Mathematical physics 2000

Mathematical Physics

2000

This page is intentionally left blank

Mathematical Physics

2000

Edited by

A Fokas A Grigoryan T Kibble B Zegarlinski Imperial College, London

^ffi

Imperial College Press

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

Library of Congress Cataloging-in-Publication Data Mathematical physics 2000 / edited by A. Fokas ... [et al.]. p. cm. ISBN 186094230X(alk. paper) 1. Mathematical physics-Congresses. I. Fokas, A. S., 1952QC 19.2.1538 530.15-dc21

2000

. II. Title.

00-037042

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

Copyright © 2000 by Imperial College Press Copyright of each article is owned by the contributors). A11rightsreserved. This book, or parts thereof, may not be reproduced in anyform 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.

Printed in Singapore by Uto-Print

V

PREFACE The International Congress on Mathematical Physics in the year 2000 is to be held at Imperial College, London. It occurred to the local organizers that, since this is a natural time to look back at the achievements of the twentieth century and forward to the opportunities of the twenty-first, it would be an appropriate occasion on which to ask a number of dLstinguished mathematical physicists to contribute their personal perspectives on some aspects of the discipline. We did not try to impose any general structure or theme; nor did we aim for comprehensive coverage. Instead, we invited a number of experts and left them to select their own topics and approaches. The result is seventeen diverse and highly individual articles on a wide variety of topics, providing many fascinating insights into our field. We are very grateful to the authors who agreed to write articles for this volume. We also wish to acknowledge the support and expert assistance of the publishers, Imperial College Press. The book is to be ready in time for the International Congress in July 2000. We hope it will be of interest to all the participants, and indeed to mathematicians and physicists in general, especially to young people just starting out on their careers.

A. A. T. B.

Fokas Grigoryan Kibble Zegarlinski

This page is intentionally left blank

V where

CONTENTS Preface

where

Modern Mathematical Physics: What It Should Be L.D. Faddeev

1

New Applications of the Chiral Anomaly Jiirg Frohlich and Bill Pedrini

9

Fluctuations and Entropy Driven Space—Time Intermittency in Navier-Stokes Fluids Giovanni Gallavotti

48

Superstrings and the Unification of the Physical Forces Michael B. Green

59

Questions in Quantum Physics: A Personal View Rudolf Haag

87

What Good are Quantum Field Theory Infinities? Roman Jackiw

101

Constructive Quantum Field Theory Arthur Jaffe

111

Fourier's Law: A Challenge to Theorists F. Bonetto, J.L. Lebowitz and L. Rey-Bellet

128

The "Corpuscular" Structure of the Spectra of Operators Describing Large Systems R.A. Minlos

151

Vortex- and Magneto-Dynamics — A Topological Perspective H.K. Moffatt

170

Gauge Theory: The Gentle Revolution L. O'Raifeartaigh

183

Random Matrices as Paradigm L. Pastur

216

Wavefunction Collapse as a Real Gravitational Effect Roger Penrose

266

Schrodinger Operators in the Twenty-First Century Barry Simon

283

The Classical Three-Body Problem — Where is Abstract Mathematics, Physical Intuition, Computational Physics Most Powerful? H.A. Posch and W. Thirring

289

Infinite Particle Systems and Their Scaling Limits S.R.S. Varadhan

306

Supersymmetry: A Personal View B. Zumino

316

1

MODERN MATHEMATICAL PHYSICS: W H A T IT S H O U L D B E

L. D. FADDEEV Steklov Mathematical Institute, St Petersburg 191011, Russia When somebody asks me, what I do in science, I call myself a specialist in m a t h ­ ematical physics. As I have been there for more than 40 years, I have some definite interpretation of this combination of words: "mathematical physics." Cynics or pur­ ists can insist that this is neither mathematics nor physics, adding comments with a different degree of malice. Naturally, this calls for an answer, and in this short essay I want to explain briefly my understanding of the subject. It can be considered as my contribution to the discussion about the origin and role of mathematical physics and thus to be relevant for this volume. The m a t t e r is complicated by the fact t h a t the term "mathematical physics" (often abbreviated by MP in what follows) is used in different senses and can have rather different content. This content changes with time, place and person. I did not study properly the history of science; however, it is my impression t h a t , in the beginning of the twentieth century, the term M P was practically equivalent to the concept of theoretical physics. Not only Henri Poincare, but also Albert Einstein, were called mathematical physicists. Newly established theoretical chairs were called chairs of mathematical physics. It follows from the documents in the archives of the Nobel Committee t h a t MP had a right to appear both in the nom­ inations and discussion of the candidates for the Nobel Prize in physics *. Roughly speaking, the concept of MP covered theoretical papers where mathematical for­ mulae were used. However, during an unprecedented bloom of theoretical physics in the 20s and 30s, an essential separation of the terms "theoretical" and "mathematical" oc­ curred. For many people, MP was reduced to the important but auxiliary course "Methods of Mathematical Physics" including a set of useful mathematical tools. T h e monograph of P. Morse and H. Feshbach 2 is a classical example of such a course, addressed to a wide circle of physicists and engineers. On the other hand, MP in the mathematical interpretation appeared as a the­ ory of partial differential equations and variational calculus. T h e monographs of R. Courant and D. Hilbert 3 and S. Sobolev 4 are outstanding illustrations of this development. The theorems of existence and uniqueness based on the variational principles, a priori estimates, and imbedding theorems for functional spaces com­ prise the main content of this direction. As a student of O. Ladyzhenskaya, I was immersed in this subject since the 3rd year of my undergraduate studies at the Physics Department of Leningrad University. My fellow student-N. Uraltseva now holds the chair of M P exactly in this sense. MP in this context has as its source mainly geometry and such parts of classical mechanics as hydrodynamics and elasticity theory. Since the 60s a new impetus to M P in this sense was supplied by Q u a n t u m Theory. Here the main a p p a r a t u s is functional analysis, including the spectral theory of operators in Hilbert space, the mathematical theory of scattering and the theory of Lie groups and their repres-

