the direct sum of the dual lattices L* and L (equipped with dual metrics): (2.2.39)

A = L* © L.27

Recall that if ( • , • )A denotes the (Lorentzian) inner product associated with Q by polarization, then, the dual lattice A* is defined to be the set of points 27

Note that if d = 1,0 = 0 and Td = S(R) = R/2TTLZ, as in the rest of this section, then L = BZ, L* = R~1Z and so A = RL 0 R~1Z. Alternately, much as in Section 2.1, one could let L = Z and L* = Z, each equipped with the appropriate dual metrics and associated inner products.

36

M. L. Lapidus

q = (q+, 2 6 C with uii > 0: Zst(io) = ZstA(u>) M

=hHr E

exp

M"(P+)2

-*(p-) 2 )]

peA

2d

= ir?Mr E

ex

p [-™*((p+)2 + ip-f)+^i((p+)2

- (p-f)],

peA

where p = (p+,p~) G A and (p*) 2 := g^ptpt(Note that in the displayed equation just above, the exponent of |/7(w)| is now —2d instead of —2. The reader can easily check that when d = 1, as in the rest of this section, the last expression for Zst(u>) = Zati\(u>) just above reduces to the last one for Zst(u) = ZstiR(uj) occurring in Equation (2.2.25).) Then, using the Poisson Summation Formula (see (2.4.11) in Section 2.4 below) along with the modular transformation (2.2.27) of the Dedekind eta function r\ = rj(uj), we obtain (see [Polc3,Eq.(8.4.23),p.252]) Z s t,A*(-l/w) =

vAZstA(ui),

M. L. Lapidus

38

where VA denotes the volume of the fundamental domain (or unit cell) of A. The self-duality of A (i.e., A* = A) implies t h a t v\ = 1 (because VA = v^}) and hence (2.2.45)

ZstA{-l/w)

=

ZstA(u),

which together with the easily verified identity (using (2.2.27a)) (2.2.45a)

Zrf,A(w + 1) =

ZstA(w),

implies the modular invariance of the partition function Zst = ZstA(= Zst^), in the present case of (bosonic, closed) string theory on a d-dimensional torus. (As is observed in [Polc3,p.252], the modular invariance of ZstA is actually equivalent to the self-duality of A, since (2.2.32') must hold for all w = uj\ +iu)2 e C with CJ2 > 0.) We refer the interested to [Polc3,§8.4,pp.251-255] for more details and for an interesting discussion of the consequences in the present higher-dimensional context of the key properties of the Narain lattice A for the T-duality symmetry (viewed as an 'enhanced gauge symmetry', see [Polc3,§8.3]), which is much enlarged from the one-dimensional case. Remark 2.2.3 ( T - D u a l i t y a n d Mirror S y m m e t r y ) . As was alluded to earlier, T-duality has many other aspects, including nonperturbative ones. (In fact, as is pointed out in [Polc3,4], T-duality is an exact symmetry of string theory, not just one valid at the perturbative level.) We will not discuss them here but refer instead the interested reader to Polchinski's second volume on String Theory, [Polc4]. Moreover, as is explained by Cumrun Vafa in his interesting recent survey article on Geometric Physics, [Va2], "the notion of duality [is] in some sense a non-linear infinite dimensional generalization of the Fourier transform" [Va2,p.537]. Circle duality—or T-duality on circular space-times—is only the simplest example of such dualities, or mirror symmetry. Further on, Vafa writes ([Va2, pp. 541-542], 1998): One of the most interesting aspects of string theory is that strings moving on one manifold may behave identically with strings moving on a different manifold. Any pair of manifolds M\ and M2 which behave in this way are called mirror pairs. Of course, this would be a trivial duality if M\ and M2 are isomorphic Riemannian manifolds. The interesting dualities arise when M\ and M2 are distinct Riemannian manifolds. In some cases M\ and M2 are topologically the same [as for circle (or torus) duality], but in some cases they are distinct even topologically ... The simplest example of mirror symmetry corresponds to choosing M\ to be a circle [of radius R] and M2 to be a circle [of radius 1/R]. This is a case that can be rigorously proven ...

