Fractional Statistics and Anyon Superconductivity

11111111111 ~Jlm 550172

FRAOIONAL STATISTICS AND ANYON SU PERCON DUalVlTY

•

FRAalONAL STATISTICS AND ANYON SUPERCONDUOIVITY Fronk Wilczek The Institute for Advanced Study Princeton

~

.

\\h World Scie . , . . . Singapore. New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. POBox 128, Farrer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

Tll~

er!i10[ ,,;1(1 rublisher are g:-ateful to the author,: and the following

!heir

assi~tanee

publi~hers for and permission tl) reproduce th ... reprinted papers found in this volume:

Association for the Advancement of Science (Science) Institute of Physics (J. }([ath. Phys.) American Physical Society (Phys. Rev. & Phys. Rev. Lett.) Elsevier Science Publishers B. V. (Nucl. Phys. & Phys. Le:t.) Kluwer Academic Publishers (Lett Math. Phys.) Societa Italiana di Fisica (n. Nuovo Omento) A.:-!'leric111

Am~rican

FRACTIONAL STATISTICS AND ANYON SUPERCONDUCTIVITY Copyright © 1990 by World Scientific Publishing Ce,. Pte. Ltd.

All rights reserved. This book, or parts thel"lo/, 'may not be reproduced in any form or by any means, electronic or mechanical, including photocvpying, recording or any information· storage and retrieval system now known or to be invented, without written perimission from the Publisher. ISBN 981-02-0048-X 981-02-0049-8 pbk

Library of Congress Cataloging-in-Publication data it;a~~ble .

.;0. "~

II

Printed in Singapore by Loi

.:t.

. "C' . .

Print~:h~.· ~'d·~ \1'-:' ~-:~.~ ; . ...... :~.. -:-',,' :

".~

I

v

Introduction Over the past few years the idea of fractional quantum statistics, and the techniques developed to study it, has proved useful in several quite diverse areas of physics, from cosmic strings and black holes to the fractional quantized Hall effect and high-temperature superconductivity. While the early papers were motivated by a spirit of exploration or simple playfulness, it now seems clear that a suprisingly powerful and coherent set of ideas has evolved around this theme. This book is an attempt to make what I see as the central concepts readily accessible to physicists with diverse backgrounds. Mathematicians and chemists may also fi!ld material of interest to them. The book consists of two parts. The first part of the book can be regarded as a self-contained monograph. It consists of two long articles. "Concepts in Fractional Statistics" is mainly concer!led with the basic formulation of fractonal statistics, and with various realizations of it. "States of Anyon Matter" is mainly concerned with the many-body theory of anyons, and its possible application to real materials. The second part of the book consists of a reprint collection, together with brief introductory paragraphs designed to put the papers into a coherent context. I have not attempted to write the ultimate anyon book. Indeed, since the subject continues to develop at an explosive ra~e, the time is not ripe for such an attempt. My goal has been the more modest one of making a beautiful and exciting subject widely accessiblt: while it is still in the making. The purpose and form of the book I hope partially justify two of its peculiarities. The first is, that in the monograph part only a few papers closely related to specific points in the text are referenced. The interested reader will of course find abundant entries into the relevant literature in the reprint part. Also, there is some redundancy between the different parts, including even repetition of some key paragraphs. I thought it better to make the text easy to read, than to aim for formal perfection. I have had the good fortune to work over the years with a large number of gifted collaborators on the subject~ treated here. They include (more or less in chronological order) Tony Zee, Dan Arovas, Bob Schrieffer, Richard MacKenzie, Fred Goldhaber, Al Shapere, X.-G. Wen, Lawrence Krauss, Mark Alford, John March-Russell, Yi-Hong Chen, Edward Witten, Bert Halperin, Y.-S. Wu, Steve Giddings and Martin Greiter. Their contributions and influence permeate the subject, and especially my account of it. I would also like to thank Al Shapere for a careful reading of the monograph part and for several useful suggestions. Finally I am happy to thank P.-H. Tham for her editorial efforts, and Maggie Best, Gisele Murphy and Val Nowak for their help in preparing the manuscript for publication.

vii

Contents Introduction ......... ... ........................... ...... ...................... ....... ....... .......... ... ..... .............. ..........

v

Part L CONCEPTS IN FRACTIONAL STATISTICS

1

1. 2. 3. 4. 5. 6. 7. 8. 9.

Transmutation and Fractionization of Angular Momentum ............................... Fractional Statistics: Trajectory Space and the Braid Group ............................. Fictitious Field Con3truction, and Quantum Mechanical Examples................... Aharonov-Bohm Scattering and its Implications.................................................. Hopf Term ................ ........... .................... ..... ........ ................... ................................. Local Field Theory; Chern-Simons Construction .................................................. Berry Phase and Anyons in the Quantized Hall Effec~ ..................... ................... Induced Quantum Numbers ................................................................................... Boundary and Coboundary .....................................................................................

Part n. STATES OF ANYON MATrER 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Definition of Anyons ............................... ................................................................ Average Field Theory .............................................................................................. Anyon Superconductivity: Basic Mechanism ....................................................... Anyon Superconductivity: Phenomenology .......................................................... Adiabatic Principle ............................................................................. :.................... More Supcrconducting States ..................... ........................................................... Quantized Hall States . ....................... .................................................................... Anyon Metal ............................................................................................................ Microscopic Origins ................................................................................................. Conclusions ............ .......................................... ........................................................

4 11 17 21 27 34 40 47 50

57 60 62 65 72 80 82 83 87 89 98

Part m. REPRINTED PAPERS

103

1.

105

Classics

D. Finkelstein and J. Rubinstein, "Connection between Spin, Statistics, and Kinks", J. Math. Phys. 9 (1968) 1762-1779 .......................................................... 107 Y. Aharonov and D. Bohm, "Significance of Electromagnetic Potentials in the Quantum Theory", Phys. Rev. 115 (1959) 485-491 .......................................... 125

J. M. Lcinaas and J. Myrhcipt, "On the Theory of Identical Particles",

n Nuovo Cimento 37 (1977) 1-23 .................................................................................. 132 2.

The Last Bastion of Rationality

157

J. Goldstone and F. Wilczek, "Fractional Quantum Numbers on Solitons", Phys. Rev. utt. 47 (1981) 986-989 ............................................................................... 159

viii F. Wilczek, "Magnetic Flux, Angular Momentum, and Statistics", Phys. Rev. utt. 48 (1982) 1144-1146 ........................................................................... 163

F. Wilczek, "Quantum Mechanics of Fractional-Spin Particles", Phys. Rev. utt. 49 (1982) 957-959 ............................................................................... 166

3.

Foundations

171

D. P. Arovas, R. Schri6ffer. F. \Vilczck and A. Zee. "Statistical Mechanics l)f Apyl)!1~··.lVILcl /-'hy.C!. B2:-il

(1~·m5~

1 U-l~1) ............... -............................................ 173

i. S. Wl!.. "G~ner,:jl Theory f()~ Quaf'tuM Stati3tics in Two Dimensions", Phys. Rev. Lett. 52 C.984) 2103-2106 ........................................................................... 183

A. S. Goldhaber and R. Mackenzie, "Arc Cyons Really Anyons?", Phys. utt. B214 (1988) 471-474 .................................................................................. 187 A. S. Goldhaber, R. Mackenzie and F. Wilczek, "Field Corrections to Statistics", Mod. Phys. Lett. A4 (1989) 21-31 ................................................ 191

.~. Ind~c.ed

J. FrOhlich and P.-A. Marchetti, "Quantum Field Theory of Anyons", Lett. Math. Phys. 16(1988) 347-358 ............................................................................. 202 A. M. Polyakov, "Fermi-Bose Transmutations Induced by Gauge Fields", Mod. Phys. Lett. A3 (1988) 325-328 ............................................................................. 214

4.

AnYOf1S in Model Field Theories

221

F. Wilczek and A. Zee, "Linking Numbers, Spin, and Statistics of Solitons", Phys. Rev. utt. 51 (1983) 2250-2252 ........................................................................... 222 A. N. Redlich, "Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three Dimensions", Phys. Rev. D29 (1984) 2366-2374 .......................................................................................................... 225 Y.-H. Chen and F. Wilczek, "Induced Quantum Numbers in some 2 + 1 Dimensional Models", Int. J. Mod. Phys. B3 (1989) 117-128 ..................................... 234

5.

Anyons in the Quantized Hall Effect

249

B. 1. Halperin, "Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States", Phys. Rev. Lett. 52 (1984) 1583-1586 (Erratum: Phys. Rev. Lett. 52 (1984) 2390) ................................................................. 251 D. Arovas, J. R. Schrieffer and F. Wilczek, "Fractional Statistics and the Quantum Hall Effect", Phys. Rev. Lett..53 (1984) 722-723 ........................................ 256

ix S. M. Girvin and A. H. MacDonald. "Off-Diagonal Long-Range Order, Oblique Confinement. and the Fractional Quantum Hall Effcct", Phys. Reu. utt. 58 (1987) 1252-1255 ........................................................................... 258

R. B. Laughlin. "Fractional Statistics in the Quantum Hall Effcct" .......................... 262

6.

Chiral Spin States

307

V. Kalmeyer and R B. Laughlin. "Equivalence of the Rcsonating-Valence-&nd and Fractional Quantum Hall States", Phys. Reu. utt. 59 (1987) 2095-2098 .......... 308 X.-G. Wen. F. Wilczek and A. Zoe, "Chiral Spin Statcp and Superconductivity", Phy.'). Rev. B39 (1989) 11~13-11423 .......................................... 312

7.

Anyon Superconductivity

325

R. B. Laughlin. 'crrhe Relationship between High-Temperature Superconductivity and the Fractional Quantum Hall Effect", Science 242 (1988) 525-533 .......................................................................................... 326 R. B. Laughlin, "Supcrconducting Ground State of Noninteracting Particles Obeying Fractional Statistics", Phys. Reu. utt. 60 (1988) 2677-2680 ........................ ..... ...... .. ............................................................................... 335 A. L. Fetter, C. B. Hanna and R. B. Laughlin, "Random-Phase Approximation in the Fractional-Statistics Gas", Phys. Reu. B39 (1989) 9679-9681 ........................ 339 Y.-H. Chen, F. Wilczek, E. Witten and B. I. Halperin, "On Anyon Superconductivity", Int. J. Mod. Phys. B3 (1989) 1001-1067 ..................................... 342

8.

Some Recent Directions

411

G. S. Canright and S. M. Girvin, "Anyons, the Quantum Hall Effect, and Two-.Dimensional Superconductivity", Int. J. Mod. Phys. B3 (1989) 1943-1963 ...................................................................................................................... 413 D.-H. Lee and M. P. A. Fisher, "Anyon Superconductivity and the Fractional Quantum Hall Effect", Phys. Reu. utt. 63 (1989) 903-906 (Erratum: Phys. Reu.ll!tt. 63 (1989) ................................................................... 434 D.-H. Lee and C. L. Kane, "Boson-Vortex-Skyrmion Duality, Spin-Singlet Fractional Quantum Hall Effect, and Spin-1I2 Anyon SuperconductivitY', Phys. Reu. utt. 64 (1990) 1313-1317 ........................................................................... 439 T. Einarsson, "Fractional Statistics on a Torus", Phys. Rev. utt. 84 (1990) 1995-1998 ...................................................................................................... 444

PART I. CONCEPTS IN FRACTIONAL STATISTICS

.1

I Concepts in Fractional Statistics The possibility and significance of fractional angular momentum is discussed, and some simple physical realizations of it are mentioned. This leads naturally to consideration of the possibility of fractional quantum statistics, which is seen to be a possibility inherent in the kinematics of 2+1 dimensional quantum mechanics. Both sorts of fractionalization are intimately relat~d to theories, and the classic considerations of Aharonov and Bohm on the significan-:e of the vector potential in quantum mechanics. The meaning and importa'lce of dis~rete gauge invariance in continuum theories is pointed out. Fractional ptatistics i~ shown to have a si.mple clyanmical realization in the dynamics of charge-flux tube composites. Fractional statistics is shown to occur very naturally in the most geometrical quactum field theories in 2+1 dimensions, that is in the nonlinear sigma model and in quantum electrodynamics. Transmutation of quantum statistics can be implemented by a simple univel'sal construction, the so-called Chern-Simons construction, at the level of local quantum field theorv. It is shown that the type of field theory involved has interesting behavior at the boundary of a finite region, and that it occurs 46 the boundary theory of a simple class of 3+ 1 dimeD.sional models. Physical ilnplications of these facts are mentioned. Quasiparticies in the fractional quantized Hall effect are shown to provide a physical realization of fractional statistics particles: or anyons.

1. Transmutation and fractionization of angular momentum 1. In two spatial dimensions the group of rotations would seem to be quite trivial; but in a certain sense this very triviality leads to richer possibilities for the quantum mechanicl:I of angular momentum. f:l t hr-:~:

f')r

rr~orc

dimens;ons the

r~)tat:~n

group is ncn-abelian, and the corre-

sponding algebra of angular iTIOmenta haa non-trivial cOITlmutation relations:. which fixes the normalizations. Consideration of the Lie algebra of infinitesim':u rotations, which are of COUfRe proPQrti-Jnal to th~ t'.ngular rnOI!lentum generators, then leads directly to the quantization of angular momentum in units of Ii.

!

Bu t in twa dimensions the rotation group is a a trivial abelian group, and there are no non-trivial commutators. Evidently then, in two dimensions consideration of infinitesimal rotations does not fix the quantum of angular :nomentum. What about finite rotations?

Since the only significant global aspect of rotations around a single axis is that a 21r rotation reduces to the identity transformation, clearly the question of interest is how that rotation is implemented as an operator in the quantum-mechanical Hilbert space. While we must demf)nd that directly physkal quantities remain invariant under the physically trivial operation of rotation by 211", it is not necessarily so for the quantum mechanical states. State vectors themselves are not directly physical, but only their overlaps. (Ultimately it is only probabilities that are observable, i.e., magnitudes of states vectors. But by considering linear combinations of states one can express overlaps in terms of magnitudes.) Thus the operator U(211") associated with a 211" rotation can have an eigenvalue ..\ which is an arbitrary complex number of modulus one, when acting on a state. The physical triviality of U(211") requires only that 6tate6 characteMzed by aifferent eigenvalue6 A "I A' have zero overlap, i.e. that sllch states are orthogonal. More generally, since rotation through 211' leaves any local observable unchanged the corresponding operators must commute; it follows from this that no local observable can connect states characterized by different values of ..\. One says that such states, which in a sense reside in entirely different worlds, belong to different 6uperlelection lector6. Within each superselection sector, labelled by A = ei9 , angular momentum is quantized in units of t.,( 2~ + integer). However there is nothing in our general considerations so far to indicate that 9 is anything but a perfectly general real number. Indeed we shall now discuss some quite simple physical systems that realize arbitrary values of 9. 2. There is an extremely simple physical circumstance where one finds fractional angular momentum. Consider a charged particle with charge q which orbits around, but does not penetrate, a thin solenoid running along the z axis. When

5

no current flows through the solenoid the orbital angular momentum is of course quantized in integer multiples of 1i.. If a current is slowly turned on the charged particle will feel an electric field in the azimuthal direction, whose magnitude is inversely proportional to the distance from the solenoid and proportional to the rate of change of the current, according to Faraday's law. This results in a torque on the charged particle. At the same time, of course, a magnetic field is generated within the solenoid. The relation between the torque, or time rate of change of angular momentum, and the change in flux is given by

(1) Thus the total ..:hange in angular momentum depends only on the final flux, ac· c0rding to L\Lz = --( f,; )~. Evidf'ntly: if the initial a.ngular momentum was quanti!.ed in integer units, the final orbital angular momentum is quantip.ed in units of

f;

j--

integer.

Thus a composite particle which is a "molecule" (.onsisting of a flux tube and a charged particle in orbit around it, will in general have a fractional total augular momentum or spin. (This example, although illuminating and correct in its essential aspects, contains hidden subtleties. In fact, e large and contentious literature has grown up around it. As an example of the kind of issue which has been raised, consider the following. In the absence of magnetic monopoles, magnetic flux lines must close. Thus t\Jgether with the concentrated flux through our thin solenoid we ought to consider a return flux. Indeed if one closes the current loop feeding the solenoid, then one does find such a return flux - very far away, if the solenoid is long, but definitely present. There is field momentum - i.e., a non-zero Poynting vector, associated with the cross product of the Coulomb field of the charged particle and the return flux. The field angula.r momentum from the return flux turns out to be finite - equal and opposite to the fractional part of the angular momentum - no matter how far away the flux returns![l] Thus if we add in this contribution to the angular momentum, the total remains quantized in integer units. However for most physical questions, which after all involve local processes, it seems appropriate to forget the contribution "from infinity". Spectfically, the question that is most relevant to our main theme of quantum statistics is the phase multiplying the quantum-mechanical amplitude when one charge-flux-tube composite winds around another. Now if one winds around another within the return Jluz of both, we get the phase associated with fractional statistics. Thus if we are to maintain a sensible spin-statistics connection, we must assign fractional angular momentum to this composite. On the other hand if one chargeflux-tube composite winds around another ouuide the return flux there are no unusual effects, neither in spin nor in statistics, again consistent with the spinstatistics connection. I believe that all the controversies that have grown up around

6

this example can be traced to confusion concerning the appropriate order of limits, and that in all cases where we consider physics on scales well within the return flux, fractional angular momentum is physically appropriate.) 3. These subtleties disappear in the case of a massive gauge field, a case which is of considerable intrinsic interest. As a definite example, let us consider the case of a U (1) gauge theory spontaneously broken to a discrete Zp subgroup. Wf! assume that some charge pe field ¢ condenses, giving mass to tLe gtL~Jge qua.nta according to the usual Higgs mechanism. \Ve aiso a8Slirn~ that tr...ere are add.iticnal unit charge particles, produced by a field "", in thp, theory. The case p = 2 is reali!ied in ordinary BeS superconductor8, where tr.e aO'lbly cha.rg"!d Cooper pair field condenses, and ther~ are additional singly charged fields to describe the "normal elet;t,ron u exdtations. Such a theory suppurts vortex solutions, where cally as a function of the azi:nuthal angle (J as

t/>( r, 8)

-+

ve iS , r

t~e

q, field behaves asymptoti(2)

-+ 00

where v is the value of fjJ in the homogeneous ground state. Along with this asymptotics for t/> we must have for the gauge potential

As(r,8)

-+ -

1

(3)

pe

=

so that the gauge covariunt derivative Ds4> (89 4) - ipeA s 4», which appears (squared) in the energy density, vauishes at infinity. In this set-up the field strengt.h also vanishes asymptotically. Indeed we note that since under !!l. gauge transformation the fields transfo':'m as

= exp(iQA(z))~(z) = ~xp(ipeA(z))t/>(~) (4) A~(z) = AI'(z) + 8/LA(x), we can formally, by making the choice A = -8/pe, remove the space depenJence of ¢'(~)

t/> in (2) and make As in (4) vanish altogether. In this way, in fact, we have formally transformed back to the homogeneous ground state. But the gauge transformation function A is not quite kosher, since the angle 8 is not a legitimate single-valued function. The correct statement is that the asymptotic field configurations for a vortex are tdvial and can be gauged away locally, but not globally. Since we can pick a well defined branch of 8 in any patch that does not surround the origin, all local gauge invariant quantities must reduce to their ground state values - this explains, if you like, why Dt/> and F vanish. However the line integral of A around a closed loop surrounding the origin, which according to Stokes measures the flux inside, cannot be changed by any legitimate gauge transformation, and it is definitely not zero for the vortex. Indeed we find the basic flux unit is. ~p 211". pe

=

Another perspective on the global non-triviality of the vortex, is that our putative gauge transformation A = -8/pe transforms a unit charge field"" into

7

something that is not single-valued: following (4) we find that "",'(8 exp( -~)""'(8).

