The Many-Anyon Problem, Like the Second Quanti

Home , Anyon

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HEP-TH-9502071 y a Hamiltonian ho ol of P v y for exotic statistics arises, and then hian and Ricardo F ; :::; N 2 t set of lectures giv h to the Quan hanics, when rst faced with a treatmen ; 1 tered an argumen (1 ts in theoretical ph H e additional inspiration. abasco, 3{7 Octob er 1994. v US MONTONEN = HU-TFT-95-12 tingly few applications to observ w to describ e a man tum mec H e encoun ons|an e ort that ev w the p ossibilit v CLA y elopmen vndal ga particles describ ed b artment of Physics, The N ted at the VI Mexican Sc t of quan ould ha Villahermosa, T Dep er all the attempts in this direction, nor will the list of references ry Group Approac FIN-00014 University of Helsinki, Finland v orth sp ending some time up on. y other dev hanics of an tum Hall e ect b eing the only candidate at the momen y Finn Ra t in the problem arose from an attempt to describ e comprehensiv THE MANY-ANYON PR y collab orators Masud Chaic tco y explaining ho ypical trait of p ostmo dern theory will not apply to an ast literature on the sub ject. I ha tening discussion. An excellen erage studen e so man olvmen Lectures presen v v tro duction ork is not discussed or cited. , the sub ject p ossesses considerable elegance and is in articles tical particles, w y trate on the problem of ho us p erhaps w ons has so far had disapp oin a y y enligh Consider a system of The a 2. The Symmet My in I will start b 1. In These lectures deal with a particular problem in the theory of an yw een. Lik w this: tical P of iden Helsinki in 1993 b man wish to thank m consulted when preparing thewhose lectures w and these notes. I o er m the statistical mec will the treatmen exhaust the v An and th concen hop e that this t the fractional quan of an ob eying statistics that is neithert Bose{ Einstein or F provided byCERNDocumentServer brought toyouby CORE

where the lab el i denotes op erators (co ordinates, momenta, spins,...) relating to the

ith particle. The statement that the particles are identical is taken to mean that

any conceivable Hamiltonian (2.1) describing a system of such particles is invariant

under a p ermutation of the op erators relating to di erent particles: Denote by the

p ermutation

1 2 ::: N

(1) (2) ::: (N)

b elonging to the group S of p ermutations of N ob jects, then the particles are iden-

tical if

U ( )H (1; 2; :::; N )U ( ) H ( (1);(2); :::; (N )) = H (1; 2; :::; N ): (2:2)

If this is the case, then according to the general principles of quantum mechanics,

the eigenstates of H should transform according to some representation of S :

U ( )j i = j iD ( ): (2:3)

j k kj

Denoting by j1; 2; :::; N i the eigenstates of a complete set of commuting one-particle

op erators, on which the p ermutation op erators act as follows:

U ( )j1; 2; :::; N i = j (1);(2); :::; (N )i; (2:4)

the wavefunctions

(1; 2; :::; N )=h1;2; :::; N j i (2:5)

transform in the following way:

1 1 1

h1; 2; :::; N jU ( )j i = ( (1); (2); :::; (N )) = (1; 2; :::; N )D ( );

j j k kj

or, by a relab elling of the arguments of the wave functions:

(1; 2; :::; N )= ((1);(2); :::; (N ))D ( ): (2:6)

j k kj

A textb o ok in group theory would tell us that there are exactly two one-dimensional

representations of any S , N 2:

- The trivial representation D ( ) = 1. Particles transforming according to this

representation are called b osons and their wavefunctions are completely symmetric.

j j

- The alternating representation D ( )= (1) , where j j denotes the number

of exchanges needed to build the p ermutation (although not unique, this number is

always either even or o dd for a given ). Particles transforming in this way are called

fermions, and their wavefunctions are completely antisymmetric.

Irreducible representations of higher dimensions do o ccur for N 3, and the term

parastatistics has b een intro duced to describ e this situation. Parastatistics will not

b e treated further in this course, b e it either b ecause it do es not seem to o ccur in 2

nature or b ecause general theorems say that parastatistics can always b e replaced

by hidden ("colour") degrees of freedom.