T-Duality, Functional Equation

39

Earlier on in his paper ([Va2, pp. 539-540], 1998) Vafa gives the following general and enlightening definition of the notion of physical duality, inspired by string theory: Consider a physical system Q (which I will not attempt to define). And suppose this system depends on a number of parameters [Aj]. Collectively, we denote the space of the parameters A, by M, which is usually called the moduli space of the coupling constants of the theory. The parameters Aj could for example define the geometry of the space the particles propagate in, the charges and masses of particles, etc. Among these parameters there is a parameter Ao which controls how close the system is to being a classical system (the analog of what we call h in quantum mechanics). For Ao near zero, we have a classical system and for Ao > 1 quantum effects dominate the description of the physical system. [In the case of circle duality, Ao is proportional to 1/R, the reciprocal of the radius; hence, the "classical limit" when Ao —> 0 corresponds to large scales. Similarly, for more general mirror symmetries, Ao can be taken to be the reciprocal of the volume of the manifold.] Typically, physical systems have many observables which we could measure. [The momentum operator P and the Hamiltonian operator H are two such observables in the above discussion of circle duality.] Let us denote the observables by Oa. Then we would be interested in their correlation functions which we denote by (Oai

. • • Oan)

= /Ql...Qn(Aj).

Note that the correlation functions will depend on the parameters defining Q. The totality of such observables and their correlation functions determines a physical system. Two physical systems Q[M,Oa], Q[M,Oa] are dual to one another if there is an isomorphism between M and M and O O respecting all the correlation functions. Sometimes this isomorphism is trivial and in some cases it is not. We are interested in the cases where this isomorphism is non-trivial. In such cases typically what happens is that a parameter which controls quantum corrections Ao on one side gets transformed to a parameter Afc with k ^ 0 describing some classical aspect of the dual side. This in particular implies that quantum corrections on one side have the interpretation on the dual side as to how correlations vary with some classical concept such as geometry. This allows one to solve difficult questions involved in quantum corrections in one theory in terms of simple geometrical concepts on the dual theory. This is the power of duality in the physical setup. Mathematics parallels the physics in that it turns out that the mathematical questions involved in computing quantum corrections in certain cases [are] also very difficult and the questions involved on the dual side are mathematically simple. Thus

40

M. L. Lapidus

non-trivial duality statements often lead to methods of solving certain difficult mathematical problems. One should note, however, that very rarely can one prove (even in the physics sense of this word) that two given physical systems are dual to one another. Often the existence of dualities between two systems is guessed at based on some physical consistency arguments. Testing many non-trivial consequences of duality conjectures leads us to believe in their validity. In fact we have observed that duality occurs very generically, for reasons we do not fully understand. This lack of deep understanding of duality is not unrelated to the fact that it leads to solutions of otherwise very difficult problems. At the mathematical level, evidence for duality conjectures amounts to checking validity of proposed solutions to certain difficult mathematics problems.

2.2.3

T - D u a l i t y a n d t h e E x i s t e n c e of a F u n d a m e n t a l L e n g t h

The quote heading this section and that heading the next one, Section 2.3 (as well as that heading this chapter) explain in detail the physical considerations t h a t lead to deduce from T-duality (in the present case, circle duality), the existence of a fundamental (or minimum) length in string theory. For this reason, we will keep our present discussion rather brief. In these quotes from [Gre], the fundamental length is referred to as Planck length (or scale), for simplicity. However, if the string tension (2na')~1 is taken into account, the fundamental (or minimum) length is equal to [a1)1!2 and is called the string length or scale. (See the footnote following Equation (2.2.10).) The string length is often taken to be of the order of Planck length ( « 10~ 3 3 cm, see footnote (4)) but—depending on the value of the string tension, the basic parameter of string theory—it may also be significantly different from it. (See, for example, [GreeSWit],[Kak], and [Polc3,4].) We assume here t h a t we have chosen units so t h a t the string length is equal to 1. We begin by citing an excerpt from [FroGa] t h a t describes this topic in more technical details. Referring to circle duality (discussed in Section 2.2.2 above 2 8 ), Frohlich and Gawedzki write ([FroGa, p. 67], 1994): 29 This phenomenon distinguishes the stringy geometry of circles from their point-set Riemannian geometry. While the space of Riemannian circles was the half line parametrized by the radius R, that of the sigma 28

and reinterpreted from the point of noncommutative geometry, as in Section 2.3 below ([FroGa], [LiSzl,2]). 29 We note that we have slightly adapted the notation of [FroGa] to our present ones.