2

entations. The main subject is the Schrodinger operator. Though the methods and concrete content of this part of MP are essentially different from those of its classical counterpart, the methodological attitude is the same. One sees the quest for the rigorous mathematical theorems about results which are understood by physicists in their own way. I was born as a scientist exactly in this environment. I graduated from the unique chair of Mathematical Physics, established by V.I. Smirnov at the Physics Department of Leningrad University already in the 30s. In his venture V.I. Smirnov got support from V. Fock, the world famous theoretical physicist with very wide mathematical interests. Originally this chair played the auxiliary role of being responsible for the mathematical courses for physics students. However in 1955 it got permission to supervise its own diploma projects, and I belonged to the very first group of students using this opportunity. As I already mentioned, O.A. Ladyzhenskaya was our main professor. Although her own interests were mostly in nonlinear PDEs and hydrodynamics, she decided to direct me to quantum theory. During the last two years of undergraduate studies I was to read the mono­ graph of K.O. Friedrichs, "Mathematical Aspects of Quantum Field Theory," and relate it to our group of 5 students and our professor on a special seminar. At the same time my student friends from the chair of Theoretical Physics were absorbed in reading the first monograph on Quantum Electrodynamics by A. Ahieser and V. Berestevsky. The difference in attitudes and language was striking and I was to become accustomed to both. After my graduation O.A. Ladyzhenskaya remained my tutor but she left me free to choose research topics and literature to read. I read both mathematical papers (i.e. on direct and inverse scattering problems by I.M. Gelfand and B.M. Levitan, V.A. Marchenko, M.G. Krein, A.Ya. Povzner) and "Physical Review" (i.e. on formal scattering theory by M. Gell-Mann, M. Goldberger, J. Schwinger and H. Ekstein) as well. Papers by I. Segal, L. Van-Hove and R. Haag added to my first impressions on Quantum Field Theory taken from K. Friederichs. In the process of this selfeducation my own understanding of the nature and goals of MP gradually deviated from the prevailing views of the members of the V. Smirnov chair. I decided that it is more challenging to do something which is not known to my colleagues from theoretical physics rather than supply existence theorems. My first work on the in­ verse scattering problem especially for the multi-dimensional Schrodinger operator and that on the three body scattering problem confirm that I really tried to follow this line of thought. This attitude became even firmer when I began to work on Quantum Field Theory in the middle of the 60s. As a result, my understanding of the goal of MP was drastically modified. I consider as the main goal of MP the use of mathematical intuition for the derivation of really new results in fundamental physics. In this sense, MP and Theoretical Physics are competitors. Their goals in unraveling the laws of the structure of matter coincide. However, the methods and even the estimates of the importance of the results of work may differ quite significantly. Here it is time to say in what sense I use the term "fundamental physics." The adjective "fundamental" has many possible interpretations when applied to the classification of science. In a wider sense it is used to characterize the research

3 directed to unraveling new properties of physical systems. In the narrow sense it is kept only for the search for the basic laws that govern and explain these properties. Thus, all chemical properties can be derived from the Schrodinger equation for a system of electrons and nuclei. Alternatively, we can say t h a t the fundamental laws of chemistry in a narrow sense are already known. This, of course, does not deprive chemistry of the right to be called a fundamental science in a wide sense. The same can be said about classical mechanics and the q u a n t u m physics of con­ densed matter. Whereas the largest part of physical research lies now in the latter, it is clear t h a t all its successes including the theory of superconductivity and super­ fluidity, Bose-Einstein condensation and q u a n t u m Hall effect have a fundamental explanation in the nonrelativistic q u a n t u m theory of many body systems. An unfinished physical fundamental problem in a narrow sense is physics of elementary particles. This puts this part of physics into a special position. And it is here where modern MP has the most probable chances for a breakthrough. Indeed, until recent time, all physics developed along the traditional circle: ex­ periment — theoretical interpretation — new experiment. So the theory tradition­ ally followed the experiment. This imposes a severe censorship on the theoretical work. Any idea, bright as it is, which is not supplied by the experimental know­ ledge at the time when it appeared is to be considered wrong and as such must be abandoned. Characteristically the role of censors might be played by theoreticians themselves and the great L. Landau and W.Pauli were, as far as I can judge, the most severe ones. And, of course, they had very good reason. On the other hand, the development of mathematics, which is also to a great ex­ tent influenced by applications, has nevertheless its internal logic. Ideas are judged not by their relevance but more by esthetic criteria. T h e totalitarianism of theor­ etical physics gives way to a kind of democracy in mathematics and its inherent intuition. And exactly this freedom could be found useful for particle physics. This part of physics traditionally is based on the progress of accelerator techniques. T h e very high cost and restricted possibilities of the latter soon will become an uncircumventable obstacle to further development. And it is here that mathematical intuition could give an adequate alternative. This was already stressed by fam­ ous theoreticians with mathematical inclinations. Indeed, let me cite a paper 5 by P. Dirac from the early 30s: T h e steady progress of physics requires for its theoretical formulation a mathematics t h a t gets continually more advanced. This is only nat­ ural and to be expected. W h a t , however, was not expected by the sci­ entific workers of the last century was the particular form that the line of advancement of the mathematics would take, namely, it was expec­ ted that the mathematics would get more complicated, but would rest on a permanent basis of axioms and definitions, while actually the mod­ ern physical developments have required a mathematics t h a t continually shifts its foundations and gets more abstract. Non-euclidean geometry and non-commutative algebra, which were at one time considered to be purely fictions of the mind and pastimes for logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely t h a t this process of increasing abstraction