The alternativebetween Bose or Fermi statistics, which the ab ove argument led

us to, can b e expressed in a simple way, which, however, is an extremely p owerful

to ol for computations. I am referring to the formalism of second quantization. Let

(x) b e the op erator creating a particle at x, and (x) the op erator annihilating a

particle at x (for simplicitywe assume that the particle numb er is conserved as e.g.

in nonrelativistic many- b o dy theory), and j0i the vacuum (no particle) state. Then

the particle statistics is all contained in the algebraic relations ([A; B ] AB BA

0 y y 0

[ (x); (x )] =[ (x); (x )] =0; (2:7)

y 0 0

[ (x); (x )] = (x x ); (2:8)

(x)j0i =0; (2:9)

where the - sign applies to b osons, the + sign to fermions. The state with particles

lo calized at x ; :::; x is represented by the state vector

1 N

y y

jx ; :::; x i = (x ) (x )j0i (2:10)

1 N 1 N

and the relations (2.7) automatically ensure the correct symmetry prop erties.

A nice and imp ortant fact is that the Bose{Fermi{alternative lo oks the same

in any representation: Indeed, in Eqs. (2.4) and (2.5) we did not sp ecify which

representation wewere using. In the language of second quantization, we see this as

follows:

Let fu (x)g b e a complete set of orthonormal functions (eigenfunctions of a one{

particle op erator):

dxu (x)u (x)= ;

m nm

0 0

u (x)u (x )= (xx):

Expanding (x); (x):

(x)= a u (x)

n n

y y

(x)= a u (x);

n n

one easily derives the algebra of the op erators a ;a , which is formally identical to

(2.7){(2.8):

y y

[a ;a ] =[a ;a ] =0

n m

[a ;a ] = : (2:11)

n nm

The fact that the second quantization is equally simple in any representation is

of crucial imp ortance e.g. in the relativistic case, where (asymptotic) states of sharp

momenta make sense, but states of sharp lo calization do not. 3

Can more exotic p ossibilities for the statistics of identical particles b e envisaged?

The answer, gleaned at in partial results for 1+1{dimensional eld theories ,was

de nitively shown to b e yes by Leinaas and Myrheim in 1977 , provided the dimension

of space is 1 or 2. Subsequently , the name anyons (in Spanish: qualquierones,

according to Eduardo Fradkin) was given to particles ob eying such exotic statistics.

It is evident that all parts of the previous reasoning cannot apply to anyons (since our

argument uniquely led to b osons or fermions), but it is of interest to investigate how

much of it can b e saved, not least b ecause of the computational ease of the second

quantized formalism.

As an example, consider generalizing the relations (2.7) to

0 0

(x) (x )= (x ) (x): (2:12)

0 1

Exchanging x and x we see that consistency requires = , i.e. = 1. Thus we

are back at the Bose-Fermi alternative. Hence cannot b e a constant, rather should

we take

0 0 0

(x) (x )= (x;x) (x) (x); (2:13)

0 1 0

with (x; x )= (x;x). But the form (2.13) is now p eculiar to x-space; in p-space,

e.g., the relation will lo ok completely di erent!

3. How Come Anyons?

In order to clearly see what new features are implied in the reasoning leading to

anyon statistics, I will here present a strict, ortho dox party line. Like all party lines,

it should b e constantly challenged, and ways to overthrow the ortho doxy should b e

sought. In this way new discoveries can b e made, and what survives of the ortho doxy

will stand on more secure ground. So here we go:

By identical particles we shall mean the following: Firstly, the con guration space

D N

for N identical particles in D { dimensional Euclidean space is not (R ) , instead

(x :::x ) should b e identi ed with (x :::x ) for any 2 S . The space ob-

1 N N

(1) (N)

D N

tained after such an identi cation, denoted by(R ) =S , has the awkward prop erty

of p ossessing p otentially singular p oints. The candidates for such p oints are the xed

D N D N

p oints of the action of S on (R ) , i.e. the diagonal f(x :::x ) 2 (R ) jx =

N 1 N i

x for at least one pairg. So, to stay clear of trouble we should remove the diagonal

(we could imagine that there is a hard core interaction b etween the particles keeping

them apart). The rst statement of our dogma is thus that the con guration space

of N identical particles in D -dimensional space is

D N

(R )

M =

: (3:1)

Secondly,we will allow as observables only symmetric op erators. This means e.g.

that we are not allowed to consider one{ particle op erators suchas H (i)= p =2m,

although in the symmetry group way of lo oking at things a quantity like hH (1)i

0 4

makes sense (it is p erfectly calculable), by symmtery, of course, it equals hH (2)i =

= hH (N)i.