+ 211")

=

After these preparations, let us consider the angular momentum of particles in the presence of the vortex. The asymptotics (2) of the scalar order parameter seems not to be rotationally invariant: a scalar field should be unchanged by a rotation, whereas in (2) it acquires a phase. However the phase of tP is gauge dependent and we cannot infer from (2) that any physical property of the vortex violates rotation symmetry. In fact it is easy to see that if we supplement the naive rotation generator J. with an appropriate gauge t.ransforrnation: KiJ

1 = J% --Q pe

(5)

then K. leaves both the action and the asymptotic scalar field configuration (2) invariant. Thus, assuming the core is invariant, a true rotation symmetry of the vortex is generated by K.. If the core is not invariant, since the asymptotic fields are invariant the solution will still have a finite moment of inertia, and upon proper quantization we will get a spectrum of rotational excitations of the vortex similar to t~e band spectrum of an asymmetric molecule. (By way of comparison, the vortex associated with a global U(l) symmetry truly breaks rotational symmetry, and has infinite moment of inertia. This is the situation ·one finds, for example, for vortices in superftuid helium He-II. In such vortices. the condensate is in the unit orbital angular momentum or p-wave state, and there is a finite angular momenturr.. per unit volume a:sociated with it.) An important consequence of the modification of J into K is that the quantization of angular momentum for quanta orbiting the vortex is modified. The angular momentum of quanta with the fundamental charge e, for example, is quantized in units of -; + integer. This modification of the quantization condition, and the associated existence of states with fractional angular mOInentum, is precisely what we were led to expect by the arguments of the previous numher. 4. As was mentioned in passing previously, an ordinary BCS superconductor realizes the p 2 case of this set-up. The condensate consists of charge 2e particles (Cooper pairs), and the flux of a vortex is quantized in units of t/>o i;. Thus an ordinary electron, or more accurately a charged quasiparticle, in the presence of a vortex has its orbital angular momentum quantized in half-integer units. The fractional orbital angular momentum actually plays an important role in determining the properties of "core states" of vortices, which in turn play an important role in the dynamics of vortey motion. Indeed, excitation of low-energy core states is a major mechanism for dissipating vortex translation - a coherent motion - into charged quasiparticle motion, which quickly thermalizes.

=

=

8 (Note that strictly speaking the quasiparticles should be regarded as mixtures of quasielectrons and quasiholes; their charge is determined only up to a multiple of 2e, since they are inevitably embedded within a condensate of charge 2e pairs. Fortunately - but of course this is no accident - the charge ambiguity does not alter the ,h,ifted angular momentum quantization condition, since the fractional part of qt/>o does not change under q -+ q ± 2e.) States of fractional angular momentum also arise naturally in the interaction of rnatter with string sclutions of unified gauge models of fundarnental particle physics. Such s·-l\'1I".i0ns represt:'lt 5table or metastable field configurat~ons that might ha.ve b~en produ;:~d in the ~arly universe, 3.nd are widely conjectured to play a role in catalyzing the formation of galaxies. For this reason, they are called "c08mk stringst'l. T.:> see the e~sence c;f th~ p!'-,enciTaenon s it suffices to lo\)k l't a simplified model. Let us consider the gauge group SU(2). and suppose that the symmetry breaking OCCIlL::l ~hrulJ.gh the condeneatbn of scalar quanta belonging ·~o the spin-~ representation of the internal SU(2). Let us write the scalar field representing the condensate as f(Ja{3-r' where the Greek indices run over ±. That is, t/> belongs to the 3-ind~:v- symmetric spinor representation of the internal SU(2). Now suppose that the component whi~h condenses is the one with 1~ = i.e.,

i,

(6) in the homogeneous ground state. Then one has minimal vortices such that the phase of (tp) wlnds through 211' as we traverse a large circle at infinity. As in the Y(l) example we discu~sed before, this vortex asymptotic behavior is not invariant under naive rotations but is invariant if we supplement the rotations with an appropriate gauge rotation. In the present case, it is clear that the transformation which does generate a symmetry in the presence of the vortex is (7)

Clearly, this corrected angular momentum generator entails peculiar fractional angular monlentum lor particles with different representation content. For example, will have have orbital angular momentum quantized in particles carrying 1~ = unit5 of - + integer and ~ + integer respectively.

i

±i

Similar phenomena occur in more elaborate models. Roughly speaking, we may expect that particles carrying charge less than that of the condensate will carry fractional angular momentum in the presence of a minimal vortex. Some dramatic physical consequences of the fractional angular momentum around cosmic strings are mentioned in Section 5. 5. The mixing of ordinary and gauge synunetries that occurs for rotations in the presence of vortices also occurs in several other context.s. One example, perhaps the most basic of all, plays a central rQle in later developments. It concerns charged particles moving in a constant uniform magnetic field. In that case, translation

9 invariance gets mixed up with gauge transformations. For example, if we choose the symmetric gauge 1 Az = --By

2

A,I

=

(8)

1 -Bz 2

then naive translation through (say) E in the z direction fails to be a symmetry, since it changes the form of Ay: Ay -+ Ay + f B. Of course since the original physical situation - uniform B field - clearly i8 translation invariant, it must be possible to correct this failure. To do this, we must add a gauge transformation which takes A,I -+ Ay - E~B. It is of course easy to find such a gauge transformation, namely A == By in (4) . The t equired transf.:>rmation also nets nunt.ri;riaHy on cnargerl ftelds, nlultiplying a charge q field by the phase factor ~ez = eif~1B,.

t

f!

For our purposes, the crucial feature of cp~~ is that it depends on y. Because of this, if we follow our translation through E in the ~ direction with translation through '1 in the y direction, we will get a tot.al phase (9)

On the other hand, if we had translated in the opposite order - first y, .then z - we would have obtained e- i9 1e'1 B for the final factor, with the opposite sign in the exponential. Thus in the presence of the B field the algebra of the proper translation operators i~ altered) precisely because the naive translations must be supplemented with gauge transformations. This seemingly esoteric phenomenon is in fact responsible for a profound aspect of the physics: it is why we can have loctdized energy eigenstates in the presence of a uniform B t:ield, even though the physlcal situation is fully translation invariant. If, on the contrary, the translation generators commuted, then we could have diagonalized them simultaneously. In tha.t case, we wouJd have found that the ~P1P!'g~' eigenstates were necessarily plane wave3. Another example where naive space-time symmp.try transformations must be supplemented by gauge transformations is for a charged particlE in the presence of a magnetic monopole. This leads to a modification of the quantization condition for orbital angular momentum. In the presence of a magnetic monopole with magnetic charge g, a particle with electric charge e is constrained have integer or half-odd integer values of orbital angular momentum, when the product is il!~eger or halfodd integer respectively. (Of course the Dirac quantization condition is precisely that this product must be half-integral; indeed one way to derive it is from the quantization of angular momentum.) In particular, the minimal Dirac value = 1 gives half-integral orbital angular momentum. Thus monopole-charge composites will have the "wrong" total angular angular momentum, compared to what one would naively guess from their constituents. For example, if the isolated monopole and the isolated charged particle were both spin 0 objects, then the composite would half-odd integral angular momentum. Yet the spin-statistics theorem is not mocked:

*

g

10

as was shown by Goldhaber [2], the composite formed from these two bosons is a fermion!

~___- - -.........·11

III 1 2. Fractional statistics: tr~jector""-----­ space and the braid group 1. The existence of fractional angular momentum raises an interesting question: what about the spin-statistics theorem? Recall that this theorem, one of the finest general results in relativistic quantum field theory, connects the angular momentum of elementary particles to their quantum statistics. Particles with half-odd integral spin (e.g. electrons) are predicted to be fermions, and particles with integral spin (e.g. photons) are predicted to be bosons. This connection is of course amply confirmed by experiment. Since some ;)f our examples of fractional angular momentum were constructed ,n the framework of relativistic qua.ntum fi~ld theory, \~l~ :1.re. compelled either to make sense of the notion of fractional quantum statistics, or to give up the general spin-statistics connection. 2. What could fractional quantum statistics possibly mean? We are accustomed to thinking that quantum statistics supplies symmetry on the wave function for several identical particles: the wave function for bosons must not change if we interchange the coordinates of two of the bOGons, while the wave function for fermions must be multiplied by -1 if we exchange the coordinates of two of the fermions. A first guess might be that interchange of the coordinates of two fractional statistics particles should be accompanied by a fixed phase factc: ei9 1= ± 1. However a moment's thought reveals that this guess does not make good mathematical sense: iterating the exchange must give back the original wave function, so one definitely needs (e i9 )2 = 1. cond~tions

Nevertheless w:e can ml'ke sense of the notion of fractional statistics; but to do so we must go back to basics. In quantum mechanics we are required to compute the amplitude for one configuration to evolve into another over the course of time. Following Feynman, this is done by adding together the amplitudes for all possible trajectories (path integral). Of course the essential dynami~al question is: how are we to weight the different paths? Usually, we take guidance from classical mechanics. To quantize a classical system with Lagrangian L we integrate over all trajectories weighted by their classical action ei f Ldt. However, essentially new possibilities arise when the space of trajectories falls into disconnected pieces. Classical physics gives us no guidance as to how to assign relative weights to the different disconnected pieces of trajectory space. For the classical equations of motion are the result of comparing infinitesimally different paths, and in principle supply no means to compare paths that cannot be bridged by a succession of infinitesimal variations. The space of trajectories of identical particles, relevant to the question of quan-

12 tum statistics, does fall into disconnected pieces. Suppose, for example, that we wish to construct the amplitude to have particles at positions :Ct, :C2, ••• at time to and again at time t.t·. The total amplitude gets contributions not only from trajectories such that the particle originally at Zt winds up at Zt, but also from trajectories where this particle winds up at some other Zk, and its place is taken up by a particle that started from some other position. All permutations of identity, between the particles in the initial and final configurations, are possible. Clearly, trajectories that result in different permutations cannot be continuously deformp.d into one ano~.her. Thus we have the situation mentioned above! t.hat ~he space ·)f trajectories fallR into disconnected pieces.

Although the clMsical limit cannot guid~ us !n the choice of weights, there i-. an important consistency condition frvrn Guant.unl ~echanics itself that severely E:nit~ the possibilities. We must respect the rule, that if we fallow a trajectory ~{)1 from to to t1 by a trajectory 012 from t1 to t2, then t.he amplitude assigned to the combined. trajectory 002 should be the product of the amplitudes for 001 and Qu. This rule is closely tied up with the unitarity and linearity of quantum mechanics - i.e., with the probability interpretation and the principle of superposition - and it would certainly be very difficult to get along without it. The rule ie of course automatically obeye~ by the usu.al e~pression for the amplitude as the exponential of i times the classical action. 3. So let us determine the disconnected pieces, into which the space of identical particle trajectories falls. For simplicity let us cons~der only closed trajectories, that is trajectories with identical initial and final configurations, and to begin with let us focus on just two particles. In two spatial dimensions, but not in any higher number, we can unambiguously define the angle through which one particle moves with lespect to the othel, as they go through the trajectory. It will be a multiple of 7i; an odd multiple if the particles ar _ interchanged, an even multiple if they at(, ilot. Clearly the angle adds, if we follow one trajectory by another. Thus a weighting of the trajectories, consistent with the basic rule stated in the preceding paragraph, is

(10) where 4> is the winding angle, and modulo 21r.

(J

is a new parameter. As defined,

(J

is periodic

In three or more dimensions, the change in the angle 4> cannot be defined unambiguously. In these higher dimensions it is only defined modulo 2",. In three or more dimensions, then, we must have eifJt/Jlf( = eifJt/J'If( if 4> and 4>' differ by a multiple of 211". So in three or more dimensions we are essentially reduced to the two cases (J 0 and 8 1r, which give a factor of unity or a minus sign respectively for trajectories with interchange.

=

=

Thus in three dimensions the preceding arguments just reproduce the familiar ca:Jes - bosons and fermions - of quantum statistics, and show that they exhaust

13 the possibilities. In two space dimensions, however, we have seen that there are additional possibilities for the weighting of identical particle paths. Because of their close relationship to the traditional notion of quantum statistics, it seems appropriate to call the new possibilities that are opened up in two dimensions new forms of quantum statistics. The case for this terminology will be strengthened shortly, as their connection to the fractional angular momentum discussed previously is elucidated. Particles carrying the new forms of quantum statistics, are called generically Q,nyon,.

4. It is int uitively obvious, and can be rigorously proved, that the disconnected pieces of the two-particle trajectory space we have identified - trajectories with different windings in two spatia) dimensions, or with different permutations in three or more dirnensione - are a complete- ::s:l.talogue. That is: tW(' tr~.jecto"ies in two dimensions can be continuously deformed into one another if and only if they have ~he same winding, and t",:,::> trajectories in higher dimens!ons can be deformed into one another if and only if ei~her they both do, or they both do not, interchange the particles.

Passing to N particles, we finu that in three or more dimensions the disconnected pieces of trajectory space are still classified by permutations. With the o~vious natural rule for composing paths (as used in our statement of the consistency requirement for quantum mechanics, above), we find that the disconnected pieces of trajectory space correspond to elements of the per~utation group Pn . Thus the consistency rule, for three or more dimensions, requires that the weights assigner a.mplitudes from different disconnected classes must be selected from sonle representation of the group Pn . In two dimensions there is a much richer classification, involving the so-called braid group Bn.. The braid group is a very important mathematical object. The elem~nts of the braid group are the disconnected pieces of trajectory space. The multiplication law, which makes it a group, is simply to follow one trajectory from the first piec~, by another from the second piece - their composition lands in a. uniquely determined piece of trajectory space, which defines the group product. The "braid" in braid group evidently refers to the interpretation of the disconnected pieces of trajectory space as t.opologically distinct methods of styling coils of hair. It may be shown (see the paper by Wu [3]) that the braid group for n particles is generated by n - 1 generators (1'k satisfying the relations (1';(1'k

=

(1'1c(1';,

Ii - kl

~

2

(1';(1';+1(1'; = (1';+1(1';(1';+1, 1 ~

i

~

n - 2.