4

will continue in the future and that advance in physics is to be associ­ ated with a continual modification and generalization of the axioms at the base of mathematics rather than with logical development of any one mathematical scheme on a fixed foundation. There are at present fundamental problems in theoretical physics awaiting solution, e.g., the relativistic formulation of quantum mechanics and the nature of atomic nuclei (to be followed by more difficult ones such as the problem of life), the solution of which problems will presumably require a more drastic revision of our fundamental concepts than any that have gone before. Quite likely these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will therefore have to proceed in a more inderect way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalise the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities. Similar views were expressed by C.N. Yang. I did not find a compact citation, but the spirit of his commentaries to his own collection of papers 6 shows this attitude. Also he used to tell this to me in private discussions. I believe that the dramatic history of setting the gauge fields as a basic tool in the description of interactions in Quantum Field Theory gives a good illustration of the influence of mathematical intuition on the development of the fundamental physics. Gauge fields, or Yang-Mills fields, were introduced to the wide audience of physicists in 1954 in a short paper by C.N. Yang and R. Mills 7 , dedicated to the generalization of the electromagnetic fields and the corresponding principle of gauge invariance. The geometric sense of this principle for the electromagnetic field was made clear as early as in the late 20s due to the papers of V. Fock 8 and H. Weyl 9 . They underlined the analogy of the gauge (or gradient in the termino­ logy of V. Fock) invariance of the electrodynamics and the equivalence principle of the Einstein theory of gravitation. The gauge group in electrodynamics is commut­ ative and corresponds to the multiplication of the complex field (or wave function) of the electrically charged particle by a phase factor depending on the space-time coordinates. Einstein's theory of gravity provides an example of a much more sophisticated gauge group, namely the group of general coordinate transformation. Both H. Weyl and V. Fock were to use the language of the moving frame with spin connection, associated with local Lorentz rotations. Thus the Lorentz group became the first nonabelian gauge group and one can see in 8 essentially all formu­ las characteristics of nonabelian gauge fields. However, in contradistinction to the electromagnetic field, the spin connection enters the description of the space-time and not the internal space of electric charge. In the middle of the 30s, after the discovery of the isotopic spin in nuclear physics, O. Klein 10 introduced the cor­ responding noncommutative group and the affiliated vector field. And apparently it was here that the censorship of W. Pauli played its killing role for very good

5

reason, based on the experimental facts. The problem was that of mass: the mass of would-be vector particles is equal to zero classically; the only known massless partcles (and accompaning long-range interaction) are the photon and graviton. There is no room for the massless quanta of the hypothetical charged vector fields, so the theory must be rejected and forgotten. Thus there is no wonder that Yang received the same reaction when he presented his work at Princeton in 1954. The dramatic account of this event can be found in his commentaries 6 . Pauli was in the audience and immediately raised the question about mass. It is evident from Yang's text, that Pauli was well acquainted with the differential geometry of nonabelian vector fields but did not allow himself to speak about them. As we know now, the boldness of Yang and his esthetic feeling finally were vindicated. And it can be rightly said, that C.N. Yang proceeded according to mathematical intuition. In 1954 the paper of Yang and Mills did not move to the forefront of high energy theoretical physics. However, the idea of the charged space with noncommutative symmetry group acquired more and more popularity due to the increasing number of elementary particles and the search for the universal scheme of their classification. And at that time the decisive role in the promotion of the Yang-Mills fields was also played by mathematical intuition. At the beginning of the 60s, R. Feynman worked on the extension of his own scheme of quantization of the electromagnetic field to the gravitation theory of Einstein. A purely technical difficulty — the abundance of the tensor indices — made his work rather slow. Following the advice of M. Gell-Mann, he exercised first on the simpler case of the Yang-Mills fields. To his surprise, he found that a naive generalization of his diagrammatic rules designed for electrodynamics did not work for the Yang-Mills field. The unitarity of the S-matrix was broken. Feynman restored the unitarity in one loop by reconstructing the full scattering amplitude from its imaginary part and found that the result can be interpreted as a subtraction of the contribution of some fictitious particle. However his technique became quite cumbersome beyond one loop. His approach was gradually developed by B. DeWitt n . It must be stressed that the physical senselessness of the Yang-Mills field did not preclude Feynman from using it for mathematical construction. The work of Feynman 12 became one of the starting points for my work in Quantum Field Theory, which I began in the middle of the 60s together with Victor Popov. Another point as important was the mathematical monograph by A. Lichnerowitz 13 , dedicated to the theory of connections in vector bundles. From Lichnerowitz's book it followed clearly that the Yang-Mills field has a definite geometric interpretation: it defines a connection in the vector bundle, the base being the space-time and the fiber the linear space of the representation of the compact group of charges. Thus, the Yang-Mills field finds its natural place among the fields of geometrical origin between the electromagnetic field (which is its particular example for the one-dimensional charge) and Einstein's gravitation field, which deals with the tangent bundle of the Riemannian space-time manifold. It became clear to me that such a possibility cannot be missed and, notwith­ standing the unsolved problem of zero mass, one must actively tackle the problem of the correct quantization of the Yang-Mills field.