T-Duality, Functional Equation

41

models with circle targets is an orbifold of the latter obtained by the identification of R with R~l. The above is an example of a more general duality phenomenon responsible for the appearance of a fundamental length in string theory. This is certainly one of the most promising features of the latter. In order to understand how such a length scale arises, let us think of how one probes the effective space(-time) geometry in string theory. As mentioned before, this should be done by looking at the low energy states of the string in which the stringy internal modes are not excited and one effectively sees quantum mechanics of point-like objects. Let us suppose that the string vacuum is described by a sigma model with the Sl target. The oscillatory modes of the string created by [the] operators a*, n < 0, have energies quantized to integer values, so we should look at the states with energies C 1. / / the radius R of the circle is much bigger than 1 (or than the Planck length [really, the string length] in dimensional units) then the low-energy spectrum of the sigma model is given by the states |po;0), Po £ Z, describing the barycentric degree of freedom x° [the position of the center-of-mass of the string]. We effectively obtain the quantum mechanics of a particle moving on the circle of radius R. If R is much smaller than 1, then the low-energy spectrum of the sigma model is given by the states \0;w) corresponding to the winding modes of the string. But this is exactly like the spectrum in the quantum mechanics of a particle moving on the circle of radius R~l. As a result, we never see circles of small radii as effective geometries! Notice that this is a quantum phenomenon as is signaled by the presence of h in the expression for the Planck length.

We close this subsection by briefly mentioning t h a t the entire discussion of T-duality—and hence, its crucial physical consequence, the existence of a fundamental length—applies just as well to toroidal (rather than circular) spacetimes of arbitrary finite dimension. (See Remark 2.2.2 just after Equation (2.2.23) for the easy extension of the setting of Section 2.2 to toroidal spacetimes.) Intuitively, this is to be expected and can be verified in much the same way as for one-dimensional tori (i.e., circles). However, the curious reader may wonder for which types of spacetimes analogous conclusions can be drawn. There are beautiful formulations of (and partial results for) the socalled "Mirror Symmetry Conjectures" for spacetimes t h a t are Calabi-Yau manifolds, for example, and these are mathematically very challenging. (See, [Va2], [Witl2], and the relevant references therein.) We do not wish to discuss further this fascinating topic, however. Instead we would like to quote once again Brian Greene [Gre, pp. 254-255] who pursues in a more general context the train of thought presented in the first quote (from [Gre,pp.249252]) heading this section (i.e., Section 2.2):

M. L. Lapidus What if the spatial dimensions are not circular in shape? Do these remarkable conclusions about minimal spatial extent in string theory still hold? No one knows for sure. The essential aspect of circular dimensions is that they permit the possibility of wound strings. As long as the spatial dimensions—regardless of the details of their shape— allow strings to wind around them, most of the conclusions we have drawn shoud still apply. But what if, say, two of the dimensions are in the shape of a sphere? In this case, strings cannot get "trapped" in a wound configuration, because they can always "slip off" much as a stretched rubber band can pop off a basketball. Does string theory nevertheless limit the size to which these dimensions can shrink? Numerous investigations seem to show that the answer depends on whether a full spatial dimension is being shrunk [as in the case of circles or tori] ... or ... [whether] an isolated "chunk" of space is collapsing. The general belief among string theorists is that, regardless of shape, there is a minimum limiting size, much as in the case of circular dimensions, so long as we are shrinking a full spatial dimension. Establishing this expectation is an important goal for further research because it has a direct impact on a number of aspects of string theory, including its implications for cosmology.

Noncommutative Stringy Spacetimes and T-Duality Distance is such a basic concept in our understanding of the world that it is easy to underestimate the depth of its subtlety. With the surprising effects that special and general relativity have had on our notions of space and time, and the new features arising from string theory, we are led to be a bit more careful even in our definition of distance. The most meaningful definitions in physics are those that are operational—that is, definitions that provide a means, at least in principle, for measuring whatever is being defined ... How can we give an operational definition of the concept of distance? The answer to this question in the context of string theory is rather surprising. In 1988, the physicists Robert Brandenberger of Brown University and Cumrun Vafa of Harvard University pointed out (in [BrVa]) that if the spatial shape of a dimension is circular, there are two different yet related operational definitions of distance in string theory. Each lays out a distinct experimental procedure for measuring distance and is based, roughly speaking, on the simple principle that if a probe travels at a fixed and known speed then we can measure a given distance by determining how long the probe takes to traverse