How can we then intro duce the concept of particle statistics, since by adopting

the ab ove dogma wehave banished all talk ab out "interchanging particles" and the

like? The key p oint is that the con guration manifold (3.1) is (for D 2) not

simply connected: There are closed lo ops in M that cannot b e continuously shrunk

to a p oint. A simple example makes this clear. M can b e constructed as follows:

2 2

The co ordinates x ; x of (R ) are replaced by the center of mass co ordinate X =

1 2

(x + x ) and the relative co ordinate x = x x . Removing the diagonal x = x

1 2 1 2 1 2

means leaving out the origin of the x{plane. "Mo dding" by S means identifying x

and x. This we can do by restricting us to the left half x{plane x 0. On the

2 2 2

x {axis we still have to mo dify the p oints (0;x ) and (0; x ), i.e. glue the negative

2 2

x {axis to the p ositive x {axis. The resulting construction is the surface of a cone

with the tip (x = 0) excluded:

2 2

M = R fcone without the tipg:

Evidently,any closed lo op on the mantle of the cone encircling the tip cannot b e

shrunk to a p oint; thus M is multiply connected.

Quantum mechanics on multiply connected spaces shows interesting new features

not present in the familiar case when the con guration space is top ologically trivial,

and the p ossibility of exotic statistics lies hidden in these new traits. I will present

the argument using Feynman's path integral formulation of quantum mechanics. If

you prefer the Schrodinger wave function formulation, I recommend the b o ok by

Morandi .

In Feynman's formulation, the propagator for a con guration a 2 M at time t

to develop into a con guration b at t is given by the path integral

q(t )=b

D qe : (3:2) K (b; t ; a; t )=

b a

q (t )=a

Here S is the action, and the integral runs over all paths q (t)inM connecting a

to b.AsM is multiply connected, there are paths which cannot b e continuously

deformed into each other. All paths which can b e deformed into each other we group

into the same homotopy class. The set of all paths from a to b is thus divided into

homotopy classes, and we denote the set of all homotopy classes by (M ;a;b).

As was rst p ointed out bySchulman , in this case we can a priori weight the

contributions from di erent classes di erently:

K = ( )K ; (3:3)

where the sum runs over all classes, and K denotes the integral over all paths in the

class .

What consistency conditions do the weights ( )have to satisfy? The answer to

this question was given by Laidlaw and Morette{De Witt . The argument is quite 5

subtle, and I shall only summarize the results here, referring the reader to the original

pap er for details.

Firstly, the weights ( )have to b e pure phases: j ( )j = 1. The reason for this

is basically that when t t ! 0, only one class contributes, and this term has to

b a

repro duce K up to a phase.

Secondly,we make the following observation. Let us cho ose a xed p oint q 2 M ,

and xed paths C ;C joining a and b to q , resp ectively. Then in each class there

a b 0

are representatives consisting of:

- the path C

- a closed lo op in M starting and ending at q

- the path C .

Now the set of all closed lo ops at q falls into classes for whichwe can intro duce

a"multiplication": The pro duct of two lo ops ; is the lo op formed by rst going

around and then around . With this multiplication the set of classes of closed

lo ops at q forms a group (the unit element b eing the class to which the constant

lo op staying at q b elongs, and the inverse of the class of the lo op b eing the class

to which traversing in the opp osite direction b elongs), the fundamental or rst

homotopy group (M )(we can drop the reference to q , since all the groups at

1 0

di erent q are isomorphic).

In this waywe establish a one-to-one corresp ondence b etween (M ;a;b) and

(M ) (whichby the way is not unique, since it dep ends on the choice of C and

1 a

C !), and can b e taken as lab elling (M ).

b 1

If t is a time intermediate b etween t and t , the propagator has to ob ey

c a b

K (b; t ; a; t )= dcK (b; t ; c; t )K (c; t ; a; t ): (3:4)

b a b c c a

For the classes this means

dcK (b; t ; c; t )K (c; t ; a; t ): (3:5) K (b; t ; a; t )=

b c c a b a

; ; =

Together, (3.4) and (3.5) imply that

( ) ( )= ( = ); (3:6)

D 8

i.e. the weights form a unitary, one-diemnsional representation of (M ) .

These groups are known:

(M )=B ; (3:7)

1 N

the N-string braid group, whereas

(M )=S (3:8)

1 N

for D 3. Both B and S are generated by N 1 generators ::: , ob eying

N N 1 N1

the constraints

= ; ji j j 2; (3:9)

i j j i 6

= : (3:10)

i i+1 i i+1 i i+1

The di erence b etween B and S arises from the fact that for S we require in

N N N

addition to (3.9) and (3.10)

= e; (3:11)

where e is the unit element.