(11)

The (1's generate counterclockwise permutations of adjacent particles (with respect to some fixed ordering). Thus in formulating the quantum m~(.hanics of identical particles, we are led to consider representations of Pn - or, in two spatial dimensions, Bn. The simplest, and from the point of view of direct physical applications probably the most important,

14 representations are the one-dimensional ones. Higher-dimensional representations correspond to particles with some sort of internal degree of freedom, intimately associated with their quantum statistics. (Higher-dimensional representations of P n are related to the old idea of parastatistics. Higher-dimensional representations of Bn have appeared prominently in recent discussions of 2-dimensional conformal quantum field theories and integrable models. Although it is out of order logically, it seems r.ppropriate to mention at this point that there is a physically natural class of higher-dimensional representations elf Eft, which I believe has a good chance of occurrin& in thp. natural world. It is rt!aii~ed by qua.s£particles that carry both charge and flux with respect tc a discrete !H'o-abelian group.) The one-dimensional representations of Pn are simply the trivial or!e, which to each p~rrnutat~on: and the one which a.~sign8 ±~ depending on whether the p~rmutatior.. b even or odd. Clearly, these are simple generalizationd of the possibilities we found for two particles. They correspond to bosons and fermions, respecti vely. ~l~RigT)(! 1

The one-dimensional representations of Bn likewise are sirnple generalizations of the possibilities for two particles. They are again labelled by a parameter 8, periodic modulo 21r. In fact the formula for the weight is just the same as before, if now 4> is interpreted to mean the total winding angle, obtained by adding the pairwise windjngs over all the pajrs. In terms of the presentation (11) of the braid group, thp. second condition forces us to ta}p. the same phase, i.e., the same statistical parameter IJ, for every pair of particles. 5. For bosons or fermions, we can incorporate the effect of quantum statistics in a very simple way, as a condition on the behavior c,f the multibody wave function under exchange of coordinates. In t.he case of bosons or fermions, one can construct t.he eigenstates of the many-particle Hamiltonian directly from the eigenstates of the single-particle Hamiltonian, simply by taking tensor products. The sole effe~t of the statistics, in these two cases, is that one restricts to the subspace of many-body wave functions either symmetric or antisymmetrk under permutations, res!>ectively. The reason why this familiar, simple procedure fully incorporates the quantum atatistics, is ultimately that the rule for assigning amplitudes to trajectories beginning at Zl, :1:2, ... and ending at ZPlt ZP2, ••• depends only on the sign of the permutation P. Thus symmetry or antisymmetry in these coordinates is a condition stable in time. Also, we can obtain all trajectories with the proper weighting from trajectories along which the particles do not change their identity, if we allow all permutations of identity, with the appropriate sign factors, in the initial state. (Indeed, we have just the same trajectories, but with p-l acting on the initial configuration instead of P on the final one.) For generic anyons, the situation is different. The ampli~ude assigned to a trajectory depends not only upon the permutation suffered by the partides as they follow the trajectory, but also on other aspects of the trajectories by which they wind around one another. We cannot infer the number of windings, which determines the

15 weight to be assigned a trajectory, merely from the initial and final configuration of the particles. Thus it is impossible to take generic anyon statistics into account purely as a condition on the many-body wave function - a conclusion we anticipated at the beginning of this Section. Yet there is a limited, but important, sense in which this can be accomplished. To see it clearly, let us first specialize to the case of two particles. If we are willing to let the relative angle coordinate run from -00 to 00, without identifying angles differing by a multiple of 211', then we CAn keep track of the number of windings - the number of windings is just the relative angle divided by 211'. Then we can incorporate the effect of statistics as a boundary condition on the wave function, just as for bosons and fermions. In fact if we take (12) then the quantum statistical weighting is taken tare of: the bountiary condition (12) is stable in time, and the amplitudes for trajectories need not have extra statistical weights. Of course real particles live in real space, where the rela.tiv~ angle coordinate 1/J defined on the extended space without care. Without attempting to determine exactly how far its use could be pushed, we can see that "" is adequate to evaluate local observables, since by choosing a branch of 4> we get perfectly definite wave functions. If we restrict say tc 0 < £ < 4> < 211', what is the remaining implication of fractional statistics - i.e., of (12) ? It evidently a boundary condition, that restricts the allowed valued of the relative momentum, to values I such that iI 211' periodic, and we cannot use

(13) or in other words 8

I =-= - -

11"

+ even .Integer.

(14)

Thus the entire content of the fractional statistics in captured, for two particles, as a restriction on the relative angular momentum. For bosons (8 = 0) the relative angular momentum must be even, for fermions (8 11') it is odd; anyons with intermediate values of 8 interpolat~ linearly between these possibilities.

=

In the following Section this observation will used to solve some simple problems in two-body anyon quan,um mechanics. Let us note, however, that for three or more particles the relative angles form a very awkward, constrained, and redundant set of variables. Not much use has been found for the n-body generalization of the construction just discussed.

6. The interpretation of fractional statistics as a boundary condition on relative angular momentum allows us to address (partially) a gap in the preceding discussion. In 88serting that the winding angle is uniquely determined, I have implicitly assumed that the paths of individual particles do not Croll. This is a point that deserves to

16 be addressed more carefully than it has been in the literature, or will be here. The point is that for statistics other than bosonic, there is a non-zero minimum relative angular momentum between any pair of particles, of magnitude 18/11"1. Thus there is a centrifugal barrier, which prevents their paths' crossing, and the winding is well defined. For bosons it is not; but for bosons the weight does not depend on the w:nding anyway. From a deeper point of view, however, the definition of quantum statistics should not be too closely tied up with the behavior of the particles as they come close together. When particles corne rea.lly close together they undergo aU kinds of interactions, Q,nd a little phase more or less is not a separable !S8ue. Th.is question arise~ a!ready in cOlLventional phyai':s. It is, for e~ample, traditjonal and useful to call a dell~erium nudeu~ a ~o6on, e v~n though Nhen deuterons com~ close tog~the: we can't reailr assign them ~nique trajectories and isolc.te quant~nl statistical weight factors. (For instance the deuterons might exchange protons - then who's to say if the deuterons' coordinates were exchanged?) Similar issues arise for Cooper pairs in a superconductor. Since realizations of anyon statistics in nature will inevitably be composite Qr collective states of bosons and fermions, we should not expect (or require) ideal pointlike anyon behavior at short distances. The proper definition of quantum statistics for particles or refers to the phases of amplitudes for trajectories where the particles remain widely separated. A direct operational definition of the quantum statistics for particles can be given in terms of low-energy scattering: see Section 4 To avoid misunderstl' "1ding, it may be well to state explicitly that the issues involved in defining particle quantum statistics are very different from those involved in formulating the commutation or anticommutation relations for local quantum fields. Whether the additional possibilities for quantum 5tatistics of particles can be used to define additional possibilities for field quantization in any direct way, I do not know. The (ol1siderat:o:ls of Section 7 do give a con~truction of pointlike anyons in local quantum field theory, but in a somewhat more roundabout way. 7. Finally, to close this Section I would like to make a remark that is very important to any consideration of possible phenomenology associated with fractional statistics. It is, that fractional statistics is generically associe-ted with violat,ions of the discrete symmetries P and T. This can be seen in several ways. Most basically, it comes about because these discrete transformations reverse the orientation of particle trajectories, Le. effectively they change the sign of the angie q,. Thus a trajectory that was weighted with ei8 ",/fr is instead weighted with the complex conjugate of this number, after P or T is allowed to act. Remembering that q, can be taken to be any integral multiple of 11", we see that aside from the cases ei8 ±1, these weights are distinct. This means that P or T cannot be valid symmetries of the system (although the combined operation PT may be). Exceptions to this conclusion could arise if there are also particles with -8 statistics, and ordinary space-time reflections are combined with changing each particle into its mirror partner; but otherwise generically fractional statistics requires P and T violation.

=

17

3. Fictitious field construction, and quantum mechanical examples 1. The physical implications of fractional statistics are elucidated by a simple dynamical realization of them. It will now ue shown that the quantum statistics of particles in two spatial dimensions can be reproduced by imagining that there is (fictitious) charge and flux attached to the particles. This "electric charge" and "magnet~c flux" need not, and generally will not, have anything to do with ordinary electromagnetism. Rather a neVi gauge field is introduced as a mathematical device, '!specially for th4! purpose. It has become conventional to borrow the terminology of electromagnetiem in describing these fields, but to r~fer to them explicitly as fictitinu,8 fields if there is danger of confu8ion. Tn equations, I ahall use a for fictitious fields and A for t.he true electfomagn~tic fIeld. First, let us recall (fame elementary properties of magnetic flux; tubes in two space dimensions. Since for a !ong solenoid the magnetic field runs through the center and vanishes outside, we are not surprised to find that in two dimensions (taking a slice) we can have flux confined to a small core, with vanishing field strength outside. The singular limit of this situation is the delta-function flux tube, realized by the vector potential

(15)

= Ii

=

z Thus in Cartesian coordinates 4 z f; Z2~,2' By Z2-+ ,2 or in polar coordinates 4,. 0, 4", From any of these forms, but particularly the last, it is easy to see that the field strength !I" vanishes away from the singularity at the origin, while the line integral of f 4"dz" is • for any path that winds around the o:igin once clockwise. Acccrding to Stokes' theorem~ this indicates a total flux J l1"dz k dz' = J b dA. ~ through the lo~p. This flux evidently rr.ust be ascribed to a delta function magnetic field at the origin.

=

= /;.

= 0,,4, - a,41c

=

Now since the magnetic field ;ranishes away from the origin, there is no classical force. Nevertheless, as Aharonov and Bohm made famous, in quantum mechanics there are importa:tt long-range effects associated with a point flux tube. From a formal point of view, this is perhaps most easily appreciated in the framework of path integrals. In the path integral, to get the total amplitude we add the contributions of all possible paths with the appropriate weights. For charged particles in the presence of a gauge field, the weight includes a factor ei Liut. de eiq ,,·a de eiq a·dx representing the interaction of the particle with the gauge field. For the singular we have q 4· dx = ga~. Thus this flux tube, represented simply by 4. = integral simply measures the angle swept out by the particle, as viewed from the origin. So the weighting factor that appears multiplying the contrioution from a given path contains a phase proportional to the winding of that path around the origin.

= J

J

!W,

J

= J

18 But this sort of weighting is just what we need to implement fractional statistics! In other words the dynamical effects of attaching fictitious charge and point flux to particles are indistinguishable from the general kinematical possibilities opened up by fractional quantum statistics. Thus ill two dimensions we may freely trade the quantum statistics of particles for special sorts of gauge interactions. We may for instance represent fermions as bosons with fictitious charge and flux attached, or bosons as fermions, or indeed anyons of any kind as any other kind. 2. It ia instructive to implelnent these observations in a Hamiltor..ian frame':.Jrk.

Let us take the simplest case of two identical nonrelativistic particles (statistic!:'; farticle going around another should give 11m as much again, or 27f1m. The preceding heuristic argumen~ can be backed up by explicit calculation. Consider again the famous (unnormalized) Laughlin wave function 'lt~ =

II

(z~ - zi)m

IT e-~IZiI2,

(69)

l$i + qA Aj

--+

.A j

+ ajA,

(85)

but the second term is not. In fact

Now the first term on the right hand side vanishes upon integration by parts (What? are we suddenly allowed to be cavalier about integration by parts? - that's how we got into trouble in the first place. The point is that here we are already dealing with a theory on the boundary, and the boundary of a bour..dary is zero!). The second term manifestly vanishes. The third term becomes, upon integration by parts,

(87) This is exactly of the right form to cancel off the offending term in the gauge transformation of the bulk theory! To enforce the cancellation, the product -qb is detcrmined to be equcl to the coefficient of the bulk Chern-Simons term. In this construction it is essential that the gauge transformation law for t/J takes the Stiickelberg form (85) . This form forbids the appearance of mass terms - e.g., terms of the form 4>2 - in the Lagrangian describing the t/J fields. Without such mass terms, the left- and right-movers are essentially independent. Also, their dispersion relation is forced to be of the form w <X Ie at small wave vectors Ie. Thus the existence of "massless" or "gapless" edge excitations emerges clearly as a n~cessary consequence of gauge invariance, in the context of a Chern-Simons description of the quantized Hall state.

S3

These considerations extend to the more general si.tuation envisaged in Eqns. (82), (83) above. In the general case, the surface Lagrangian (84) reads (88) where

in is the component of i normal to the boundary.

4. In the previous three numbers, we have seen that Chern-Simons interactions in bounded regions are necessarily associated with interesting surface phenomena. There is an amusing and potentially significant dual to this: these interactions also arise in a very natural way ell surface terms. To start the. discussion, let us consider the 3 + I-dimensiona1 action

J '" J J

115 =

d3 xdt eo{lT'8u A{l8TA, tfxdt 4E . B

=

(89)

tfxdt 8o (e°{lT' A{l8TA,).

Here a is a gauge field, which mayor may not represent electromagnetism. The second equality in (89) shows that we are considering quite a sir.'tple sort of interaction. It is fully gauge invariant, though P and T violating (this no longer bothers us, of course!), and of quite low order in gradients. In fact it is manifestly of the same order as the standard Maxwell kinetic energy i(E 2 + B2). The third term shows why the possibility of such an interaction is usually ignored: since it is a total derivative, it would seem not to alter the equations of motion and thus to be negligible. In non-abelian gauge theories the corresponding terms play a central role in the saga of anomalies and instantons. The reason such terms are important in that context is closely related to our considerations in Section 3. There we found that the Aharonov-Bohm terms implementing fractional statistics were formally total derivatives and thus did not effect the classical equations of motion. Yet because the quantities (relative angles) of which they are total derivatives are not legitimate globally defined quantities, the Aharonov-Bohm terms can and do have a dramatic impact on the quantum theory. Likewise in the non-abeli&, gauge theories the quantity that tr( e . b) is the total derivative of is not a legitimate globally defined quantity. (To be more precise, it is the relevant surface integrals which fail to exist.) Here I want to discuss a much simpler phenomenon, that exists even at the classical level. It is simply that if ·we insert into the integral appearing in (89) the characteris+ic function of a bounded region, as we should do for an effective interaction arising in a bounded sample, then it is no longer true that the integrand is a total divergence. In fact the difference between· this modified integrand and a

S4 total divergence -

- is exactly a 2 + 1 dimensional Chern-Simons interaction living on the boundary! To see this more clearly, let us specialize to the case in which the boundary is the z - y plane z O. Then 8a Xs is, for Q Z, a Dirac delta function for the z 0 plane. The final expression in (90) then becomes e&fJ.,' AfJ8.,A, ~ efJ'" AfJ8.,A, with a 2 + 1 dimensional £ symbol now understood, which is nothing but the density of the Chern-Simons interaction in the plane.

=

=

=

5. In this way we see that the sorts of 2 + 1 dimensional theories which support fract.ional statistics, and alJ the interes~ing behavior that goes wlth them: arise very naturally in another way, quite different from what we cave discussed before. That i3, they arise as part of the appropriate description of surfaces on, or interfaces between, bulk materiais at least one of which has P and T violation. The possible phenomenology assodated with such structures has not been much explored to date. I will venture one modest suggestion, which lies close to the center of the circle of ideas we have been discussing. Let us suppose that term!!) of the form (89) arise in the effective Lagrangian (or, if you like, free energy) describing the bulk state of some P and T violating type II superconductor. When ouch a superconductor is subjected to a magnetic field H in the range Hel < H < Hc2 this magnetic field penetrates the sample in vortices. Now frollA the point of view of the bounding surface, the end of the vortex looks like a localized flux point. Those of you who have persevered to this point have become very familiar with the fact that the Chern-Simons term associates electric charge with such a flux point. The sign of this charge depends on the direction of the magnetic flux. Thus, in this situation, we are led to predict the existence of charge inhomogeneities at the end of flux tubes. Of course this charge will be screened, but there ought to remain a significant difference in the profile of the electric potential across vorticeR with magnetic fields pointing in different directions. It is widely suspected that some of the heavy fermion superconductors have P and T violating order parameters. Recently it has become possible to study the electric potential profile of superconducting vortices, using scanning tunneling microscopy. Clearly, the possibility that vortices and antivortices will have markedly different profiles should be kept in mind.

ss

REFERENCES 1. M. Peshkin, Ph'll'. Rep. 80, 376 (1980).

2. A. Goldhaber, Phy,. Rev. Lett. 38 , 1122 (1982). 3. Y. S. Wu, Phy,. Rev. Lett. 62, 2103 (1984). 4. R. P. Feynman and A. R. Hibbs, Quan.tum Mechar.ic, an.d Path Integra" (McGraw-Hill, Inc., New York; 1965) Section 7.4 5. J. March-Russell and F. Wilczek Ph'll'. Rev. Lett. 81, 2066 (1988). 6. M. Alford, J. March-Russell and F. Wilczek Nud. Phy, B328, 140 (1990). 7. L. Krauss and F. \Vilczek Phy,. Re!J. Lett. 82, 1221 (1989).

8. E. Witten, Nud. Phy,. B223, 433 (1983). 9. E. Witten, Comm.. Mo,th. Phy,. 121, 351 (1989). 10. R. Prange and S. Girvin, eds. The Quantum Hall Effect (Springer-Veriang, Inc., New York; 1987). 11. Y.-H. Chen and F. Wilczek} Int. J. Mod. Ph'll'. B3, 1 (1989). 12. A. MacDonald and S. Girvin .Phy,. Rev. Lett. 68, 1252 (i9&7); ~. Rezayi and F. D. M. He.ldane, Ph'll'. Rel1. Lett. 81, 1985 (1988); S. Zhang, T. H. Hansson, and S. Kivelson Phy,. Rel1. Lett. 62, 82 (1989); N. Read Phy,. Rev. Lett 82, 86 (1989).

PART II. STATES OF ANYON MATTER

S9

II' States of Anyon Matter In this part, the many-body theory of anyons is considered. First, the basic definition and elelnentary properties of anyons are briefly reviewed. Average field theory is introduced as a method to roughly assess the qualitative properties of aggregates of anyons, and is argued to be reliable in certain limits. The basic mechanism of anyon superconductivity is discussed in simple physical terms. The mechanism is argued to be fundamentally different from spontaneous symmetry breaking, and specifically frQrn BeS pairing. The formal principle involved, previously discussed under the rubric "spontaneous fact violation", is given a new more precise mathematical formulation: it is the spontaneous projectivization of a linear symmetry. Tight conceptual connections are made between the fractional quantized Hall effect and anyon superconductivity, basically by inverting the average field procedure. Values of the quantum statistical parameter are identified at which anyon superconducting states are likely to occur. Values of the filling fractions are identified, as a function of the quantum statistics parameter, at which anyon quantized Hall states are likely to occur. The existence of a new state of matter, the anyon metal, is suggested, and some of its propertks a." roughly assessed. Possible generalizations of the ideas to case~ involving inhomogeneous, three-dimensional or finite systems are discussed in a preliminary but reasonably definite fashion.

60

1. Definition of Anyons 1. In two spatial d!mensions, there are new possibilities for quantum statistics, that interpolate continuously between bosons and fermions. Particles carrying these generalized statistics, are called generically anyon". Thf" new possibilities arise. fundamentally, because in two spatial dimensions (but not in three or more) the winding number of particle trajectories around one another is well defined. This means that it is sensible to weight these trajectories by a phase factor e 2i6w , where UJ is the winding number. We should also include, for identical particles. oriented "half-windings" where the particletJ are interchanged in pi'..8sing from the ir.itial to the final state; thus Ui is allowed to be half-integral. It should be clea.r that (J is periodic lnodulo 21r, and that (J = 0 corresponds to bosons, (J 1r to fermions.