6

The geometric origin of the Yang-Mills field gave a natural way to resolve the difficulties with the diagrammatic rules. The formulation of the quantum theory in terms of Feynman's functional integral happened to be most appropriate from the technical point of view. Indeed, to take into account the gauge equivalence principle one has to integrate over the classes of gauge equivalent fields rather than over every individual configuration. As soon as this idea is understood, the technical realization is rather straightforward. As a result V. Popov and I came out at the end of 1966 with a set of rules valid for all orders of perturbation theory. The fictitious particles appeared as auxiliary variables giving the integral representation for the nontrivial determinant entering the measure over the set of gauge orbits. Correct diagrammatic rules of quantization of the Yang-Mills field, obtained by V. Popov and me in 1966-1967 14 ° 15 , did not attract the immediate attention of physicists. Moreover, the time when our work was done was not favorable for it. Quantum Field Theory was virtually forbidden, especially in the Soviet Union, due to the influence of Landau. "The Hamiltonian is dead" — this phrase from his paper 16 , dedicated to the anniversary of W. Pauli — shows the extreme of Landau's attitude. The reason was quite solid, it was based not on experiment, but on the investigation of the effects of renormalization, which led Landau and his coworkers to believe that the renormalized physical coupling constant is inevitably zero for all possible local interactions. So there was no way for Victor Popov and me to publish an extended article in a major Soviet journal. We opted for the short communication in "Physics Letters" and were happy to be able to publish the full version in the preprint series of newly opened Kiev Institute of Theoretical Physics. This preprint was finally translated into English by B. Lee as a Fermilab preprint in 1972, and from the preface to the translation it follows that it was known in the West already in 1968. A decisive role in the successful promotion of our diagrammatic rules into phys­ ics was played by the works of G. 't Hooft 17 , dedicated to the Yang-Mills field interacting with the Higgs field (and which ultimately led to a Nobel Prize for him in 1999) and the discovery of dimensional transmutation (the term of S. Coleman 18 ). The problem of mass was solved in the first case via the spontaneous symmetry breaking. The second development was based on asymptotic freedom. There exists a vast literature dedicated to the history of this dramatic development. I refer to the recent papers of G. 't Hooft 19 and D. Gross 20 , where the participants in this story share their impressions of this progress. As a result, the Standard Model of unified interactions got its main technical tool. From the middle of the 70s until our time it remains the fundamental base of high energy physics. For our discourse it is important to stress once again that the paper 14 based on mathematical intuition preceded the works made in the traditions of theoretical physics. The Standard Model did not complete the development of fundamental physics in spite of its unexpected and astonishing experimental success. The gravitational interactions, whose geometrical interpretation is slightly different from that of the Yang-Mills theory, is not included in the Standard Model. The unification of quantum principles, Lorentz-Einstein relativity and Einstein gravity has not yet been accomplished. We have every reason to conjecture that the modern MP and its mode of working will play the decisive role in the quest for such a unification.