T-Duality, Functional Equation

it. The difference between the two procedures is the choice of probe used. The first definition uses strings that are not wound around a circular dimension, whereas the second definition uses strings that are wound. We see that the extended nature of the fundamental probe is responsible for there being two natural operational definitions of distance in string theory. In a point-particle theory, for which there is no notion of winding, there would be only one such definition. How do the results of each procedure differ? The answer found by Brandenberger and Vafa is as surprising as it is subtle. The rough idea underlying the result can be understood by appealing to the uncertainty principle. Unwound strings can move around freely and probe the full circumference of the circle, a length proportional to R [the radius of the circle]. By the uncertainty principle, their energies are proportional to 1/R (recall [from quantum mechanics] the inverse relation between the energy of a probe and the distance to which it is sensitive). On the other hand, we have seen that wound trings have minimum energy proportional to R; as probes of distances the uncertainty principle tells us that they are therefore sensitive to the reciprocal of this value, 1/R. The mathematical embodiment of this idea shows that if each is used to measure the radius of a circular dimension of space, unwound string probes will measure R while wound string probes will measure 1/.R, where, as before, we are measuring distances in multiples of the Planck length. The result of each experiment has an equal claim to being the radius of the circle—what we learn from string theory is that using different probes to measure distance can result in different answers. In fact, this property extends to all measurements of lengths and distances, not just to determining the size of a circular dimension. The results obtained by wound and unwound string probes will be inversely related to one another. [Italics added.] If string theory describes our universe, why have we not encountered these two possible notions of distance in any of our day-to-day or scientific endeavors? Any time we talk about distance, we do so in a manner that conforms to experience of there being one concept of distance without any hint of there being a second notion. Why have we missed the alternative possibility? The answer is that although there is a high degree of symmetry in our discussion, whenever R (and hence 1/R as well) differ significantly from the value 1 (meaning again, 1 times the Planck length), then one of our operational definitions proves extremely difficult to carry out while the other proves extremely easy to carry out. In essence, we have always carried out the easy approach, completely unaware of there being another possibility. The discrepancy in difficulty between the two approaches is due to the very different masses of the probes used—high-winding-energy/low-

M. L. Lapidus

vibration-energy, and vice versa—if the radius R (and hence 1/R as well) differs significantly from the Planck length (that is, R = 1). "High" energy here, for radii that are vastly different from the Planck length, corresponds to incredibly massive probes—billions and billions times heavier than the proton, for instance—while "low" energy corresponds to probe masses at most a speck above zero [in Planck units]. In such circumstances, there is a monumental difference in difficulty between the two approaches, since even producing the heavy-string configurations is an undertaking that, at present, is beyond our technological prowess. In practice, then, only one of the two approaches is technologically feasible—the one involving the lighter of the two types of string configurations. This is the one used in all of our discussions involving distance encountered to this point. This is the one that informs and hence meshes with our intuition. Putting issues of practicality aside, in a universe governed by string theory one is free to measure distances using either of the two approaches. When astronomers measure the "size of the universe" they do so by examining photons that have traveled across the cosmos and happened to enter their telescopes. No pun intended, photons are the light string modes in this situation. The result obtained is [equal to] 1061 times the Planck length [or about 15 billion light-years] ... If the three familiar spatial dimensions are in fact circular and string theory is right, astronomers using vastly different (and currently nonexistent) equipment, in principle, should be able to measure the extent of the heavens with heavy wound-string modes and find a result that is the reciprocal of this huge distance. It is in this sense that we can think of the universe as being huge, as we normally do, or terribly minute. According to the light string modes, the universe is large and expanding; according to the heavy modes it is tiny and contracting. There is no contradiction here; instead, we have two distinct but equally sensible definitions of distance. We are far more familiar with the first definition due to technological limitations, but, nevertheless, each is an equally valid concept ... Since we now have two concepts of distance—using light or heavy string probes—we must compare measurements made in the same manner ... If one does stick to measuring distances "the easy way"— that is, using the lightest of the string modes instead of the heavy ones—the results will always be larger than the Planck length. To see this, let's think through the hypothetical big crunch for the three extended dimensions, assuming them to be circular. For argument's sake, let's say that at the beginning of our thought experiment, unwound string modes are the light ones and by using them it is determined that the universe has an enormously large radius and that it is shrinking in time. As it shrinks, these unwound modes get heav-