=

2. The effect of these unu3ual quantum statistics is best understood in a shnple dynamical model. Imagine particles carrying a charge q and magnetic flux. with respect to some gauge field a. Note that in two dimensions magnetic flux is naturally associated with point-like objects, with a vector potential but no magnetic field outside a singular core. The gauge field here is assumed to have no independent dynamics, and in particular it is not to be identified with ordinary electromagnetism; it is common to call a the fictitious gauge field. Now as one particle winds around another, it acquires phase according to the factor e iq JB'" dt = e iq J(I.dr, an elementary fact made famous by Aharonov and Bohm (and vice ver.ta). Wh~n one particle winds all the way around the other, the total phase is clearly eiq~. Thus the effect of the fictitious charge and flux is precisely to implement a change in the statistical parameter, accordint; to ~8 qfJ/2. Thus there is a simple dynamical construction which allows us to transmute the statistics of particles in two dimensions at will. We can represent bosons as fermions with fictitious charge and flux attached, or fermions as bosons, or either of them as anyons with any given () ....

=

3. This simple construction can also be implemented in great generality, at the level of local quantum field theory. Indeed let there be given a conserved current P', and imagine adding to whatever Lagrangian you had before the two terms (1) where ai' occurs nowhere else in the Lagrangian. Then variation with respect to ai' yields the field equation

(2) Eq. (2) indicates that the gauge invariant content of a is totally enslaved to j, so that a has no independent dynamics. Furthermore the a 0 component may be

=

61

integrated to give the relation Q = 1'9 between the fictitious charge and flux carried by a particle. Thus we see that addition oi ilL has essentially implemented the construction discussed in the previous number, and therefore that it has transmuted the statistics of particles carrying ;0-charge Q by il8 = Q2/21'. This construction is known as the Chern-Simons construction, after two mathematicians who discussed the second term of Eq. (2) in an entirely different context. As a method of transmuting quantum statistics, it was introduced by Tony Zee and me. 4. Generically, fractional statistics - that is, the existence of anyons that are not bosons or fermions - requires violation of the discrete symmetries P and T. A sure formal sign of this is the appearance of an E symbol in Eq. (1). It also follows directly from the definition of statistics in terms of winding of paths. Perfonning either a reflection in a spatial axis, or a rf!versal of the motion, changes the sense of winding for a trajet.:tory. The transforrr:ed trajectory (in the minimal case of a halfwinding) should be weighted by e- i9 , which is different from the original weighting ei9 unless 8 0 or 11' modulo 211', i.e. bosons or fermions.

=

Conversely, a variety of examples indicate that once P and T are violated in a 2+1 dimensional field theory, the emergence of anyon statistics is almost inevitable.

62

2. Average Field Theory

1. It is a fascinating problem, to figure out the behavior of anyon matter. It has been more than sixty years since physicists have been confronted with new forms of quantum statistics, and the corresponding new problems of many-body statistical mechanics. After aixty years, perhaps bosons and fermions are getting a little stale, and an infusion of fresh ideas will be welcome. However, it seems to be quite a difficult problem til figure out the statistical mechanics even of ideal anyon gases. The basic difficulty, which make!; this problern much more difficalt i~r generic anyons than fo: bosons or fermions, is that for generic anyons the Dlany-body Hilbert space is in no sense the tensor product of the oneparticle Hilbert spaces. This circumstance can be understood in various ways. Its root is that in the general case the weighting supplied by anyon .statistics depends not only on the initial and final states, but also on a (topological) property of the trajectory connecting them. This means that in the general case it is impossible to summarize the effect of quantum statistics by projection on the appropriate wE;ighted states, as we do for bosons and fermions - wh~re, of course, we project respectively on symmetric and antisymmetric states. Another viewpoint is afforded if one considers what the fractional statistics means as a restriction on the relative angular momenta of a set of particles. For bosons, the relative angular momentum is an even integer; for fermions, an odd integer; these rules are encoded as symmetry or respectively antisymmetry requirements. In the general case the rule is that relative angular momenta 2~ + even integer are allowed. This requirement generally (i.e. for 8 ;;j:. 0,11' ) cannot be stated in any simpler way. It is simple enough for a single pair of particles, and indeed many problems in two-anyon quantum mechanics are easily solved in closed form. For tbree or more particles, however, the relative angle coordinates are a redundant set, and they are related by complicated constraints. Thus these coordinates cannot be used in any straightforward way to study anyon statistical mechanics. 2. Instead, the most fruitful app.roach has been to employ what I call the average field approximation. (This approximation is commonly called the mean field approximation in the literature, but I think that it is sufficiently different from mean field theory as usually understood to merit a new name.) The essence of the average field approach is to replace the actual fictitious field, which configuration by configuration is rapidly variable in space - indeed, it consists of delta functions at the particle positions - by its uniform average. To be more precise, one replaces the anyons with statistical parameter 8 by the nearest fermions, and repla:es the residual flux per particle

(3)

63

where

81'('0'.

= ?r( -1r8 -

8

Int( -)) 1r

(4)

measures the deviation from fermi statistics, and I nt( z) denotes the nearest integer to z. Two questions immediately arise: Why perturb around fermions? and Why does one expect this approximation to be reasonable? I will now address these two questions, in turn. 3. Why fermions? In the first place, of course, because a lot is known about fermions, and so by reducing the problem to fermions we can hope to benefit from the wisdom of past sages. The real questions is, therefore, lNhy not bosons? In fact there have been attempts to learn about aoyon! by pe:turbing around boson!. However, they have met with limited success, for instructive reasons. As we shall soon see, the crucial features of the ground atate, for low-energy properties, are the incompre"ibilityand ab,ence of low-lyin.g ezcitation" (equivalently: existence of a gap) under appropriate circumstances. Tliese are features that are easy to see in a fermion picture, where they occur for when bands are exactly filled. They seem to be quite difficult to see ill a boson picture. In Feynman's famous papers on liquid helium, one of the main points was to make it plausible that for the boscn gas at low temperatures there are no low-energy excitations that do not change the density, i. e. nothing besides sound waves. His argument, although brilliant and surely physically correct, involves furious hand-waving and does not form a secure useful basis for further deductions. Concretely, we shall find that the low-energy, long-wa1.r elength properties of the anyon gas depend crucially and in a highly non-analytic fashion on 8. This dependence arises from commensurability effects: it is absolutely crucial whether the Landau bands in the average field theory are exactly filled or not. To make the argument for bosons, it would be necessary to develop some generalization of Feynman's argument that applied to bosons in the presence of E. magnetic field, and to find that it works (to prove the absence of low-energy excitations) only for certain values of the density. This may not be impossible, but at the very least it seems a difficult way to proceed. 4. When is the average field approximation likely to be correct? I can identify at least two circumstances. Let us first consider why this approximation isn't manifestly ridiculous. The point is that while configuration by configuration - i.e., in a basis where position states are diagonal - the fictitious magnetic field looks extremely inhomogeneous, it is not necessarily so for the ground state. The particles' position is affected by quantum-mechanical uncertainty, and it is not ridiculous to think that the fictitious magnetic they carry around with them is, in the ground state, smeared into its average. tAt suflidently high energies and momenta, of cour~e, one would certainly be able to resolve the structure, .and the correlation functions in this regime will surely not be well described by the average field approximation).

64

To assess this possibility, let us consider its self-consistency. To be specific, let us suppose that ~ is close to, but slightly less than, one. Then the average field is related to the density by b

= 211"p(1 -

9 - )/q.

(5)

11"

Now let us evaluate the cyclotron radius, to see how spread-out the particles are. The cyclotron radius is, in general mv

r C'·C. .

= --:"'"b q '

and if we substitute for v the nominal fermi velocity numbel" of particies within a cyclotron radius 2 p7rrr;yc.

1

= (1 _ !)2·

(6) vJ

~ we find for the m

(7)

When the quantity on the right-hand side is > > 1, we find that the size of a single particle's orbit contains many other particles - the magnetic length is much larger than the interparticle spacing - so that the mean field approximation should be very reasonable. Thus this approximation is self-consistent when the st·atistics is sufficiently close to fermions. 5. Also, the average field should get better as the range of the fictitious flux increases. In o"r treatm~nt so far, based on the pure Chern-Simons construction Eq. (1), this range was zero: the flux is concentrated in deltc:. functions at the particle positions, according to the field equation Eq. (2). This is because the mass of the fictitious gauge field a is effectively infinite; indeed, we left out a kinetic energy term fo" a, so its quanta can't propagate! We can modify the construction slightly by including a kinetic energy term proportional to the Maxwell term f~~, for the the fictitious field. Then a will have a finite IOMS (proportional to the coefficient I' of the Chern-Simons tt:rm), and the range of the ~orresponding flux tubes attached to the particles will be finite. Of course in adopting this modification we hav~ introduced a new degree of freedom, because the a field is no longer completely enslaved to j. The existence of this degree of freedom mayor may not be appropriate, depending on the intended application. If the mass of the a quantum is not too small - I won't attempt to be precise about how small - it will not be excited in the low-energy, long-wavelength regime of primary interest in any case. It will, however, spread out the flux even at the classical level, and thus it will tend to make the average field approximation more appropriate.

6S

3. Anyon Superconductivity: Basic Mechanism t

=

1. For ~ (1 - ~), n integer, we find that the average field approach directs us to replace the anyons, to first approximation, by a gas of fermions moving in the presence of a fictitious external magnetic field, whose magnitude is such that n Landau levels are ezactly filled. One nlight expect that exact filling of these magnetic bands is especially favorable energetically and produces a particularly stable or rigid state, similar to what o~curs for noble gas atoms, magic nuclei, or insulating solids. To test this expectation, we Inust find out what it c'osts to have configurations for which it is no longer true that n levels are exactly filled. Since the magnitude of the fictitious field is directly tied to the local density according to the field equation Eq. (2), to, wind up in a situation where we don't have n Landau levels exactly filled, we must introduce an extra real magnetic field. These are very fundamental points, to which we shall refer repeatedly below. If the real field is in t he same direction as the fictitious one, the density of states per Landau level will be somewhat greater, and we will not quite completely fill n levels anymore. Let us denote the fractional filling of the highest level by 1- z. Then from the conservation of particle number we derive

(b+ B)(n - z)

= bn;

(b+ B)z

= Bn.

(8)

(In this Section it is convenient to set the quantum q of fictitious charge equal to the quantum e of real charge.) For the total energy we then find

(9)

Thus the energy relative to the ground state is positive, and grows linearly with B for small B. If the real field is in the opposite direction from the fictitious one, the density of states per Landau level will be smaller, and we will have to promote some particles to the n + 1st level. Denoting the fractional filling of this level by z, we have from particle conservation

(b - B)( n + z)

t

= bn;

(b - B)z

= Bn ,

(10)

This section, and the next, draw heavily upon the treatment in [1] and [2]

66 and for the energy

E=e(b-B) e(b-B) 211" m

{I:(l+!)+(n+!)z} !=O

2

2

(11 )

Thus in this case too the energy relative to the ground state is positive and grows linearly with B for small B. Despite the asymmetry of the situation, the coefficients of the terms linear in B are equal in the two cases. The quadratic terms differ. 2. These ~rguments though simple are quite significant. They suggest that the anyon gas, dot t~e statistics considered, will strive to exclude external magnetic fields. Thi~ 's the germ of the Meissner effect, a hallmarlt of superconductivity. At th~ same ti!lle they suggest thp. existence of a~ energy gap in the charged particle spectrum. Indeed, the energy to create a separated particle-hole pair should be just the energy to excite a fermion into the lowest empty Landau band, viz. Epair

21fp = -eb = -. m mn

(12)

Considered more closely, these arguments also suggest a close connection between vortices and fermion excitations that seems to be something new in the theory of superconductivity. This connection is characteristic of anyon superconductivity, and will playa key role behw both'in its deeper theory and in its phenomenology. The point is this: since the fictitious field is uniquely tied to the particle density, and is appropriate to n Landau levels being exactly filled, to accommodate any additional real magnetic field we will necessarily have to excite particles across the gap. (Or to create holes, a process which we have seen is also charactedzed by a gap.) Conversely, if the particles do not fill the Landau levels exactly, there must be a real magnetic field present to account for the mismatch. Anticipating that the filled Landau level state and its possible adiabatic modulations is the su::>erfluid component, we are led to conclude that in An.yon !uperconductivity, chArged qua!i-

particle! and vortice! do n.ot con!titute two !eparo,te !ort! of elementary ezcitation! - they are one and the !ame. We can also infer the value of the flux quantum from this identification. Adding a single fundamental unit 211"1 e of real flux increases the number of available states by one per Landau level. Thus, for n filled Landau levels, the act of piercing the material by a unit flux tube creates n holes. Clearly this is not the most elementary excitation. The most elementary excit~tion is to produce just one hole. Thus the elementary fluxoid is lIn of the fundamental unit, or 211" lne. 3. Although these simple arguments have taken us a long way, there remains a central feature of superfluidity that is not at all obvious, or even true, in the simple approximation described thus far. This feature is the existence of a sharp NambuGoldstone mode, or concretely an excitation with the dispersion relation ",2 oc 1c2 at low frequency and small wave vector. It does exist. It was discovered in a remarkable

67 calculation by Fetter, Hanna, and Laughlin. They calculated the effect of adding back the residual interactions in the random phase approximation, and found that these interactions produced the necessary pole in the current-current correlation function. In physical terms, this means that there are particle-hole bound states at zero energy. Unfortunately t"'e calculations do not by themselves make it clear why the massless morle exists. Besides being emotionally disturbing, it is objectively Ullsatisfactory to lack such understanding. Vlithout it, one may be left uncertain whether this central qualitative feature of the anyon gas is robust, or an artifact of the approximations employed in the calculation. Similarly, one may be left uncertain whether small changes in the modp.1 Hamiltoniar.. itself - which after all, is· high Iy idealized - might change this feature. Fortunately, the existence of the massless mode can be understood conceptually, from several different viewpoints. One path to understanding follows the trial blazed by Landau and Feynman the theory of superfluid He 4 , and by Loncon in the theory of supe:-conductivity. in The main goal of Feynm8O's arguments was to explain the ab,ence of low-energy excitations that do not change the density. If that tlb,ence is established, superftuidity is a very plausible cClnsequence. Indeed, there will always be a sound wave mode with the dispersion relation w 2 <X k 2 , and if long-wavelength sound waves have nothing they can possibly deca.y into, they will represent dissipationless superftows! Along similar lines, London emphasized the possibility of understanding superconductivity starting from the concept of an en.ergy gAp in the charged particle spectrum, which he argued would lead to "rigidity" of the wave function and explain the Meissner effect. This simple physics has been somewhat lost in the modern emphasis on spontaneous symmetry breaking, which produces the massless boson - i.e. the sound wave - as a Nambu-Goldstone boson of broken number symmetry. While the spontaneous symmetry breaking point of view leads to profound insights for many physical systems, we should not forget that it can be a complicated and artificial approach for others. Indeed, at present the Feynman-London arguments seem to provide the best heuristic understanding of anyon superconductivity. The burden of the previous number was precisely to show the existence of an energy gap. On the other hand, the anyon gas is definitely compre"ible, and therefore supports sound waves with w 2 <X 1c2 at smaH k. That the compressibility of the 8Oyon gas is finite can be argued in many ways, at various levels of rigor. At the lowest level of rigor, we may appeal to the famous Gold berger method of proof by reductio Ad Ab,urdum. Assume that the the anyon gas is incompressible. But that's absurd. Therefore, it's compressible. (You might ohject that this rather begs the question: Why is it absurd? Well the worst case is fermions, since they have the .largest repulsion and should exert the greatest pressure. But we know the ideal fermion gas.is compressible.)

68 By way of contrast, a filled Landau level of true fermions has a fixed preferred length scale - the magnetic length. In that case the energy cost of a variation in density (with the magnetic field· fixed) does not go to zero as the wavelength of this variation is increased to infinity. In the language of the previous number, one must produce a fixed number of particle-hole pairs to accommodate a density ·/ariation, regardless of its wavelength. The crucial difference between the anyon gas and this fermion system, which is missed in the average field approximation, is the fundamental faet that the fictitious magnetic field adjusts itself to the local density. Thus, in contrast to what happens for fermions in a fixed magnetic field, particle-hole pairs do not have to be excited in order to accommodate anyon density gradients. At a higher level of rigor, one may appeal to a sum rule derived frorr. the fundamental cOl1lre.utator identities between the Hamiltonian and the momentum generators. Let us define a spectral weight,

W(k,w)

= E l(llpkIO)1 2 w- 16(w -

,

E,

+ Eo),

(13)

where PI; is the density operator at wave vector k, 10) is the ground state of the system, and the sum is over all excited states Il), while Eo and E, are the respective energy eigenvalues. For a system of non-relativistic particles of mass m, with forces that are velocity-independent, there is a well-known sum rule:

(14) This sum-rule, which is obtained by evaluating the quantity (01 [[P-I;, H], Pk] 10), is easily derived for the anyon system using a representation in a singular gauge, where the wave function is multivalued and the kinetic energy has just the free-particle form. It is a version of the famous f-sum rule used in atomic theory. At the same time, we know that

J 00

W(k,w)dw =

Aoo(k),

(15)

o where Aoo( k) is the density response function evaluated at w = o. The k -+ 0 limit of this function is the compressibility, which is finite for our system since the ground state energy is an analytic function of p. (For non-interacting anyons, the energy per particle is simply proportional to p.) From the formulas, it follows that the root-mean square value of the energy in the spectral density at wave vector k is given, in the limit Ie -+ 0, by, Wk Vo

= vok, = (p/mAtMJ(O)) 1/2 •

(16)

Now there are two possibilities. The spectral density may be exhausted by a single mode, in which case the frequency of that mode must be precisely equal to vol:.