7

Indeed, the new generation of theoreticians in high energy physics have received an incomparably higher mathematical education. They are not subject to the pressure of old authorities maintaining the purity of physical thinking and/or ter­ minology. Futhermore, many professional mathematicians, tempted by the beauty of the methods used by physicists, moved to the position of the modern mathem­ atical physics. Let use cite from the manifesto, written by P. MacPherson during the organization of the Quantum Field Theory year at the School of Mathematics of the Institute for Advanced Study at Princeton: The goal is to create and convey an understanding, in terms con­ genial to mathematicians, of some fundamental notions of physics, such as quantum field theory. The emphasis will be on developing the intuition stemming from functional integrals. One way to define the goals of the program is by negation, excluding certain important subjects commonly pursued by mathematicians whose work is motivated by physics. In this spirit, it is not planned to treat except peripherally the magnificient new applications of field theory, such as Seiberg-Witten equations to Donaldson theory. Nor is the plan to consider fundamental new constructions within mathimatics that were inspired by physics, such as quantum groups or vertex operator algebras. Nor is the aim to discuss how to provide mathematical rigor for physical theories. Rather, the goal is to develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra. I propose to call the intuition to which MacPherson refers that of mathemat­ ical physics. I also recommend the reader to look at the instructive drawing by P. Dijkgraaf on the dust cover of the volumes of lectures given at the School 21 . The union of these two groups constitutes an enormous intellectual force. In the next century we will learn if this force is capable of substituting for the traditional experimental base of the development of fundamental physics and pertinent physical intuition. References 1. B. Nagel, The Discussion Concerning the Nobel Prize of Max Planck, Science Technology and Society in the Time of Alfred Nobel (New York: Pergamon, 1982). 2. F. Morse and H. Feshbach, Methods of Theoretical Physics, (New York: McGraw-Hill, 1953). 3. R. Courant and D. Hilbert, Methoden der mathematischen Physik, (Berlin: Springer, 1931). 4. S.L. Sobolev, Nekotorye primeneniya funktsional'nogo analtza v matematicheskoi fizike (Some Applications of Functional Analysis in Mathematical Physics), (Leningrad: Lenigrad. Gos. Univ., 1950). 5. P. Dirac, Quantized Singularities in the Electromagnetic Field, Proc. Roy. Soc. London A 133, 60-72 (1931).

V

6. C.N. Yang, Selected Papers 1945-1980 with Commentary, (San Francisco: Freeman, 1983). 7. C.N. Yang and Ft. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191-195i (1954). 8. V. Fock, L'equation d'onde de Dirac et la geometrie de Riemann, J. Phys. et Rad. 70 392-405 (1929). 9. H. Weyl, Electron and Gravitation, Zeit. Phys., 56, 330-352 (1929). 10. O. Klein, On the Theory of Charged Fields: Submitted to the Conference New Theories in Physics, Warsaw, 1938, Surv. High Energy Phys., 1986, 5 269 (1986). 11. B. De-Witt, Quantum Theory of Gravity II: The manifest covariant theory, Phys. Rev., 1967, 162, 1195-1239 (1967). 12. R.P. Feynman, Quantum Theory of Gravitation, Acta Phys. Polon. 24, 697722 (1963). 13. A. Lichnerowicz, Theorie globale des connexions et des groupes d'holonomie, (Roma: Ed. Cremonese, 1955). 14. L. Faddeev and V. Popov, Feynman Diagrams for the Yang-Mills Field, Phys. Lett. B, 25, 29-30 (1967). 15. V. Popov and L. Faddeev, Perturbation Theory for Gauge-Invariant Fields, Preprint, National Accelerator Laboratory, NAL-THY-57 (1972). 16. L. Landau, in Theoretical Physics in the twentieth century, a memorial volume to Wolfgang Pauli, ed. M. Fierz and V. Weisskopf, (Cambridge, USA, 1956). 17. G. 't Hooft, Renormalizable Lagrangians for Massive Yang-Mills Fields, Nucl. Phys. B, 35, 167-188 (1971). 18. S. Coleman, Secret Symmetries: An Introduction to Spontaneous Symmetry Breakdown and Gauge Fields: Lecture given at 1973 International Summer School in Physics Ettore Majorana, Erice (Sicily), 1973, Erice SubnuclPhys., 1973. 19. G. 't Hooft, When was Asumptotic Freedom Discovered? Rehabilitation of Quantum Field Theory, Preprint, hep-th/9808154 (1998). 20. D. Gross, Twenty Years of Asymptotic Freedom, Preprint, hep-th/9809080 (1998). 21. V. Dijkgraaf, Quantum Fields and Strings: A course for mathematicians, vols. I, II (AMS, IAS, 1999).

9

N E W APPLICATIONS OF THE CHIRAL ANOMALY*

J U R G F R O H L I C H AND BILL P E D R I N I Theoretical Physics, ETH-H6nggerberg, E-mail: [email protected];

CH-8093 Zurich, Switzerland pedrini&itp.phys.ethz.ch

We describe consequences of the chiral anomaly in the theory of quantum wires, the (quantum) Hall effect, and of a four-dimensional cousin of the Hall effect. We explain which aspects of conductance quantization are related to the chiral anomaly. The four-dimensional analogue of the Hall effect involves the axion field, whose time derivative can be interpreted as a (space-time dependent) difference of chemical potentials of left-handed and right-handed charged fermions. Our fourdimensional analogue of the Hall effect may play a significant role in explaining the origin of large magnetic fields in the (early) universe.

1

What is the chiral anomaly?