T-Duality, Functional Equation

ier and the winding modes get lighter. When the radius shrinks all the way to the Planck length—that is, when R takes on the value 1—the winding and vibration modes have comparable mass. The two approaches to measuring distances would become equally difficult to carry out, and, moreover, each would yield the same result since 1 is its own reciprocal. As the radius continues to shrink, the winding modes become lighter than the unwound modes and hence, since we are always opting for the "easier approach", they should now be used to measure distances. According to this method of measurement, which yields the reciprocal of that measured by the unwound modes, the radius is larger than one times the Planck length and increasing. This simply reflects that as R—the quantity measured by unwound strings—shrinks to 1 and continues to get smaller, 1/R—the quantity measured by wound strings—grows to 1 and gets larger. Therefore, if one takes care to always use the light string modes—the "easy" approach to measuring distance—the minimum value encountered is the Planck length [which explains the existence of a fundamental length or "minimum size" in string theory]. Brian R. Greene, 1999 [Gre, pp. 249-252] A quantum theory of point particles on [the spin, Riemannian manifold] M naturally supplies [the Hilbert space of spinors] H. = L 2 (spin(M)) and can therefore be thought of as describing an ordinary spacetime M. In the case of string theory then, we can anticipate that the full stringy spacetime is described by a noncommutative geometry, i.e., a spectral triple [denoted by T] with a noncommutative algebra A. In this case the notion of a "space" breaks down, as it is in general impossible to speak of points. In the low-energy limit of the string theory, the vibrational modes are negligible and the model describes ordinary pointlike particles corresponding to an ordinary (commutative) spacetime. Thus in the low-energy limit of the string theory, T should contain a subspace TQ [= (L 2 (spin(M),C(M),D), where D is the Dirac operator on M] representing an ordinary spacetime manifold M at large distance scales. The symmetries of the quantum spacetime T then determine the stringy effects on the geometry of the spacetime 7"o represented by ordinary quantum field theory. In the following we construct the spectral triple T associated with the stringy geometry and show that the full duality group of spacetime is determined in a very simple fashion by the group of automorphisms of the appropriate algebra A. The duality symmetries of the spacetime emerge as the possibility of assigning two independent Dirac operators to the spectral triple T associated with strings, so that duality is simply a consequence of a change of

45

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M. L. Lapidus

metric representing a redefinition of distances in the string spacetime. These automorphisms leave the spectral triple T invariant and correspond to internal symmetries of the quantum geometry. We shall also construct the low-energy projections [onto] TQ and illustrate how the duality isomorphisms of T correspond to the more familiar notions of target space duality [i.e., T-duality] in string theory. These projections then illuminate the full structure of the stringy modification of classical general relativity. Fedele Lizzi and Richard J. Szabo, 1997 [LiSzl, pp. 3581-3582]

2.3.1

Target S p a c e D u a l i t y a n d N o n c o m m u t a t i v e G e o m e t r y

We continue considering closed string theory on a circular spacetime of radius R, as in the previous two sections, viewed as S(R) or as the unit circle M := S1 equipped with the Riemannian metric ds2 = R2dx2 and Haar measure dx/\^2n. Because the setting is identical and the formulas involved are very similar, we will also consider higher-dimensional target spaces, as in [FroGa,LiSzl-2]; namely, toroidal space-times M = Td := Rd/2irL, where L is a (Euclidean) lattice of rank d (for example, M may be a finite product of circles). This should help better understand the special case of circular space-times and place it in a broader framework. (We refer to Remark 2.2.2 where we have indicated the simple changes to be made in order to extend the discussion of Section 2.2 from circular to toroidal target spaces.) We could also work with spacetimes t h a t are Calabi-Yau manifolds, as is suggested in [LiSz2]; we will refrain from doing this, however, since our main focus—from the point of view of the applications—will be circular (and possibly, toroidal) target spaces. Our goal is to provide a geometric description of the circular (or toroidal) spacetime from the point of view of string theory. The underlying quantum spacetime is often referred to as a stringy spacetime (or string 'spacetime') in the physics and mathematical physics literature. A very interesting noncommutative geometric description of a circular (or toroidal) stringy spacetime has been provided by Frohlich and Gaw§dzki in 1994 in [FroGa] and pursued in a number of papers, including those of Chamseddine and Frohlich [ChaFro], Chamseddine [Chal,2], as well as Lizzi and Szabo ([LiSz2], announced in [LiSzl]). A more systematic treatment—giving, in particular, an abstract noncommutative differential geometric version of Witten's classic approach to supersymmetric quantum mechanics (see, for example, [Witl,2,7])—can be found in the interesting recent papers by Frohlich, Grandjean and Recknagel [FroGrRel,2]. References to other approaches—