69

(This is what happens in superfluid He4 , or in a neutral fermion superconductor, such as He 3 .) Alternatively, there may be a spread of energies entering the spectral weight at wave vector Ie. In this case there will be some excitations with energies greater than vok, while others must have energy less than vok. This is the case in a normal fermi liquid, where there are particle-hole excitations throughout the interval 0 < W < vFk, where VF is the fermi velocity. For anyons, we have argued that there is no continuum of particle-hole excitations at low energies. Thus we are no longer surprised to find that there is an isolated boson mode with energy w vok.

=

4. Another perepective on the superfluidity of the neutral anyon gas, which becomes superconductivity if the anyons are charged t is provided by the concept of spontaaeous f!l.ct violation. Very roughly, we Inay say that the Nambu-Coldstone phenomenon occur! whp.n there is, in an effective theory, some symmetry of the microsccpic theory that has been lost. Alternatively phrased, the associated charge operator does not commute with the Hamiltonian, even though the local conservation law is g11aranteed on microscopic grounds. Then the mismatch is repaired by the addition of massless particles to t.he theory. These can generate terms which excuse the mismatch, by making the integration by parts that would otherwise lead directly to a contradiction, subtle. Anyon superfluidity does not fit easily into this mold. This fact is reflected in the nature of the arguments I presented-for anyon superfluidity, which made no reference to a broken symmetry. It emerges even more clearly when one considers the mechanics of the more formal RPA calculations, which demonstrate the superfluidity. (I will not discuss the details of these calculations here; I only want to make a very general point ab~ut their structure.) We would expect that in constructing a broken symmetry ground state we would have to make some arbitrary choice a.mong a set of energetically degenerate possibilities. Thus for in~tance i~ a ferrOITlagnet we would hE!.ve to choose a definite direction for the magnetization; in a BCS superconductor we would have to choose a phase for the condensate, and so forth. However in the RPA calculations onE; simply starts from the average field approximation and treats the remainder of the Hamiltonian straightforwardly as a perturbation, and it is not at all obvious where such a choice has be4l!n made. Having convinced ourselves that anyon superfiuidity does fit easily into the familiar mold of spontaneous symmetry breaking, we are led to ask whether there is some similar algebraic characterizatior.. of these superflaid states - some generalization of the classic Nambu-Goldstone mechanism. I shall now argue that there is a related, but significantly different, general principle at work. In the average field approximation translation symmetry, although not broken, is realized in an unusual way. The algebra of this symmetry is altered. This alteration is what has been called "spontaneous fact violation". In the context of anyon superfluidity the fact that is spontaneously violated is the commutativity of

70 the translation generators ti, [ti' tj] :F O. More precisely, within the average field approximation to the anyon gas, where the behavior of this gas is approximated by that of a gas of electrons in an external magnetic field, we have the following situation. The system is translationally invariant, in that there are unitary operators which implement translations and commute with the Hamiltonian. However, these generators do not commute with each other. This behavior is related to the fact that aithough the background b field, and all gauge invariant quantities, are invariant under translations, yet the gauge covariant derivatives which generate translations in the two directions do not commute with one another. We can pick a gauge in -by, a y 0 - then tra.nslation symmetry in the z direction is manifest; which ax or a gauge in which ax = 0, a, bz - then translation symmetry in the y direction is 'manifest; Dui; it is iInpossible to choose a gauge in which both are manifest at the same time. Again, we can choose single-particle eigenfunctions which are plane wave~ in the x direction but localized in the y direction, or 'Vice 11ersa; however genuine plane waves are definitely not eigenfunctions.

=

=

=

Thus, similarly to what happens in the classic Nambu-Goldstone phenomenon, certain commutators related to sy"mmf;tries are altered; they have "impossible" matrix elements in the ground f'tate. Yet, as I have been at pains to emphasize, no symmetry - certainly not translation invariance - seems to have been lost. How can symmetry commutators be altered, without the symmetry being lost? The mechanism at work is related to a famous tbeorem of Wigner concerning the realization of symmetry in quantum mechanics. The deep point of Wigner'd theorem is that it is not amplit~des themselves, but absolute squares of amplitudes, which appear in the physical interpretation of quantum mechanics. Thus, when we realize symmetries by unitary transformations in Hilbert space, these transformations are only determined up to a phase. In particular, if SI and S2 are symmetry operations which are realized by the unit.ary operators U(St}, U (82 ), thc!n the unitary operator for the combined operation 8 1 . S2 will in general be (17) where 71(Slt 8 2 ) is a phase factor, i.e. a complex number of unit modulus. Sometimes the Us may be redefined, multiplying by phases, in such a way that '1 reduces to the identity. Then we have a normal, linear representation of the symmetry algebra. However it is also possible that 71 cannot be defined away. In that case, we say we have a (properly) projective representation. For a continuous symmetry, the infinitesimal form of Eq. (17) will give a modification of the commutation relations among the symmetry generators. If the ground state is such that the symmetry algebra is realized by a projective representation, we can repeat the arguments used in the ordinary Nambu-Goldstone context, to show the necessary existence of massless particles. The integrated charge operator does not commute with some operator, even though the local conservation law is guaranteed on microscopic grounds. Th~ mismatch is repaired, by the addition of massless particles to the theory. Such particles can generate terms which excuse

71

the mismatch. They make the integration by parts, which would otherwise lead directly to a contradiction, subtle. 5. I believe that this mechanism of generating massless particles by spontaneous projectivization is a proper generalization of the usual Nambu-Goldstone mechanism, and that it may be of use elsewhere in physics. In a certain sense it allows us to have our cake and eat it too, in that we get the massless particles while retaining the symmetry. 6. The elusi veness of the order parameter is a familiar story in some of the other 2 + 1 dimensional systems in which fractional statistics playa role. In particular, there has never been a fully sat.isfactory description of the relevant order parameter in the fractionally quantized Hall effect - a description, that is, of what is the general class of things of which the celebrated Laughlin wave function is an example. It seer.r.1S to me that spontaneous projectivi~ation is the right concept. here too. For electrons in a magnetic field the translation algebra is projectively realized to begin with. The commutator of translation in the z direction through a with translation in the y direction through b is multiplication by the phase factor exp(iabeB). There is a subalgebra, generated by translations satisfying ab 21r/eB, which is realized linearly. The same phases that occur "microscopically" also occur for the translation algebra acting on the translation symmetric state where a Landau level is completely filled.

=

On the other hand, the corresponding phase factors for translations of quasiparticles around the Laughlin l/m states are exp(iabeB/m). Thus part of the algebra which is linearly realized microscopically, is realized only projectively on these states. Of course there is no massless particle associated with this spontaneous projectivization, because the symmetry which has been spontaneously projectivized is discrete. Nevertheless, the projectivization of the symmetry algebra reflects deep physical properties of the states. It is,. for instl'.nce, closely tied up with the fact that the basic unit of charge for elementary excitations is e/m, and that the correspo&ding elementary unit of flux is 21rm/e. Indeed, the subalgebra of translations that is linearly realized is generated precisely by translations whose commutators correspond to transport around a rectangle enclosing an elementary flux unit, so modification of the algebra goes hand in hand with modification of the flux unit.

72

4. Anyon Superconductivity: Phenomenology 1. The arguments above lead us to expect that the electromagnetic response of the charged anyon gas (for 8 = ?r(1 - lin)) at low frequency and wave vectors is dominated by a single collective mode, essentially by a sound wave. Ther~ are no other low-energy excitations. If these expectations are correct., it should be possible to form a particularly simple effective Lagrangian describing the electromagnetic response in full detail. The electromagnetic field and a scalar field representing the sound wave are the only degrees of freedom that need to appear. Furthermore, as long as we are intereated in the behavior at low frequencies and wave vectors, we can restrict ourselveH to terms of low order in 8pGCe and tinle derivfitives. Guided by these principles, together with the general principles of Galilean invari:mce and gauge invariance, we are led to postulate an effective Lagrangian of the form

L=

21 (tP. -

2

v

2

GAo) - 2"(0,, tp e 4»). It can be shown (Ref. S, p. 126) that for any tp., f1~ in c]) (connected), '"a(, flJo) ~ '"3(, 91;). Accordingly, when concerned with only the group structure of '"a(c]), fIJ.), we shall denote this group by WI(~). fIJ.(x) = tp., for all x belongs to the iderttity of "a(~, tp.), and if cp(x) is in Q", then tp( -x) is in Q;I (or. -Q", in the additive notation).

7. Since wa(tl» is Abelian, if it is finitely generated (Ref. S, X, Corollary 8.3) its elements can be labeled by a set of numbers

(nl' •.. , nk , nl'+1t ...• n,) 55 n,

(3)

~uch

that n.,· ..• nk range over the integers and ~~. , ... , n, racge k) has arisen also in different context,' but is not supported by experimental evidence so far. From now on we shall discuss only 4»'s for which 1- k in (3), but in general our results are valid. We shall say that fIJ(x) is one "kink. of type if ,,(x) is in Q.\'f' where N, - (0, 0, ••• , 1, .•. , 0), (S)

10. Some homotopy groups of spheres are tabulated in Ref. 3, Table I , and many more are given by Toda.7 Table I shows the first five homotopy groups for the underlying manifolds of classical Lie groups; it has been compiled from se ..era! sources. 7 The relation w,,(. @ 'I") - w,,(.) e w.('I') ;s a useful tool for the construction of Qd hoc spaces (Sec. 1.9). IL QUANTIZATION

I being in the ith place; then cp( -x) is in Q-XI and will be called an antikink of type I. Any field ,(x) in Q. with" given by (3) can be considerecf as composed of a certain number of kinks or antikinks of the various "ypes N,.

1. When a classical field theory is quantized, the statement "the measurement of the field at time 1 will give the values fIJ(x, I)" is renlaced by "the probability amplitude for getting the result tp(x) when measuring the field at time 1 is 'I"[fIJ(X)](I):' 'l" bejng a complex-vatued time-dependent functional of ,,(x). continuous in t and ". Notice that in this picture (Schradinger) and representation [diagonal in f'{x)), the argument of the functional is a c-number field, an element of the function-set ~x of Sec. 1.2.

,.". - (1I1u

+ n~I), ••• , n:u + n~ll) == n cu + n(ll;

r

~ In the literature tbe nth homotopy grciJP of CI> is variously defined as the set of homotopy claS5CS of mappings (sa,P.) - (., ".) of an" sphere S· into , with p~~S-) - "., of mappings (/", ar) - (CI>, ".) of an" cube r into •• with its boundary a/" going into tp., etc. All those definitions are equivalent among themselves (this is proven by showing that after proper identifications the domains of the mappings are homeomorphic) and to ours for " -= 3: we can, for instance, deform X into a 3-balJ: x - x/l + lxi, which is homeomorphic to a 3-sp~ere upon identification of its boundary Ixl - I into one point P•• Notice that this is allowed by the boundary conditions introduced in Sec. 1.2.

• See. for .....nce, M. OeD-Man, Phya. Letters I, 214 (1964).

20 We shall assuane that, at. any time, 'Y[cp(x)Xt)

-=

°

if 9'<x) ,~x( fIJ.)

&

Q,

(6)

so we can use Q -= ~x (fIJ.) instead of ~x as our configuration space; this implies restrictions on the dynamics of the system, but we shall not elaborate

Gro.,.

, H. Toda, COlllpo6illoll M~I/aods In Ho"",op~ 01 S,"~ra (Princeton University Press. Princeton, NJ., 1962); A. Borel, in Srlllilltlln~ H. Ca"an (2) (Ecole Normale SUperieUR, Paris. 1956). L Pontrjagin, Topological Grollp. (Princeton University Press. Princeton, NJ., 1946); V. o. Boltyanskii, Transl. Am. Math. Soc. 7, 135 (1957).

111 D. FINKELSTEIN AND J. RUBINSTEIN

1766

this point in the present work. Since for 9' in Q we cali write 'Y[".](/) = 'f(q, t), a continuous complex-valued function on Q @ R (R = real number space spanned by t), another way of putting (6) is

fol'l'(q, ')1 1 dq = 1,

(7)

of

provided that the concept integral on Q can be defined; even if not. we shall use expressions of the type in (7) as symbols for expressions of the type in (6). We call" the state function(al).

J. If we accept the Feynman scheme Cor quantization, which uses integrals over continuous histories f'{x, I) (in our case paths in Q), it can be sho~n (Ref. 3, Sec. (!lol) that

( ,'I'(q, 0)1' dq - I implies

JQrI

r l'l"(q, .)!' dq .. 1;

Jo.

(8)

the flow is called closed. Its paths are then closed paths or loops. (II) is the mathematical expression of the "leaving the system unchanged" used above. Since until now we have considered 'I" to be singlevalued, (9) and (11) imply

'I'(F09)

= 'I'(F q). 1

(12)

Now we wish to> consider that the state functi('n '1'('1) might be multiple-valued; this point is elaborated upon in Sec. 11.5. and (12) no longer holds.

5. Our first problem is to find whether the domain Q. of the state funcJon'l"(q) admits multiple-valued continuous functions. Intuitively speaking, these have more than one value at nch 'I in Q.. (the value set changing co.ltinu:Jusly with 'I), such tbat any two values 'r. and 'If1 at q can be CO~lnccted by a continuous succession of values "·(q(s)] by traveling some closed i>Ath q(s)[q(O) - ttl) ~ 'I]' It is evident that no simply connected Q. admits such functions. I Moreover. since the value of a multivalued function depends on the point q and the way, is reached from some standard point. sach a function on Q. can be defined as a (single-valued) function on CQ., the universal cov~ring space of Q. We define CQ. as follows:

then, if the ··charges" defined in (Sec. 1.1) are inter· preted as nonlocal observables in the quantum theory, (8) tells us that they are conserved in time. In the following, whenever we talk about a homotopic conservation law it shall be understood in the quantum sense given by (8). In short, (6) and (8) allow us to extend the results of Sec. I to a quantum the:lry of We choose a base point 'I.. in Q. and consider the " in which ,,(x) can still be treated as having a c· number field of eigenvalues. We assume for simplicity paths ,(s) in Q. (0 ~ s ~ J) such that '1(0) - 'I".; two paths ,(s) and (s) are equivalent if i(1) that the particle numbe:s ", obey a superselection (1) - 'I (4Uly) and there exists a homotopy relative rule,I i.e., that the support of any realizable physical to {O, I} between q(s) and q'(s). i.e., a continuous state '1'('1) is one of the Q... mapping ,(s, II): [1- Q. (0 S.r S 1, 0 ~ II ~ 1) .:. In what foUows we shall be concerned with two such that kinds of discrete operations that leave physical ,(s,O) - ,(s), q(s, I) - (s), systems unchanged but may multiply the state vecton by -I: a 2" rotation and the exchange of two '1(0, II) - ' .. , ,(I, II) - 'I. identical subsystems. The 2" rotatioa can be realized The equivalence classes of paths ,(s) are the points continuously by a succession of infinitesimal trans. of CQ., if we introduce the following topology: formations in an obvious way, and we shall give in Sec. IVan analogous realization for an operation giyen '('J and an an:wise FIG. 16. Schematic or a derormation. The starting point (not shown) ii a rigid rotation or two identical kinks about the line of centen. Then the axis of rotation of each kink is twined perpendicular 10 the line or centers. siving the derormation (a), where each kink is represented by a segment, whose endpoints traverse the intertwini"l helices shown. The intermediate step (b\ is obtained in an obvious way from (a). In going (rom (b) to (c), the eXlendability or Fig. 12 into the 4-cube is applied to the upper and lower halves of (b) independently. Since (c) represents XI, the derormation 10 the identity (d) has already been shown.

ACICNOWLEDGMENT We thank Charles Misn~r for criticisms of the manuscript which helped our own understanding, as w::11 as the presentation. APPENDIX A: PROOF OF LEMMA 3

=

Let q 0 q, EE Q _, q(s) te a path between q q(O) and q, -= q(I), wacs) a 21TS extrinsic rotation operator around· some axis, and tl1(S)" W(s)· tI a homotopically trivial loop (so W· f, .. f1). Then q(s, I) Wacs) .q,

q(s, t)

-=

W'(s) . q,

q,

w.(d). t - 2t q,

o ~ s < 1, 1-0.

O~S~I

)

q,

I-t~s~1

q[(4t - 1)3s],

O~s~l

W'(3s-I)·q(4t - 1), l~s~1 q[(4, - lX3 - 35)), l~s~1

)

O~s~l

q(3s), W a(3s - 1) . q(l)

-= W(3s - 1) • q1 .. f,(S), l~s~! q(3 - 3s), f~s~l O~s~l

q(3s), tI,(s, a):q1(S, I) -= q1~S), Q1(S,t) .. ql

01-

w

I

IS

~ 2~

} P - (1 - 2~) u - Us u. - u,

f(x, y,O, It u) .. f{x, y, 0, I, u.) q;(x, " 1, I, u) .. f(x, y, 1, I, r:J 9'" fP{x - ., y, s, I, u.) fP{x - ., y, 0, I, II.) f(x - ., y, 1, I, II.) fP{0, " I, t, u) - fP{0, y, s, I + ., u.)

tp(x, y, s,O, u) f{x, Yt S, 1, II) f{x, I, 0, I, II) f{x, " 1, t, II) -

J(s) .. s,

O~ I ~ I - I I

-'1-11+«, - (I - II + «) g(s) .. I 1 - 2a I,

I(s - (l

+ II -

II

+I

I-

1- 2a 2a - - - (s - (l 2(a - c)

- I -a-« (s -

II -

I + II

+1, -

II -

f

~I ~

I-

II

I,

+ ~,

I», I - a + ISS ~ I + a -

~,

I+II-I~S~I+II+C,

(l

+ a + «»,

Then f(x. y, s, 0, II) - cp(x - p/(I), , + pg(s), s, u, 9(x, y, s, 1, II) - cp(x, y + p/.(,), I, 1, II,) f(x, y,O, I, u) - f{x, " 0, t, U.,) . fP{x, y, 1, I, u) .. cp(x, y

~

O~s~I-It-c,

- 1- 2c.,

-0, 1-2a

I

I+II-C~d,

c»,

-II-C

- 1-

~

+ P(1 -

24), 1, t, u,)

I

+ a +. ~ s ~ 1.