The chiral abelian anomaly has been discovered, in the past century, by Adler, Bell and Jackiw, after earlier work on 7T°-decay starting with Steinberger and Schwinger; see e.g. [1] and references given there. It has been rederived in many different ways of varying degree of mathematical rigor by many people. Diverse physical implica­ tions, especially in particle physics, have been discussed. It is hard to imagine that one may still be able to find new, interesting implications of the chiral anomaly that specialists have not been aware of, for many years. Yet, until very recently — in the past century, but only two to three years ago — this turned out to be possible, and we suspect that further applications may turn up in the future! This little review is devoted to a discussion of physical implications of the chiral anomaly that have been discovered recently. Before we turn to physics, we recall what is meant by "chiral (abelian) anomaly". In general terms, one speaks of an anomaly if some quantum theory violates a sym­ metry present at the classical level, (i.e., in the limit where h —> 0). By "violating a symmetry" one means that it is impossible to construct a unitary representation of the symmetry transformations of the classical system underlying some quantum theory on the Hilbert space of pure state vectors of the quantum theory. (By "vi­ olating a dynamical symmetry" is meant that it is impossible to construct such a representation that commutes with the unitary time evolution of the quantum theory.) It is quite clear that understanding anomalies can be viewed as a problem in group cohomology theory. A fundamental example of an anomalous symmetry group is the group of all symplectic transformations of the phase space of a classical Hamiltonian system underlying some quantum theory. The anomalies considered in this review are ones connected with infinitedimensional groups of gauge transformations which are symmetries of some classical Lagrangian systems with infinitely many degrees of freedom (Lagrangian field the­ ories). Thus, we consider a theory of charged, massless fermions coupled to an 'THIS REVIEW IS DEDICATED TO THE MEMORY OF LOUIS MICHEL, THE THEOR­ ETICIAN AND THE FRIEND.

10

external electromagnetic field in Minkowski space-time of even dimension In. Let 7°, 71>• • • > 7 2 n _ 1 denote the usual Dirac matrices, and define 7 := i 7 ° 7 1 . . . 7 2 n - 1 .

(1.1)

Then 7 anti-commutes with the covariant Dirac operator D := 17" {dp-iAp)

,

(1.2)

where A is the vector potential of the external electromagnetic field. Let tp(x) denote the Dirac spinor field and ip{x) the conjugate field. We define the vector current, J1*, and the axial vector current J^, by

j " := ^ y v , J" ■= fa»vl> •

(i.3)

At the classical level, these currents are conserved, d^J"

= 0 , d^J"

= 0,

(1.4)

on solutions of the equations of motion, (Di[> = 0). The conservation of the vector current is intimately connected with the electromagnetic gauge invariance of the theory, e~ix^

4>{x) ^ e'X^VOc) , i>{x) ->■ rp(x) Apix) *+A^x) x

+ dpX(*) ,

(1-5)

ls a

where x( ) test function on space-time. When x is constant in x the trans­ formations (1.5) are a global symmetry of the classical theory corresponding to the conserved quantity

Q = [dxj°(x°,x)

(1.6)

which is the electric charge. The conservation of Q (independence of x°) follows, of course, from the fact that the Noether current J*1 associated with (1.5) satisfies the continuity equation (1.4). The conservation of the axial vector current J^, in the classical theory, is con­ nected with the invariance of the theory under local chiral rotations 1>(x) -¥ eiQ{x) H+ 4>{x) eia(~x^ +-yd^x)

,

(1.7)

where a(x) is a test function on space-time. In particular, when a is a constant the transformations (1.7) are a global symmetry of the classical theory corresponding to the conserved charge

Q = J'dxj°(x°,x)

(1.8)

(which, according to (1.4), is independent of a;0). It turns out that, in the quantum theory, the local chiral rotations (1.7) do not leave quantum-mechanical transition amplitudes invariant, and the axial vector current J^ is not a conserved current, for arbitrary external electromagnetic fields. This phenomenon is called chiral (abelian) anomaly.

11

Let us see where the chiral anomaly comes from, for theories in two and four space-time dimensions. We start with the discussion of two-dimensional theories. We consider a quantum theory which has a conserved vector current J^ and — if the external electromagnetic field vanishes — a conserved axial vector current J^, i.e., ^JM

= 0 , drj" = 0 .

(1.9)

In two space-time dimensions, J1* and J^ are related to each other by J»

= e"" Jv

(1.10)

where e 00 = e11 = 0 , el)1 = —e10 = 1. The continuity equation d„J" = 0 has the general solution J"(x) = j-e^frtfix),

(1.11)

Z7T

where "{dv(x) and its conjugate field 4>(x) = ^ ' ( Z J T O - We study the effect of coupling these fields to external vector- and axial-vector potentials, A^ and Z^, respectively. The theory of these fields provides an example of Lagrangian field theory, the action functional being given by := jd2nx

S(rl>,rP;A,Z)

i>{x)DA,z^{x)

,

(1.25)

where the covariant Dirac operator is DA,z

= Widn-iAp-iZtf)

,

(1.26)

5

with 7 (= "7 ") as in eq. (1.1). The fields A^ and Z^ are arbitrary external fields (i.e., they are not quantized, for the time being). We define the effective action, Sen(A,Z), by const fvi>V4> eiS{*,*;A,z) _

eiS,„(A,Z)

^

27)

where the constant is chosen such that S(A = 0, Z = 0) = 0, and h and c have been set to 1. After Wick rotation, t = x° -)■ -ix°, A0 -¥ iA0, Z0 -»• iZ0, 7° -> -*7° -

(1-28)

eq. (1.27) reads -SS,(A,Z)

=

\[VxpVrl,e-SE(.WXP MX)

Fxp{x)

■

(L60)

Thus eqs. (1.47) read d,(J?/r(x))A

= ^ * ( F A F ) W ,

(1.61)

and, from eqs. (1.55), (1.56) and (160), we conclude that

= ± i j-j a well known result; see [1].