T-Duality, Functional Equation

Al

motivated by (nonperturbative) M(atrix) or M-theory rather than by ordinary (super)string theory—are provided in the above papers (e.g., [LiSz2]); some of these—particularly the work of Ho and Wu [HoWu] as well as that of Michael Douglas and his collaborators (see, for example, [Dou,DouKPS])— along with aspects of [ProGa] and [LiSzl-2], have been synthetized recently in the paper by Connes, Douglas and Schwarz [ConDouSc] and pursued in a number of works, including the paper by Landi, Lizzi and Szabo [LanLiSz], entitled String Geometry and the Noncommutative Torus. We will follow here especially the approach of the paper [ProGa], Conformal Field Theory and Geometry of Strings, pursued and expanded in the work [LiSz2], Duality Symmetries and Noncommutative Geometry of String Spacetimes. From the point of view of Connes' noncommutative geometry, a noncommutative space (or 'quantum space') is described by a spectral triple T = (H, A, D), where H is a complex Hilbert space, A is a suitable (normed or topological) *-algebra30 represented on H, and D is a suitable unbounded self-adjoint operator (acting on H), viewed as an abstract analogue of the Dirac operator, see [Con5,6].31 (As was alluded to in Section 1.2, further axioms for noncommutative geometry are required by Connes in [Con7,8]. It is noteworthy that they are all satisfied here, as was recently shown in [LanLiSz], but we will not be concerned by those for now.) One original aspect of [FroGa] (along with [LiSzl,2] and [FroGrRel,2], for example), is that from the point of view of noncommutative geometry, a string spacetime (underlying a circular or more generally, toroidal spacetime, say) should be described by two distinct spectral triples; namely, T = (TC, A, D) and T = (H, A, D), where the algebra A and the Hilbert space Ji are the same for 3" and T, but the 'Dirac operators' D and D are distinct (although they are constructed according to a very similar procedure). A crucial fact is that these two spectral triples are isomorphic, as noncommutative spaces. The above setup is ideally suited to give a natural noncommutative geometric description of T-duality—also called target space duality, and specialized to circle duality in the setting of Section 2.2. It can also be used to 30 For example, a C*-algebra or a von Neumann algebra (or often, in practice, a dense subalgebra thereof); see, for instance, [Con6] and [KadRi] for the definition of these terms. 31 Here and thereafter, as in [Con6], we assume that for any spectral triple T = (H,A,D), A,li and D satisfy the following two compatibility conditions: (i) the commutator [D, a] (a e A) is bounded for all elements a in a dense subalgebra of A. (Here, [D, a] is defined via the representation of the elements of A as bounded linear operators of the Hilbert space H.), (ii) D has compact resolvents; that is, for all A £ C\K, (D — A ) - 1 is a compact operator on H. (We note that these conditions are satisfied by all the spectral triples, T,T,TQ and 2~o—constructed in [LiSzl,2]—which will be used in this section.)

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describe more general examples of mirror symmetries as well as other types of symmetries encountered in string theory; we refer to [FroGa, LiSzl-2] for more detailed information. Remark 2.3.1. In [FroGa], the two spectral triples are defined as (7i,A,H) and (H,A,H), where H and H are the Hamiltonian operators instead of 9

the Dirac operators D and D, with H = D2 and H = D . We will prefer, however, the approach of [LiSzl,2]—relying on the use of Dirac operators and suggested in part by the work in [Wit 1-3,7], [SeiWit] and [ChaFro,Chal-2], in addition to [FroGa]—because it is more flexible (enabling one, for example, to study a suitable noncommutative analogue of the de Rham complex, see Section 2.3.3) and is closer to Connes' original point of view. This approach is abstracted and developed in [FroGrRe2]. We postpone until later on in this section a more precise description of the Hilbert space 7i, the noncommutative algebra A and the abstract Dirac operator D (in the case when M = S1 or, more generally, M = Td = Rd/27rL is a d-dimensional torus; see Section 2.3.2. We only mention that the definition of Ji involves the use of Fock spaces and that A is the algebra of vertex operators, representing physically—the space of string interactions or the algebra of observables on the stringy space-time. Further, D (along with D) is a suitable version of Witten's Dirac-Ramond operator (acting on loop space). We assume for now that M is a compact, Riemannian (d-dimensional) spin manifold, with Riemannian metric g = {g^}^=i and real-valued gamma matrices {7M}f=i generating the Clifford algebra (associated with the spin structure of M) and satisfying the standard anti-commutation relations (2.3.1)

{7M,7,} = 2