",») II - u

p---', u. - u.,

U,~II~II.,

124 CONNECTION BETWEEN SPIN, STATISTICS, AND KINKS 9'<x, y, s, 0, u) = 9'<x, y, (IS, 0, u.), s~ tp(x, y, (I(s - 1) + 1, 0, u.), s ~

=

1779

i} i

Y • 1 u) (I = 1 - u - u. 2(a - E), U. ~ U < 1. oJ, , • 1 - u. q:.(X, y, 0, t, U) = tp(X, y, 0, t, u.) cp(X, y, 1, t, u) = qJ{x, y, 1, t, u.) Let (a + 2E, a, 0) = e and W(s) be a rotation operator around the z axis (Sec. III). Collecting the above expressions we get (Fig. 10) 9'(x.y, s, O. J) = f1't(W(s)· (x - e», q:(x, y. of, 1, 1) = !fleX -- c), cp(x,y. O,!. 1) = 'P\(x - e), Q.E.D. fPtX,y, 1, t. 1) :a 9'l(X - c), ..Jx 't'\,

Y s 1 u) = ,

"

..Jx Y'\"

125

THE

PHYSICAL REVIEW cA jowf'fltll of nperimmlal aNti '''eordical SECOND SERIES, VOL.

P"7UCS ~sla6Iu"~tI 6, Eo L. Nic"ols iN 1893

115, No. 3

AUGUST I, 1959

Significance of Electromagnetic Potentials in the Quantum Theory Y. AHARONOV AND D. Bou.. 11.1/. Wills Ph)'sics LabortJIory, U"ifoersily of Bristol, Bristol, England (Received May 28, 1959; rmsed manuscript received June 16, 1959) In this paper, "'c discuss some interesting properties of the elertromaglletic potentials in the quantum domain. We shall show that, contrary to the conclusions of dass:cal mechanics, there exist effects of IlOtcn· tials on charscd particlcs, even in the region when' all thc fields (and therefore thc forces on thc llarticl\.'S) vanish. \\'~ shall thcn discm:~ possible experiments to test these (:onclusioas; and, I1nall)', wc Sh:lJl su~cst furthcr IlOssihlc dC\'clopmcnts in the interpretation of tne potentials.

I. INTRODUCTION

I

N classical clectrCldynamics, the vector and scalar potentials were first introduced as a convenient mathematical aid for calculating the fields. It is true that in order to obtain a classical canonical formalism, the potentials arc needed. Nevertheless, the fundamental equations of motion can always ~ expressed directly in terms of the fields alone. In the quantum mechanics, however, the canonical formalism is necessary, and as a result, the potentials cannot be eliminated from the basic equations. Nevertheless, these equations, as well .as the physical quantities, are all gauge invariant i so that it may seem that even in quantum mech~mics, the potentials themselves have no independent significance. In this paper, we shall show that the above conclusions are not correct and that a further interpretation of the potentials is need(.,() in the quantum mechaniFs.

assume this almost everywhere in the following discussions) we have, for the region inside the cage, 11=110+ Veo where 110 is the Hamiltonian when the generator is not functioning, and V (I) = '4> (t). If "'o(x,t) is a solution of the Hamiltoni.m 1/0, then the solution for 11 will be

"'="'or

iS/A

,

s=

f

V(t)tlt,

which follows from

as)

01/1 (01/1. ih-ih-+~.,- r,sl'=[H.+ V(t)}/t=Ht/I. at

at

at

The new solution differs from the old one just by a phase factor and this corresponds, of course, to no change in any physkal result. Now consider a more complex experiment in wl:ich a single coherent electron beam is split into two parts and 2. POSSIBLE EXPERIMBNTS DEMONSTRATING each part is then allowed to enter a long cylindrical THE ROLE OF POTENTIALS IN THE metal tube, as shown in Fig. 1. QUANTUM THEORY Mter the beams pass through the tubes, they are In this section, we shall discuss several possible ex- combined to interfere cohere!'!tly at F. Dy means of periments which demonstrate the significance of potcn- time-determining electrical CCshulters" the beam is tials in the quantum theory. We shall begiu with a· chopped into wave packets that· are long comparetf simple example. with the wavelength A, but short compared with the Suppose wc= have a charged particle inside a "Faraday length of the tubes. The potential in each tube is detercage" connected to an external generator which causes mined by a time delay mechanism in such a way that the potential on the cage to alternate in time. This will the potential is zero in region I (until each packet is add to the Hamiltonian of the particle a term V(x,l) well inside its tube). The potential then grows as a ,!bich is, for the region inside the cage, a function of function of time, but difterently in each tube. Finally, tUne only. In the nonrelativistic limit (and we shall it falis back to zero, before the electron comes near the 485

126 486

Y.

AHARONOV

AND

1).

nOHM B

....1

Metal foil

Ftc;, 1. Schematic experiment to demonstrate interference wilh time-dependent scalar potential. A, B, C, D, E: suitable devices to separate and uivert beams. W" IV.: wave packets . .41" M!: .:;ylindrical meLal tubes. F. intcrrertnce region.

FIG.

2. Schematic experiment to demonstrat~ interference with time-independent vector potentl:!.I,

suggests other edge of the tube, Thus the pot("lItial is nonzero onh· whit,: the electrons are '.vdl inside the tuLt (r(:hion II)~ When tb electron is in region III, there is again no jX)rentiai. The purpo!:c of th:s a~!'angemcnt is to ellsu~c :hi\t ~he electron ;s in a. t;:nc-varyin~ potcl1tj,d ',vithout cv::r being ir.. a field (becaus_ the field doc:> nct po=

f

H·ds=

f

A·dx.

To demonstrate the effects of the total flux, we begin, It is evident that the interference of the two parts at as before. with a coherent beam of electrons. (But DO~ F will depend on the phase difference (SI-S2)/". Thus, there is no need to make wave packets.) The beam IS there is a physicai effect of the potentials even though split into two parts. each going on opposite sides of the no force is ever actually exerted on the electron. The solenoid, but avoiding it. (The solenoid can be shielded effect is evidently essentially quantum-mechanical in from the electron beam by a thin plate which casts a nature because it comes in the phenomenon of inter- . shadow.) As in the former example, the beams are fl!rence. We are therefore not surpri~ed that it does not brought together at F (Fig. 2). appear in classical mechanics. The Hamiltonian for this case is From relativistic considerations, it is easily seen that tile covariance of the l" bnve conclusion demands that [P- (elc)A]Z H there should '>e similar results involving the vector 2m potential, A. The phase difference, (Sl-St)/li, can also be exIn singly connected regions, where H=VXA=O, .we pressed as the integral (ell), tpdl around a closed can always obtain a solution for the above HamiltoDilD circuit in space-time, where fJ is evaluated at the place by taking ~=~or'81', where ~o is the aolution when of the center of the wave packet. The relativistic generA=O and where VS/1= (elc)A. But, in the experiment alization of the above integral is discussed above, in which we have a multiply cXmnected region (the region outside the solenoid), ~.rUIII is. non-single-valued function1 and therefore, in ~ DOt a permissible solution of SchriSdinger's equatiOL Nevertheless, in our problem it is still possible to where the path of integration now goes over any closed such solutions because the wave function splits iDUt circuit in space-time. two parts "'=Ih+~t, where represents the beam OD As another special caSe, let us now consider a path in space only (1= constant). The above argument • Unless where • is integer.

if(~-~.4K),

"-_,e,

"'I

an

127 E LEe T R 01\1 A G NET Ie PO 'f E N T I A LSI N

"'I

one side of the solenoid'and the beam on the opposite side. Each of these beams stays in a simply connected region. We therefore can write "'1 ="'I°e-'B&I.,

"'I = "''lOr'BIIA ,

where Sl and S2 are equal to (e/c)fA·dx along the paths of the first and second beams, respectively. (In Sec. 4, an exact solution for this Hamiltonian will be given, and it will confirm the above results.) The interference between the two beams will evidently depend on the phase difference,

(Sl-S~/fJ= (e/hc) f A·dx= (e/IIC)~o. This effect will exist, Poven thC'ugh there are no magnetic forces acting in the places where the electron bear:.\

~~rder to avoid fully any possible question of contact of the electron with the magnetic field we note that our result would not be changed if we surrounded the solenoid by a potential barrier ~hat reflects the electrons perfectly. (This, too, ;s confirmed in Sec. 4.) It is easy to devise hypothetical experiments in which the vector potf!ntial may influence not only the interference pattern but also the mom~ntum. To see this, consider a periodic array of solenoids, each of which is shielded from direct contact with the beam by a small plate. This will be essentially a grating. Consider first the diffraction pattern without the magnetic field, which wiD have a di!K:rete set of directions of strong constructive interference. The effect of the vector potential will be to produce a shift of the relative phase of the wave function in different clements of the gratings. A corresponding shift will take place in the d!!"ections, and therefore the momentum of the diffracted beam. 3. A PIlACTlCABLE EXPERlMBNT 1'0 TEST FOR TID BFnCTS OF A POTENTIAL WBBRB TJIER.B ARB NO mLDS

As yet DO direct experiments have been carried out which confirm the effect of p:>tentials where there !s no fie1d. It would be interesting therefore to test whether such effects actually exist. Such a test is, in fact, within the range of present possibilities.I Recent experiments'·c have succeeded in obtaining interference from electron beams that have been separated in one case by as much u 0.8 mm.1 It is quite possible to wind solenoids which lie IIDaller than this, and therefore to place them between the separate beams. Alternatively, we may obtain localized lines of flux of the right magnitude (the

-M.=-

~~. 0wDben I. DOW makinc a prelimiaary aperimeDtai . " .If this questioa at Bristol. ~ Ph)'L Rev. 85, IOS7 (1952); !!o 490 (1953) . • G.!"'IIt ~, .:ad Suddeth, Rev. Sci. Iastr. Q , 1099 (1954). • MoDeastedt, NaturwiaseDlCbaften· 42, 41 (1955) j G. ~t &ad H. Dtlker, Z. Physik 145, 377 (1956).

QUA N TUM

457

'f HE 0 R Y

magnitude has to be of the order of t/Jo= 2..cA/,,,,4X 10-7 gauss eml ) by means of fine permanently magnetized "whiskers".' The solenoid can be used in Marton's device,· while the whisker is suitable for another experimental setup4 where the separation is of the order of microns and the whiskers are even smaller than this. In principle, we could do the experiment by observing the interference pattern with and without the magnetic flux. But since the main effect of the flux is only to displace the line pattern without changing the interval structure, this would not be a convenient experiment to do. Instead, it would be easier to vary the magnetic flux within the same exposure for the detection of the interference patterns. Such a variation would, according 10 our previous discussion, alter the shupness and the general form of the interference bands. This alteration would ti'en constitute a verification of the predicted phenomena. "·hen the magnet.ic flux is altered, there will, of c.:Ol!rSe, ue a.n induced elcctrk field oUH:ide the solenoid, but the effects of this field can be made negligible. For example, suppose the magnetic flux were suddenly :lher~l in the middle of an exp03ure. The electric field would 1hen exist only for a very short time, so that only a small part of the beam would be affected by it. 4. EXACT SOLUTION FOR SCATTERING PROBLEMS

We shall now obtain an exact solution for the problem of the scattering of an electron beam by a magnetic field in the limit where the magnetic field region tends to a zero radius, while the total flux r(.;mains fixed. This c:Jrrespollds to the setup described in Sec. 2 and shown in Fig. 2. Only this time we do not split the plane wave into two parts. The wave equation outside the magnetic field region is, in cylindrical coordinates,

f

alia-+-1 ( -+ia a )1+kI =0, [-+a,z , ar r as

(1)

where k is the wave vector of the incident particle and :1= -,./ch. We have again chosen the gauge in which A ~= 0 and A.=~/2r'. The general solution of the above equation is

.

tt= E

...+-

e'...[a..J ..... (kr)+bJ_< ..... )(kr)],.

(2)

where a", and b. are arbitrary constants and J ..... (kr) is a Bessel functiOn, in general of fractional order (dependent on .). The above solution holds only for ,.> R. For ,.
The lower limit of the integration is determined by the requirement that when " goes to zcro, 1/11 also goes to to zero becAuse, as we have seen, 1/11 includes Bessel functions of positive (\!'dcr only. In order to discuss the asymptotic behavior of t/lh let us write it as 1/11= A (II-I!], where

11 = faD

(-;)I-+oIJ 1-+ o1 " ...·

~i"-'[J ..+I-;eilJ.. }i,',

o

(9)

I t is convenient to split y, into the following three parts: >/t="-',+"-'-z+'/-a, where The first of these integrals is known':':

00

Y,I=

L

(-i)"'+"J",+"e imf ,

1ft-I -I

1/Iz =

L (- i)"'+"J ",+..e''', In our cases, fj=cosfJ, k= 1, so that 00

=

L

(-i)"'-"J_Qe- i "",

(5)

garl

e'''
exp(iz!)dz.

(23)

0

This functior. vanishes on the line d=1f·. It can be seen that its asymptotic behavior is the same as that of Eq. (2) with a set equal to n+l. In this case, the singlcvah.ieulless of t/I is evident. In gt:neral, however. the behavior of t/I is not so simple, since t/I does not become zero on the line 8=r. S. DISCUSSION OF SIGNIFICANCE OF RESULTS

The essential result of the previous discus.4\ion is that III Cjl.antum theory, an electron (for ex.ample) can be influ ...nced by the potentials ~ven if all the field regions are excluded from it. In other words, in a field-free multiply-connected region of spar:e, the physical properties of the system still depend '.>n the potentials. It is true that all these effects of the potentials depend only on the gauge-invariant quantity j"A·dx= fH·ds, so that in reality they can be expressed in terms of the fields inside the circuit. However, according to current relativistic notions, all fields must ·interact only locally. And since the electrons cannot reach the regions where the fields are, we cannot interpret such effects as due to the nelds themselves. U Se-:, for eum~~ D. Bohm, QtuuiIu_ Tlrur, (Prentice-Hall, Inc., Englewood eliDs, New Jersey, 1951).

AND

D.

BOHM

In classical mechanics, we recall that potentials cannot have such significance because the equation of motion involves only the field quantities themselves. For this reason, the potentials have been regarded as purely mathematical auxiliaries, while only the field quantities were thought to have a direct physical meaning. In quantum mechanics, the essential difference is that the equations of motion of a particle are replaced by the SchrOdinger equation for a wave. This SchrOdinger equation is obtained from a canonical formalism, which l·;HlIlCIt. h(" l·:'Vre:::'!,t",1 in il"rms oi the In:ids alont:, but whid\ ;tlso require:;; lhl: potentials. Indecd, thc potentials playa ro;(', in Sduudingcr's equation, which is analogous te) that IIf the index of rcfration in optics, The Lorel;l/. icm.:c [rE+ymme'~l'.i.c wave functions, appear in a. natural way in thp formalism. But this is only the case in which the particles move in three- or higher-dimensional space. In one &nd two dimensio1ls a continuum of posRible intermediate cases COllllcCts the boson and fermion cases. 'rile effect of particle spin in the present formalism is discussed.

1. - Intn.:ductioL.

In the quantum description of a system of identical particles, the indistinguIshability of the particles has consequences which deeply affect the physical lu.&;:';ure of the systeln. Usually, the indistinguishability is expressed in the theory by inlposing symmetry constraints on the state fWlctions and on the observabl~s. Thus, the state functions can be either symmetric or antisymmetric with respect to the interchange of two particle co-ordinates, and all the observables must be invariant under such an operation. The physical consequences of this postulate Heem to be in good agreement with the experimental facts. However, the theoretica.l justification of the postulate, as found, for example, in standard textbooks (1-a), often seems unclear, and several authors have attempted a (1) (I) (3)

A. MESSIAH: Quantum Mechanics, Chap. XIV (Amsterdam, 1962). L. I. SCHI~F: Quantum Mechom,ics, Chap. 10 (New York, N. Y., 1968). E. MERZBACHER: Quantum Mechanics, Chap. 18 (New York, N. Y., 1961). 1

133 J. M. LEINAAS

2

and

J. MYRHEIM

more careful analysis (4-8). It seems to us, however, that no completely satisfactory discussion on the consequences of indistinguishability, in the context of nonrelativistic quantum mechanics, has emerged so far. The problems start with the introduction of particle indices. This step brings elements of nonobservable character into the theory and tends, therefore, to make the discussion more obscure. Thus, the meaning of the particles being identical is often explained by saying that the physical situation is unchanged if the particles are interchanged. This is expressed by the equation

wh~l'e

p is any vel'IDutation of the N particle co-ordinates. The above statenlent has (~ol'r(·('tly been criticized (8, by pointing Ollt that the 'word « interchange» heru ha.s 110 physical meaning. The two quantities in the equation have no separu.te meaning, and the ~quation, therefore, at most refl~cts the redundancy in the notation, i.e. that the same particle configuration can be described in different ways. In the present work we want to present a formulation, which seems to be conceptually more :simple, in which this redundancy in notation is eliminated in a very natural way. First we discuss, in sect. 2, the classical dedcription of a system of identical particles, starting with the consequences of indistinguishability in classi\}al statistical mechanics. We go on and study in detail the classical configuration space. Our point is that the configuration space of a N-particle system is not the Cartesian product of the single particle spaces, but rather an identification space which has, in fact, a different global topological structure, although it is locally isometric to the product space. The quantum descriptjon is discussed in sect. 3. It is introduced in terms of quadratically integrable functions defined on the classical configur~tion space of the system. Since the indistinguishability of the particles is taken into account in the definition of the configuration space, no additional restriction, corresponding to the symmet:rization postulate, is put on the state function&. The quantization is studied in detail for a two-particle system in one-, two- and three- or higher-dimensional Euclidean space, with an emphasis on the physical effect of the global curvature of the configuration space. It is shown how a translation can be made to the traditional description in terms of complex wave functions on Euclidean space. The restriction on wave fnnctions, to be either symmetric or antisymmetric, then appears in a natural way from the formalism, without having to be introduced as an additional cunstraint. This is, however, only the case provided space is at least three-dimen(') A. M. L. (6) (I)

M. D.

R.

and O. W. GREENBERG: PAys. B61i., 136, B 248 (1964). PAys. Rev., 139, B 500 (1965). N'lMYVo Oimmto, 18 B, 110 (1973).