(£(*,*) • V) 6 (x-y)

,

(1.62)

18

The key fact reviewed in this section, from which all other results can be derived, is eq. (1.41), i.e., f d2nx a(x) A{x) .

S%t{A + dx, Z + da) = S^n{A,Z)-1i

(1.63)

In order to describe a system in which only the left-handed fermions are charged, while the right-handed fermions are neutral, one may just set A = -Z

= a

(1.64)

in eq. (163), where a is the electromagnetic vector potential to which the lefthanded fermions are coupled; see (1.25), (1.26) and (1.46). Denoting the effective action of this system by Wt(a), we find from (1.63) and (1.64) that Wt{a + dx) = Wt{a)+ 2» f d2nxX(x)A(x)

.

(1.65)

Similarly, 2n Wr (a + dX) = WT (a) - 2 i f d x x(x) A(x) ,

(1.66)

for charged right-handed fermions. Eqs. (1.65) and (1.66) show that a theory of massless chiral fermions coupled to an external electromagnetic field is anomalous, in the sense that it fails to be gauge-invariant. But let us imagine that space-time, M 2n , is the boundary of a (2n + l)-dimensional half-space M, (i.e., dM = physical space- time SIR 2 "). Let A denote a smooth U(l)-gauge potential on M which is continuous on dM, with A

a

\dM=

-

(167)

2n+l

Let u) (-;A) denote the usual Chern-Simons (2n + l)-form on M. The ChernSimons action on M is defined by Scs(A)

u;2n+1(Z;A)

:= if

,

(1.68)

JM

where £ denotes a point in M. It should be recalled that w 2 n + 1 (-;i4 + dx) = "2n+H-\A) Since d(*A) = 0 , dx^(*A) SCs(A

= d(x(*A))

+ dXA(*A)

.

(1.69)

, and hence, by Stokes' theorem,

+ dx) = Scs(A) + i f JdM

X(x)(*A)(x)

= SCs(A) + if d2nxX(x)A(x) JdM

.

(1.70)

It follows that Wi/r(a)

=F Scs(A)

is gauge — invariant.

(1.71)

This result has a (2n + l)-dimensional interpretation (see [5]): Consider a (parity-violating) theory of massive, charged fermions described by 2 n -component Dirac spinors on a (2n + l)-dimensional space-time M with non-empty boundary

19

dM. These fermions are minimally coupled to an external electromagnetic vector potential A. We impose some anti-selfadjoint spectral boundary conditions on the (2n + l)-dimensional, covariant Dirac operator DA- The action of the system is given by S(i>,1>;A) := /

d2n+}W(Z)(DA+m)rP(t;),

(1.72)

JM

where m is the bare mass of the fermions. The effective action of the system is defined by eS5t(A)

= det ren (DA + m) ,

(1.73)

where the subscript "ren" indicates that renormalization may be necessary to define the R.S. of (1.73). Actually, for n = 1, no renormalization is necessary; but, for n = 2, e.g. an infinite charge renormalization must be made. It turns out that, for n = 1 and n = 2 (after renormalization), S^(A)

= Wt/r (A \dM)

T

Scs(A) + O ( £ )

,

(1.74)

up to a Maxwell term depending on renormalization conditions, where the correc­ tion terms are manifestly gauge-invariant; see [5,6]. (Whether the R.S. of (1.74) involves Wt or Wr depends on the definition of DA)The physical reason underlying the result claimed in eq. (1.74) is that, in a system of massive fermions described by 2 n -component Dirac spinors confined to a space-time M with a non-empty, 2n-dimensional boundary dM, there are massless, chiral fermionic surface modes propagating along dM. This completes our heuristic review of aspects of the chiral abelian anomaly that are relevant for the physical applications to be discussed in subsequent sections. The abelian anomaly is, of course, but a special case of the general theory of anomalies involving also non-abelian, gravitational, global, . . . anomalies. In recent years, this theory has turned out to be important in connection with the theory of branes in string theory and with understanding aspects of M-theory. But, in this review, such applications will not be described. In Sect. 2, we describe physical systems, important features of which can be understood as consequences of the two-dimensional chiral anomaly: incompressible (quantum) Hall fluids and ballistic wires. In Sect. 3, we describe degrees of freedom in four dimensions which may play an important role in the generation of seeds for cosmic magnetic fields in the very early universe. This will turn out to be connected with the four-dimensional chiral anomaly. In Sect. 4, a brief review of the theory of "transport in thermal equilibrium through gapless modes" developed in [7] is presented. In Sects. 5 and 6, we combine the results of this section with those in Sect. 4 to derive physical implications of the chiral anomaly for the systems introduced in Sects. 2 and 3. Some conclusions and open problems are described in Sect. 7.

20 2

Quantized conductances

T h e original motivation for the work described in this review has been to provide simple and conceptually clear explanations of various formulae for quantized con­ ductances, which have been encountered in the analysis of experimental d a t a . Here are some typical examples. E x a m p l e 1. geometry. Let and the outer direction. T h e

Consider a q u a n t u m Hall device with, e.g., an annular (Corbino) V denote the voltage drop in the radial direction, between the inner edge, and let /// denote the total Hall current in the azimuthal Hall conductance, G//, is defined by GH

One finds t h a t if the longitudinal

= IH/V

resistance

•

(2.1)

vanishes

(i.e., if the two-dimensional

electron gas in the device is "incompressible") then GH is a rational multiple of ^-, i.e.,

G

"

=

1 'J '

n =

°'1,2

d = 1 2 3

' - '-- •

C2-2)

In (2.2), e denotes the elementary electric charge and h denotes Planck's constant. Well established Hall fractions, }H '■= ^ , in the range 0 < / / / < 1 are listed in Fig. 1; (see [8]; and [9, 10, 11] for general background). E x a m p l e 2. In a ballistic (quantum) wire, i.e., in a pure, very thin wire without back scattering centers, one finds t h a t the conductance Gw — I/V (I: current through the wire, V: voltage drop between the two ends of the wire) is given by Gw

= IN

e

- , N = 0,1,2,..., (2.3) h under suitable experimental conditions ("small" V, temperature not "very small", "adiabatic gates"); see [12, 13]. E x a m p l e 3 . In measurements of heat conduction in q u a n t u m wires, one finds t h a t the heat current is an integer multiple of a "fundamental" current which depends on the temperatures of the two heat reservoirs at the ends of the wire. If electromagnetic waves are sent through an "adiabatic hole" connecting two half-spaces one approximately finds an "integer quantization" of electromagnetic energy flux. Our task is to a t t e m p t to provide a theoretical explanation of these remarkable experimental discoveries; hopefully one that enables us to predict further related effects. Conductance quantization is observed in a rather wide temperature range. It appears t h a t it is only found in systems without dissipative processes. When it is observed it is insensitive to small changes in the parameters specifying the system and to details of sample preparation; i.e., it has universality properties. — It will turn out t h a t the key feature of systems exhibiting conductance quantization is t h a t they have conserved chtral charges; (such conservation laws will only hold approximately, i.e., in slightly idealized systems). Once one has understood this

21 r—

i

i

3

>

• | »/->

1

80 TeV, the c/iira/ charges, N( and yVr, defined in eq. (1.50) of Sect. 1, are approximately conserved for electrons. They are related to an approximate chiral symmetry of the electronic sector of the standard model. Among other results, we shall attempt to show that if, in the very early universe, the chemical potentials of left-handed and right-handed electrons are different from each other, this may give rise to the generation of large, cosmic magnetic fields, [15]; (see also [7] for a similar, independent suggestion). This effect is, in a sense explained in Sects. 4 and 6, an effect in equilibrium statistical mechanics. However, this is precisely what may make it appear quite unnatural and implausible: The chiral charges, Ni and /Vr, are not really conserved; leptons are massive. The very early universe is not really in an equilibrium state, and the chemical potentials of left-handed and right-handed electrons neither have an unambiguous meaning, nor would they be space- and time-independent. It may then be wrong, or, at least, misleading, to invoke results from equilibrium statistical mechanics to explore effects in the physics of the very early universe. A way out from these difficulties can be found by seeking inspiration from an analogy with the quantum Hall effect: Consider a quantum Hall fluid (QHF), confined to a strip of macroscopic width £ in the plane. If the QHF is incompressible then there are no light (gapless) modes propagating through the bulk of the sample; but, as shown in the last section, there are gapless, chiral modes propagating along the boundaries of the sample. Let Cl denote the space-time of the fluid; it is a slab of width I in three-dimensional Minkowski space. The two components of the boundary, 0; we are studying a system in a compact region of physical space.) T h e equilibrium expectation of a bounded operator, a, on % is defined by {a)p,t

■■= Tr (p/j.^a)

.

(4.4)

Let J{x) = (J°(x), J_{x)) be a conserved quantum-mechanical current density of S, where x = (x_, t), t is time and x_ is a point of physical space contained inside S. We are interested in calculating the expectation values of products of components of J in the state ppy, in particular, we should like to calculate {J_(x))j}^. Of course, if the dimension of space is larger than one, (J(x))g^ vanishes unless rotation invariance is broken by some external field. If J_{x) is a vector current then {J_{x))plfi vanishes unless the state pptll is not invariant under space-reflection and time reversal. This happens if some of the charges N\,..., Ni are not invariant under space-reflection and time reversal, i.e., if they are chiral. To say t h a t J is conserved means that it satisfies the continuity equation d^J"

= 0,

(4.5)

where x° = I denotes time, and d^ = d/dx^. If the space-time of the system S is topologically trivial ("star-shaped") then eq. (4.5) implies t h a t there is a globally defined vector field 31.20. For reference purposes we write explicitly the GNS equations with the OK41 cut-off:

iifc

-riR

^2

M*, k^kHk^

I4.JI