MESSIAH

GIRARDEAU:

MIRMAN:

134 ON THE THEORY OF IDENTICAL PARTICLES

3

sional. In one and two dimensions it is shown that also intermediate cases between boson and fermion systems are possible. In sect. 4 we discuss how to modify the formalism in order to take particle spin into account, and which further restriction is implied by the spin-statistics relation. Finally, we give in sect. 5 a brief outline of how our discussion on the two-particle case can be generalized t& a system of N particles.

2. - Tdentical particles ill classical luec!tanlcs.

As 3, pI'cpa,ration for the next section) on qU3:ntizati<Jn, we will discuss, in this section, how th(~ ei.nflsir.al dt'scriptioll. of ~t ma!ly-pal'Licle system is aff~cted l"~' the indisting'ui>3h!:I,hiWy o·~ the p~tl'ticle::;. \\7e W~,ilt to a.l'gue. that the pror er

classical description of a system of identical particles must be the basis for its quantization. 2'1. Gibbs' paradox. - The «( principle of indistinguishability of identical particles», togtther with its effects, is usually considered to be one of the characteristics of quantum mechanics as opposed to classical mechanics. Howeve~, the principle is older than quantum mechanics. It was recognized by GIBBS and it is, in fact, the root of his entropy parauox in classical statistical meehallics ('). In order to correctly calculate the zero-entropy c~'ange in a process of mixing of two identical fluids or gases (at the same ternperature etc.), GmBs postulated that states differing only by permutations of identical particles should not be counted as distinct. LANDE has argued that this postulate is not ()nly sufficient, but also necessary to explain the zerc-entropy change (8). We would like to illustrate this point hy a simple example, and refer to the literature for more general treatments COlO). Consider a system of N particles of mass m confined to the volume V and with a total energy E. Its entropy 8 is

(1)

8(N, E, V) = kIn (1,0/1,00) ,

where k is Boltzmann's constant, to is the available volume in phase space and 1,00 is an (arbitrary) scale factor. By now we ignore the identical-particle effects. As far as the interactions between the particles do not limit the phase ~pace volume, this is a product to = 'Uv of the co-ordinate space volume v = VN and the momentum space (7) J. W. GIBBS: Elementary P1'inciple8 in Statistical Mec1umic8, Chap. XV (New York, N. Y., 1960).

A. LANDE: New Foundatio'ns 01 Quantum Mechaln,ics (Cambridge, 1965), p .. 68. D. HESTENES: Ame1'. JO'U1'n. Phys., 38, 840 (1970). (10) T. H. BOYER: Amer. JO'Um. Phys.,-SS, 849 (1970). (8)

(8)

135 J. M. LEINAAS

and

J. MYBHEIIrl

volume u, which is the «area &of the surface of a sphere of radius v'2mE in a 3N-dimensional space, (2)

11

= A 8N (2mE)(8N-I>/B.

The constant Aft is determined by the recuraion relations

A.= 2n,

(3) 80

that, when

fI,

is large, we have approximately ,,/1

(4)

In Aft ~ (11,/2) In n- ~)n i ~ (11,/2)( -In (11,/2)

+ In n + 1) .

I-I

Thus we find 8(N, E, V) = kIn (w/wo) ~ Nk In (OV(E/N)')

(5)

with a as a constant independent of N, E and V. However, this formula is wrong because it implies an entropy increase of 8(2N, 2E, 2V) - 28(N, E, V)

(6)

~

2Nk In 2

when two such systems are allowed to mix. The way of correcting it is to note that, if all particles are identical, each microstate of the N -particle system is counted N! times in the abo":,,e calculation of the phase-space volume. The correct volume is, theref('re, w' = w/(N !), giving the entropy

(7)

8'(N, E, V)

= k In (w'lw~) FI::S Nk In (O'(VIN)(EIN)')

and, thereby, (8)

8'(2N, 2E, 2V)- 28'(N, E, V)

FI::S

O.

The above discussion concerns the macroscopic description of a system of identical particles, but it should be kept in mind when turning to the microscopic description. 2'2. The configuration space of a 8ystem of identical particles. - In what

follows we denote by X the co-ordinate space of a one-particle system, and we assume N identical particles to be moving in this space. The possible configurations of the N-particle system are usually described as the points in XN,

136

5

ON THE THEORY OF IDENTICAL PARTICLES

which is the Cartesian product of the one-particle spaces. However, since the particlas are identical, no physical distinction can be made between points in XN that differ only in the ordering of the particle co-ordinates. Thus, the two points in XN X,

(9)

{

=

(Xt"~ XN)

x - p(x) -

(Xp - 1 (11'

(x,eX fllr i = 1, .. " N), ""

Xp-l(N)) ,

\\"hore p i'l :t. permutation of the pn.rticle indices, both describe the same physieal configurat.i().n of the system. Therefore the true configur:ttion spa;ce of the .N -parti,:Je system is not tho Ca,rtel:iia.n prGduct XN, but the space obtaUltd by identifying points in _¥ll representing the sa.me physical configuration. The physical .3ignificance of such an identification of points is illustru.tcu by Gibbs' pal'adox, as discussed above. We denote the identifcation space by XN/BN , since it is obtained from XN by «dividing out)) the action of the symmetric gr~llP BN. Since BN is a discrete, indeed finite, transformation group in XN, the space XN/8N is locally isomorphic to XN, except at its singular points. The difference between the spaces lie~ in their global properties, more precisely in the singularity structure of XN/8ii • Whereas XN has only regular points when X is regular, those points which correspond to a coincidence of the positions of two or more particles are singular points in XN/8N, The fact that the configuration space of the N ..:particle system is XN /8N , and not XN, nas usually been overlooked, either unconsciously or deliberately. As far as the microscopic description of a classical many-particle system is conoerned, the reason is easy to understand. The dynamics of a classical system involves only the local properties of the configuration space, so that the choice between XN/8N and XN is largely a matter of convenience. To be more precise, the time evolution of the classical N -particle system is a continuous curve in X N/8N with time t a~ a parameter. The one curve in X N/8N corresponds to N! curves in XN. But if X is many-dimensional, the N! curves in XN will not normally cross, thus one is free to pick one single continuous curve in XN at random and call it the time evolution of the system. If two or more curves in XN cross at a given time, that is if two or more point particles collide, then it might still be possible to define a unique curve through the collision point by requiring continuous derivatives. Thus, one is normally able to lift the identification of points in XN by following the continuous time evolution. What makes this lifting possible is that 8 N is a discrete transformation group, while the time evolution is continuous. The quantum case is quite different, as the global properties of the configuration space then become essential. We discuss this point in the next section, assuming X to be Euclidean, and show there that the usual symmetry constraints O!l wave functions defined on XN can be derived as consequences of the topological difference between XNI8N and XN.

137 J. H. LEINAAS

6

and

J. HYRREIH

Now we want to study in more detail the configuration space XN/BN for various one-particle spaces X. Assuming X to be the n-dimensional Euclidean space 8 n , we can introduce the centre-of-mass (c.m.) co-ordinate N

(10)

X=N-I Lx,e8n , 1-1

where %1, 00., XN e 8" are the co-ordinates of the N particles. The c.m. coordinate is invariant under BN and, obviously, the N-particle space is a Cartesian product, (11)

of the c.m. space 4 n and some «relative ~ £space r(n, N) representing the ",N - n degrees of freedom of the relative motion of the particles. The space r(n, N) is obtained from the Euclidean space t!nN-n by identifying points connected through an element from BN • Let us limit our discussion to the cal:le of a two-particle system, N = 2. The relative space r(n, 2) is the result of an idc;ntification of the points % = Xt - s. and - s = %.- Xl in 8 n' The identification space has one singular point X = 0, corresponding to a c'.·.l.acidence of position of the two particles. When its singular point is included, r(n,2) is simply connected. If we instead exclude the sin gular point 0, we are left with a Oartesian product, (12)

r(n, 2.)- {O} = (0, oo)X~n-u

of the positive real line (0, 00), giving the length Ixi of a vector X in 8 n, and the (n - I)-dimensional (real) proje~tive space gJn-l for the direction ± x/lxl of s. flJo is a point, fIJI is a circle which is infinitely connected, whereas flJn-1 for n:;> 3 is doubly connected. Since r(3,2) without its singular point is doubly connected, one might be led to suspect a relation between the topology of the conflgnration space and the familiar symmetry or antisymmetry conditions on wave functions. We take up this point in the next section. Note here the essential difference among the one-, two- and three- or higher-dimensional Euclidean spaces. We will look more closely at these three cases. The very simplest case is when the two particles move on a line, that is N = 2 and X = 8 1 : In some respect this is a special case as compared to a higher-dimensional space, since one-dimensional particles cannot interchange positions without passing through each other. The configuration space is obtained by identifying pairwise the points (iDu "'.) and ("'.'~) in ~ = I •. Thus, H .1s the half-plane with, say, "'I>"'., as illustrated in fig. 1. The identification is singular along the line ~ = "'•.

138 ON THE THEORY OF IDENTICAL PARTICLES

. 7

Consider the free-particle motion. A coincidence of position of the two particles is described as a reflection from the «wall & of the half-plane. Apparently, the momentum is not a constant of motion. However, this is a matt.er

Fig. 1. - IUS. is the oonfiguration space of two identical particles on a line, illustrated here by the nonshaded half-plane. We also ill!lstrate how a tangent veot.or v = (VI' 'Va) at some point % = (~l' ~2) is parallelly transported along a curve reflecting from the boundary Zl = Z ••

of definition, since we can choose to define parallel transport of a tangent vector " along a curve in the half-plane by the rule that the vector component norma,l to the edge Xl = $. is inverted every time the curve reflects from the edge. Thus, the parallel transport of a tangent vector around a closed curve may give a ve(\tor different from the original one, if the curve is reflected from the edge. This kind of definition may seem artificial here, but becomes much less artificial when we go to the higher-dimensional one-particle spaces. The configuration space of two particles moving in I. is (13)

tt:/8. = I, x,,(2, 2).

The relative space ,,(2, 2) is the plane I. with points ~ and - ~ identified. This space is seen, from fig. 2, to be a circular cone of half-angle 30°. A cone is globally curved, although it is locally flat everywhere ex.cept at the singular vertex. This somewhat peculiar property manifest itself in the parallel transport of tangent vectors. To see how a tangent vector" is parallelly displaced along curves on the cone, we may map back into the plane. The mapping is isometric (by definitIon), and parallel transport on the cone becomes the familiary parallel transport in the plane. See fig. 3. Note that a tangent vector" at the point

139 J. M. LEINAA.8

8

and

J. MYRBEIM

_ "E 8" is identified with the vector - v at x. Therefore the parallel transport around a closed curve on the cone r(2, 2) changes v into (-1)'" v, 'In being the number of revolutions of the curve around the vertex.

y

x

c

Fig. 2. - The relative space r(2, 2) of two two-dimensional identioal partioles is the with pairs of opposite points #& and - % identified. The identifioation may be effected by outting the plane along a line I from the origin 0 and then foldi.Ilg it into a oiroular oone of half-angle 30°. A oirole 0 in the plane centred at 0 then revolves twice around the cone.

pla~e

The abovp- argument for n = 2, illustrated in fig. 3, is seen to be more general. Thus, for X = eft, n = 1, 2, 3, ... , there are two classes of equivalent closed curves with respect to the parallel transport of a tangent vector v in the relative space r(n,2). One class does not change v, while the other class changes v into - v. A closed curve of the first class connects a point (Xl' X.) E tI! continuously with itseH, while a closed curve of the second class connects (Xl' x,,) continuoysly with (XI, Xl)' in tI!. y

v _-+-_ _ _...0_ _ _ _ _

~

o

Fig. 3. - The parallel transport of a tangent veotor v around two different olosed ourves 0 1 and O. on the cone of fig. 2. When the oone is mapped isometrioally into the plane, 0 1 becomes a olosed ourve in the plane and, therefore, leaves v invariant. 0" i~ a ourve from the point % in the plane to - %. The ve~or v at -x is, however, the same vector as -vat s, through the identifioation. Hence the parallel transport around O. changes v into -v.

140

9

ON THE THEORY OF IDENTICAL PARTICLES

The physically most interesting case is of course X = tla • To illustrate r(3, 2) we note that ~'I.' the projective plane, is the surface of a 3-dimensional sphere with diametrically opposite points identified. Equivalently, fJ''1. is the northern hemisphere with opposite points on the equator identified, fig. 4.

/1'" , V2 \

"""

' .)--i\ ~-:;--"\ /; I('\ ----- -----i-,r-----c~---~' -- --_.

I

........

{

/ ,c

....

J

".

I

I

"\,'

CJ

, \

Fig. 4. - 9'1' the projeotive plane, piotured as the northern hemisphere with opposite equator points identified. It is doubly oonneoted. 0 1 ,Ot !l.nd Oa are olosed ourves. 0 1 oan be oontinuo'USly deformefi into a point, O. oannot be so deformed, but 0: oan. 0: is the olosed ourve O2 passed twice. Os may be one intermediate stage in the oontinuous deformation of into a point.

0:

Ia 1"(3,2) = (0, 00) X ~I' a tangent vector v is changed into - v by the paralle transport around a closed curve which is a closed curve in ~. connecting opposite points on the sphere. Such a curve encircles the singular point ~ = 0 once. Notice that a closed curve encircling the singularity twice can always be continuously contracted to a point without having to pass through the singular point. This is in contrast to the two-dimensional case, whel"e curves revolving a different number of times around the vertex of the cone cannot be deformed into each other without passing through that point. In other 'Words, the relative space minus its singular point is doubly connected in the three-dimensional case, but is infinitely connected in the two-dimensional case. The Euclidean case is rather exceptional in that the centre-of-mass coordinate splits oft in a trivial way. Let us consider two particles moving on a circle 0 as a final example of the difference between the spac~s XN and X N /8N • If the particles are not identical,. the two-particle space is the torus as. If the particles are identical, the two-partie:;le space is instead a Mobiu8 band! To see this, we introduce the centre-of-mass angle ¢> and the relative angle qJ, fig. 5. The rectangle in the (¢>, qJ)-plane defined by OO. A real wave function describes the relative motion and it is a standing wave bec~use of reflection against the boundary z = o. The parameter t'J has a physical significance thruugh a time shift of the reflected wave. A «stationary phase)) argmilent (11) gives, for the time shift of a wave paP,ket peaked at the relative momentum k, (18)

T

= lik('ll + kl) •

The time shift vanishes if and only if 'TJ = 0 or '1- 1 = O. In addition to the eigenfunctions (17) there exists another one if

'TJ

< 0,

(19)

describing a bound state of the two-particle system. There is only one bound state like this for a given '1}. As the eigenfun6tion shows, the effect of the boundary condition here corresponds to a zero-range attractive force betwoon the particles. 3·2. The two- and thr66-dimtm8ionaZ CaB6S. - When we go beyond the onedimensional case, the configuration space is no longer flat, even if all the singular points are simply excluded. The presence of singularities "is revealed through a global curvature, which has been studied in sect. 2 in terms of the parallel transport of tangent vectors around closed curves. In a similar way, in order to discuss the significance of the singularities for the quantum. description, we introduce the concept of parallel displacement of state vectors. (11)

c.

EOXART: Rev. Mod. Phg•• , 20, 399 (1948).

144 13

ON THE THEORY OJ!' IDENTICAL PARTICLES

First, we introduce for each point x in the configuration space a corresponding one-dimensional complex Hilbert space ks ' (In modern mathematical tetminology k* is a «( fiber »,) And we assume the state of the systE»m to be described by a continuum of v€ctors 'I'(x) E ks (a «( cross-section of the fib€r bundle »)). That is, 'I' is assumed to be a single-valued function over the configm·at.ion space, ":Vhose function value 'I'{x) at the point x is a vec.tor in h*. Tf a llorn1ed basis vector X:c is introduced in ea,('h space hx' then the complex·valued wave function 1p(x) is just the co-ordinate of the vector \I'{x) relative to that basis: \f(x) =- 'VJ(x) l.c· Taus, the funct.ion 'y;(x) \\ill dep(·ud on the. set of basi& ye('.t0r~, or gauge {X.c} ,

and a change in this Ret causes a gauge transformation of the second kind: (21)

lp(x) -+ lp'(x)

= exp [iqJ(x)] lp(x) •

The concept of parallel displacement of vectors from h.c is needed in order tv defule a gauge-invariant differentiation of the functions lp(x). Let us denote by P(x', x): k* -+ h.c' the linear operator which transports parallelly the vectors of h* into k*, along some cont.inuous curve joining x to x'. Parallel displacement in general may depend upon the curye from x to x', but we a,ssume tha:t the infinitesima.I paranel displacement P(x dx, x) from x to x dx is uniquely defined. P(x', x) is assumed to be always a unitary operator. Finally, we as-

+

+

sume that it is possible, at least locally, to choose the gauge {X.c} in such a way that the rule of infinitesimal parallel displacement is of the form (22)

P(x

+ dx, x) Xs = (1 + i~bl:(x)) X.c-Hs·

In this gauge, the gauge-invariant differentiation operator becomes

(23) The functions ble are determined partly by the dynamics of the system and partly by the choice of gauge {x.}. They must be real so as to make P(x + dx, x) unitary. The gauge-independent quantity (24)

measuring the nOI:commutativity of the components of the gauge-invariant differentiation, corresponds to the curvature tensor in the case of parallel transport of tangent vectors. The form (23) of the differentiation operator is

145

J. M. LEINA..A..S

14

and

J. MYRHEIM

even more well known from the similar formulation, introduced by WEYL (11), of the quantum theory of a charged particle in a magnetic field. What corresponds there to the antisymmetric tensor fTe,(X) is the force field, whose vector potential corresponds to bA:(x), In the present case, we do not wanli the « vector potential)) bA:(x) to deecribe a force field. Therefore we assume that fTeI(x) = 0 for all x, except for the singula,r points of the configuration space where f Tel is undefined. As a consequence, a vector 'I' E h. will be unchanged by parallel transport around any closed curve which does not encircle the singularity. However, if the vector 'I' is para.I1elly transported m times a.round the singularity, it will be transformed into P;'¥, where P z i~ a linea.r) unitary operator a.r;ting in h•. Sj.uce· tl1iR Hilbert space is one-dimensional, p. is ju.st a phase factor (25)

where E is real. Because (26)

p.' = P(x', x) p.P(x', X)-l = exp [iEJ ,

the parameter E must be independent of ~he point x, thus its value is characteristic of the given two-p~icie system. The field bA\(x) has a dynamical effect through the gauge-invariant differentiation operator D Te • However, in the case we are consi~_ering, bTe can be transformed into zero by choosing the basis vectors X. in a particular way. Let t:be basis vector X. at some arbitrary point x be given, and define the basis vectors at all other points by parallel displacements of this X.' Whe:u. f Tel vanishes, this procedure defines a gauge where bTe vanishes. On the ott.er hand, when exp [iEl =1= 1, the I)omplex wave function 'I'(.e) will be multiva.lued. in this gauge, since ail the basis vectors X., exp [± ~J X., exp [± i2E] X., etc. at x will be generated by parallel transport of X. around different closed curves. In this way the dynamical effect of the singularities of the two-particle space may be transferred from the difterentiation operator, and therefore from the Hamiltonian, to the multivalue character of the wave function lp(x). We will elaborate the two-dimensional case in some more detail to illustrate the two different approaches. We neglect the c.m. co-ordinates. In the relative space we use polar co-ordinates rand l(J, with l(J e [0, 2n). The free-particle Hamiltonian is given by (27)

4 al ) -+--+-ar ar ,-1 al(J1 ,

iii ( alIa

H- - -m -

l

(11) H. WEYL: Zetts. Phy•. , 56, 330 (1929).

,

146 15

ON THE THEORY OF IDENTICAL PARTICLES

when the wave functions satisfy tho oondition (28)

1p(r, rp

+ 2n) =

exp [iE]1p(r, rp) •

The parameter Etells the many-body nature of the system. For a boson system we have ~ = 0, and for a fermion system E= 1l:. There is, however, no obvious rea,son to restrict the allowed values of ~ to these two. And, therefore, as for 1! 3, the fundamental group is just the symmetric group 8N • For N:> 2, 8 N has exactly two one-dimensional replesentations, the trivial or completely symmetric representation p -+ 1 and the completely antisymmetric representation p -+ signum (p) = ± 1. Thus, when n>3 and N> 2, there are exactly two possi~le kinds of -vave functions. These correspond to boson or fermion systems, respectively, as is seen when we proceed to define wave functions on the Euclidean space I!, generalizing the method of subsect. 3-2. The wave functions on will be single-valued and either completely symmetric or completely antisymmetric. Let us make a final remark concerning our assumption that ttl = 0 identically, which excludes the possibility of a nonzero magnetic field_ First note that, because of the relation between 1:/86 and I:, any tensor field on 1:/86 can be looked upon as a uniquely defined tensor field on whereas, the other way round, a tensor field on tf! in general defines N! tensor fields on 1:/86 • Now let us assume, more generally, that the tensor field til on 1!/8. may be nonzero, but is still nonsingular as a field on this time including all those points that are singular points of 1!/811 • Then ttl can be derived from a nonsingular vector potential b~ on I:. Moreover, b~ can be chosen as to become single-valued on 1:/86 • If necessary, simply replace b~ by the average

I:

I: ,

I:,

154 ON THE THEORY OF IDENTICAL PARTICLE8

23

of the N! vector potentials it defines on 1:/88 • Therefore, given a vector potential hre on 8!/8. with a nonvanisbing, nonsingular fre" we make the same analysis as before, but, instead of hre, we consider h: = hi - h~, whose corresponding field = O. In this case, h~ must appear in the Hamiltonian in the usual way, whereas the wave functions still are either symmetric or antisymmetric. This short remark should indicate that nlagnetic fields can easily be incorpora.ted in to the theory, with no modifications to our conclu~ions as to the symmetry properties of the· ffiftny-body wave functions. In this section we have not discussed intrinsic spin or other internal degrees of freedom. They bring r~o ess~ntial (~ompEcatiol1, because a one-dime.nsional l"t::pl'esentation 01 the fWldamental group is stili the only arbitrary element of t.he theory. The d~scussion of this point wOlud be a rathel' obvious generalization of sect. 4. ~"nd we do not go into more details.

t:,

2. The Last Bastion of Rationality J. Goldstone and F. Wilczek, "Fractional Quantum Numbers on Solitons", Phys. Rp.v. ult. 47 (1981) 986-989 ............................................................................... 159 F. Wilczek, "Magnetic Flux, Angular Momentum, and

StatistiC8~,

Phys. Rev. utt. 48 (1982) 1144-1146 ........................................................................... 163 F. Wilczek, "Quantum Mechanics of Fractional-Spin Particles", Phys. Ret'. utt. 49 (1982) 957-959 ............................................................................... 166

157

2. The Last Bastion of Rationality My own discovery of fractional statistics was born of two parents. One was an interest in fractional internal quantum numbers, arising out of my earlier work [1] with Goldstone. (This work in turn was directly inspired by work of Su, Schrieffer, and Heeger [2] on soliton~ in poly-acetylene, and influenced by earlier work of J ackiw and Rebbi [3] on zero modes in quantum field theory.) In the short paper reprinted here, we provided a general method for calculating quantum numbers induced by vacuunl polarization. Upon applying this method, it quickly b~carr'1c clear tila.t fractional and even transcendental values of interna.l 'quantum' num[,~n were tht': generit: case, not at all difficult to realize in simple model qualitunl field theories. Inspired by this, I was motivated to investigate whether a quantized space-tlme symnletry, namely angula.r monlelltun1, could be sin-li1arly fractionalized. The second was a desire to understand some ancient lore attached to magnetic monopoles in an elementary way. Apparently it was Tamm [4] who first realized that in the presence of a monopole carrying the minimal Dirac charge an electron would acquire half-integral orbital angular momentum. Many years later, Goldhaber [5] demonstrated that the statistics of monopole-electron composites had the peculiarity necessary to maintain the spin-statistics connection in the face of this development: If the monopole is a boson, and the electron a fermion, the composite particle is a b06on~ 'tHooft and Hassenfratz [6], and Jackiw and Rebbi [7], showed how the Thmm result fit into the modern context of non-abelian gauge theory monopoles_ A central element of their work was the locking of gauge and rotation symmetries, emphasized above in "Concepts". These two lines of thcught converged at the investigation of gauge-theory strings. Since such strings are the natural point-like defects for gauge theories in a two-dimensional world, they have~many properties related to monopoles, but squNlhed one dimension and thereby in some respects !timpler. With the earlier work on monopoles in mind, it was easy to analyze these and to discover the possibilities for fractional angular momentum. By the way it i6 very easy to understand the peculiar properties of magnetic monopoles in a very simple way from this point of view, as is indicated in [8]. One day I while I was visiting Cal Tech to give a seminar (on another SUbject) I mentioned some of this stuff to John Preskill, and he immediately asked me "If you have fractional angular momentum, what about the statistics?" Amazingly enough though part of the reason I started to look at the problem in the first place was to elucidate the quantum statistics of magnetic monopoles, it had never occurred to me to worry about it in the two-dimensional case. Anyway this question threw me at first. On the drive home, however, I saw how the notion of fractional statistics could make perfer:tly good sense, and soon proceeded to work some simple problems to check that it was so [9]. I remember being very pleased, in a perverse way, to realize that quantum

158

statistics could become fractional. Although it had been somewhat startling at first to realize that ordinary internal quantum numbers could be broken into bits, one quickly became accustomed to it. After all the quantization of charge is quite a deep and somewhat mysterious fact itself. It seemed a wonderful widening of the circle of ideas around fractional quantum numbers to encompass the last, superficially impregnable bastion of rationality.

REFERENCES 1. .1 Coldstone and F. Wikzek Ph!I,. R~1J. Lett. 47 (1981) 985. •

2.

Vv~.

P. Su, J. R. Schrieffer, and A. J. Heeger Phy,. Rev. Lett. 42 (1919) 1698; PhY3. Rev. D22 (1980) 2099.

3. R. Jackiw and C. Rebbi Phy!. Rev D13 (1976) 3398. 4. I. Talnm Z. Phy!. 71 (1931) 141. 5. A. Goldhaber Phy,. Rev. Lett. 36 (1982) 1122. 6. P. Hassenfratz and G. 't. Hooft Phy,. Rev. Lett. 36 (1976) 1119. 7. R. Jackiw and C.

R~hbi

Phy,. Rev. Lett. 36 (1976) 1116.

8. F. Wilczek Phy,. Rev. Lett. 48 {1982} 1144. • 9. F. Wilcrek Phy,. Rev. Lett. 49 (1982) 957.

*

159

FrlcUooft' Quantum Numbers en Solitons JeIUey Goldlitonc,a I Sial/ford Li"."r AcceleYa'nr

C,,,t~r.

Star.tonllhr;I·,'Tsil.v. Slarr{nm. Calif".,."," '''JOS

Frank Wilczek Irtstitul,/or n,oreUcal Pflysics. U"iversil.\· 0/ Calif",."ia. Sartla Barbara. Call/o.,."f4 93106

(Received 9 July 1981) A method is proposed to calculate quantum :lumbers on solitons In quantum field theory. The method Is checked OD previously known examples and. in a special model. by other methods. It Is found. for example. that the fermion number on kinks In one dimension or on magnetic mODOpoles In three dimensions Is. in general. a transcendental funcUon of the coupllDg constant of the theories.

PAa; numbers: 1l.lO.Lm. 1l.lO.Np Peculiar quantum numbers have been found to be associated with solitons in several contexts: (1) The soliton pr~vides, of course, a different background than the usual vacuum around which to quantize other fields. The difference between these Hyacuum polarlzations" may induce unusual quantum numbers localized on the sOliton. 1 - ' (11) SoUtons may require unusual boundary condit~ons on the fields interacting with them, in particular leading to conversion of internal quantum numbers into rotational quantum numbers ... -e 986

@

(iii) In the case of dyons, there ~"" classically a family nf solitons with arbitrary electric charge. The determination of which of these are in the physicalspt:ctrum requires quantum-mechanical considerations and brings in the II parameter of non-Abelian "auge theories. T.' At present all these phenomena seem disUnct although there are suggestive relationships. In this note, we shall concentrate on (1), proposing a general method of analysis &lid working out a few examples.

1981 The AmeriCl''l Physical Society

160 PHYSICAL REVIEW LETTERS

VOLUME 47, NUMBER 14

An intuitively appealing, and perhaps physically example of the phenomena we are addressing are the fractionally charged solitons on polyacetylene. 2• 3 • 8 A caricature model of a polyacetylene molecule is shown in Fig. l(a)--in the ground state we have alternating single and double bonds, which may be arranged in two inequivalent but degenerate forms A and B. U there is an imperfection, ad shown in Fig. t(b), we [;0 (rom A on the lcft-hand side to B on the right-hand Jide. This configurati~m cannot be brl)ugh~ to either pure A or pure B by any finite r~l.rr~mgement of electrons, ard so it will relax to a stable configuration--a. soliton. If we put t;';"J impE:rfe~~'on9 together, a~ in Fig. 1((;), \I.'e hnd a cunCigu ..-ativn which begins ailCl ends as A. Compared to the corresponding segment of pure ..4,. it iR missin~ I)re bt'nd. IT we ~dd an e1.ectron to the two-imperlection strand, we C~.l deform U,~s configuratiun by a finite rearrangement into a pure A strand. (We are pretending, for simplicity, that each bond represents a single electron instead of a pair.) Interpreting this, we see tha! a two-soliton state is equivalent to the ground state if we add an electron. Thus, by symmetry, each separated soUton must carry electron number -~ (a..td electric charge + ~e). We can relate these stick-figure pi.::tures of polyacetylcne to field theory as follows: Let d l > d z be the internuclear distances characterizing single and double bonds, respectively. Define a scalar Held which is a function of the link i by (,O,=(-1)i(d--ldl-~d2)' where dis the internuclear distance for link i. Thus in the A configuration fP i =~(d I - d z) (independent of i), in the 8 configuratiol" ~f) I =-~ (d. - d2), and in the soliton configuration fP. interpolates betwee"\ these values. Now we can show that it makes sense Lo approximate fP. by a continuum field and the interactions nf the electrons with '/' (a charge-density wave) by £'1 =g'fYsfP~ furthermore the electrons reali~able,

••• /\\/\\ / \\/\\... • • • //\1/\//\//\ ..•

A

8 (0.1

S OcTOB.. 1981

can be treated for present purposes (near the Fermi energy) as relativistic particles. In this formulation, we make contact with the work of Jackiw and Rebbi" They found that the dpectrum of the Dirac equation in the presence of a soliton contains a zero-energy solution. By symmetry, this solution is composed of (projects onto) half a positive-energy and hal! a negatlveenergy solution with respect to the normal ground state. Thus if we fill the zero-energy level., we h;l\'e a soliton state with electron number + i; ir we lea vc it empt:i. the electron number is - t. Su and SchrieHer have described a generali~a­ tion, !O \I:hich occurs in a chain with a repeating unit of ~ingie-single -double ilonds, as in Fig. 2. A slight modification of U.e discussion of Fig. 1 ~hows that Wp now have solituns which can be d in 'lIlits .

(B) The same conclusion· may be reached more formally as follows. Although the magnetic Held vanishes outside the solenoid o[ course the potential. does not. Rotations around the z axis, l.e., changes In the azimuthal angle." are generated by the covariant angular momentum I =-,'8.. - qA... In a nonslngular gauee, A .. •/2. and the azimuthal dependence of the electron wave function is." a: e''''', .. -lnteger for continulty. 'roen: • .JI". (n - q. /2.,.", In agreement with the previous conclusion. (C) Finally one may ellmlnate the potential outside the solenold altogether by a singular gauge transformation from (8), namely, A'. A- VA with A· • ., /271. 'Ibis Is slngular because., is a multlvalued function. The charged-particle wave function. now obeys a free SchrOdlnger equation but with an unusual boundary condition that .' (., + 2.) =e - •••• ' «(/') following from the gauge trans[ormation A. This boundary condition requires 1/1' (01' ) a:: exp{ i (integer - q+ /21r )q1]. Now there is no vector ootential, and the angular momentum is identified as usual, 80 that I. = integer /271. If we interchange flux-tube-charged-partlcle composItes we wlll have, in addition to the usual [actors, a phase factor appearing in all gaugelnvariant observables.' This is because in the motion depicted in Fig. 1, each composite must be covariantly transported In the gauge potential o[ the other. The resulting phase is simply (since A .... /2.). If :. =integer - q+ /271 1& an Integer, the phase factor is unlty and the staUstics is normal (e.g., If the composlte is a flux tube plus electron it

© 1982 The American Physical Society

:s.

q.

e"·

164 VOLUME

48,

NUMBER

26 APRIL 1982

P H Y 5 I CAL REV I E \~I LET T E R 5

17

B (a)

(\

\

t0 • 9

1

4>,q

4>,q

\~

(Ill

/ ..

--------

roTG. 1. Tntt'rdtangl? or two o(HII.l!S by a rOlatlon.

f1U)(-lul~-l'lc'ic\e

com-

)lh~:;: Fermi sLlti:;tics). If 1= =·integer -q~/2;i ts hal[ an odd il1teg~r then thp. norm:u statistics is reversed. In intermediate cases, the composites cannot be described as (ermions nor as bosons. In a scattering experiment the direct and exchange terms will interfere with a coefficient cos(q+). Notice that in this case the plssible ambiguity of the sign of the phase is unimportant (i.e., we can reverse the direction in Fig. 1). This is not true for assemblages of more than two of these particles, which are therefore complicated to describe. Some applications of th.e above discussions are now presented. (a) Superconducting vortex. -For a uni. vortex in a superconductor the magnetiC' flux is quantized as • =2.- /2e, where e is the electron charge (and 2e the charge of the condensate). Therefore the orbital angular momenblm of an electron around a unit vort...x is i + integer and the composite· is a boson. (b) Dirac c011dition.-We can repeat the Faraday law argument (A) above as W~ imagine turning Oil a magnetic monopole (Fig. 2). For a charged particle orbiting inCi!1itesinlally above the plle the flux is g /2 while Cor a particle infinitesimally below it is -g /2. The corresponding shilts in angular momentum are I. :r-gq/4w; the allowed I. values are integer:r-gq/4.-. For these two spectra of I. values to agree one must have gq/4w" =-1 + integer, which is the Dirac quantization condition. (c) Charge-spin rewtion.-As a consequence of this argument when gq /4. is half odd Integer' the angular momentum spectrum has been shifted by t unit. I find this explanation of the relationship between the Dirac condition and the spin shift of

"·IG. 2. (a) Sl~n of magne!lC' flU'C 1~pends on whether orbit 'S Infl!liVs!maUy above cr hel"'" :he pole. (h) SJ)'!C·tru.., nf :.ngular mOmC:lt1 mov .~s up or down as thE" nux is turu,-'