The connection to particle statistics comes through recognizing that the class of

closed lo ops corresp onds to an interchange of particles i and i + 1. In the plane

(D = 2) this can b e done in two homotopically inequivalentways, which can b e

represented by the lo op where these two particles move on the circumferenceofa

circle passing through their original lo cations either counterclo ckwise (corresp onding

to ) or clo ckwise (corresp onding to )interchanging their places, whereas all other

particles stay put. In three or more dimensions these lo ops can b e deformed into each

other e.g. by rotating the circle around a diameter, i.e. = in accordance with

(3.11), but not in two dimensions.

The elements of the group B (S ) are formed by taking all p ossible pro ducts of

N N

all p ossible p owers (p ositive and negative) of the generators , taking into account

the constraints (3.9), (3.10) ((3.9), (3.10), (3.11)). B is a group of in nitely many

elements, but the inclusion of the p owerful constraints (3.11) reduces the number of

elements of S to N !.

Our general result, Eq. (3.6), instructs us to lo ok for unitary, one-dimensional

representations of B or S .Posing

N N

( )=e ;

we see that Eq. (3.10) requires = , i.e. all phases are equal. It is customary

i i+1

to write

( )=e

1 i

( )=e ; (3:12)

where 2 [0; 2) is the statistical parameter.

In three or more dimensions, Eq. (3.11) requires ( ) = 1, i.e. =0;1 are

the only p ossibilities (wehave in fact derived the result on the one- dimensional

representations of S mentioned in section 2!). But for D = 2 there is no restriction

on , and anyon statistics is p ossible.

This, then, is one version of the accepted ortho doxy on unortho dox statistics.

Parts of it can b e challenged. For instance, one might ask what happ ens if we do not

D N

remove the diagonal but work with (R ) =S as the con guration space (which then

is no longer a manifold, but rather an "orbifold"). As is evident from our example M ,

this will change the fundamental group of the con guration space, and our previous

argument breaks down. In simple cases, at least, it seems that Hamiltonians on M

can b e extended to self- adjoint op erators on R =S ("colliding anyons") in many

ways . What this means is still uncertain. 7

Although we in the sequel will b e exclusively concerned with exotic statistics in

two-dimensional space, let us brie y stop to consider what happ ens in one dimension.

In this case the con guration space is simply connected:

(M )=0;

and our previous argument seems to imply that no statistics is p ossible. In a certain

sense this is true: To exchange two particles on a line, they havetobemoved past

each other, and what then happ ens dep ends on any contact interaction b etween the

particles. In other words, there is a p ossibilityofintro ducing statistics through the

b oundary conditions to b e imp osed at the edges of M . The following example, taken

from Leinaas and Myrheim , illustrates this p oint:

Taketwo particles on a line, with co ordinates x and x . M is e.g. the region

1 2

to the right of the diagonal x = x of the (x ;x )-plane, and wehave to decide

1 2 1 2

what b oundary conditions to imp ose on the diagonal. The normal derivative of the

wave function on the b oundary is the partial derivative with resp ect to the relative

co ordinate x = x x , and a natural condition on the wave function would b e that

1 2

the probability currentvanishes at the b oundary (no probability " ows out of" M ):

j / i(

)j =0:

x=0

@x @x

The solution to this condition is that

j = (x =0)

x=0

for any real . Cho osing = 0 allows to b e extended to an even function of x in the

whole plane, i.e. Bose statistics; if = 1; (x =0)=0,and can b e extended

into an antisymmetric function, i.e. we get Fermi statistics. For any other value of

we get statistics intermediate b etween Bose and Fermi; i.e. what corresp onds to

anyons in D =1.

4. The Transmutation of Statistics intoaTop ological Interaction

A direct attack on the anyon problem using the b oundary conditions on the wave

function of N anyons implied by the propagator (3.3) with weights (3.12) is easy for

N =2 , but already for N = 3 it b ecomes a very dicult problem, and signi cant

progress in solving the three-anyon problem was achieved only very recently . An-

other approach, whereby the exotic statistics is transformed into a p eculiar interaction

between ordinary b osons or fermions, has b ecome more p opular. In this section, we

shall derive this statistical interaction, following Wu , and in the next section we

shall show that this same statistical interaction can b e generated byintro ducing a

gauge eld with a very sp ecial kinetic term, the famous Chern{Simons term.

De ne

1 1 2 2 i

z =(x x )+i(x x )=jz je : (4:1)

ij ij

j i j i 8

As a lo op representing the class we can take the lo op in M where particles i and

i + 1 exchange their places by rotating counterclo ckwise through around the center

p oint of the line joining their original p ositions, whereas all other particles stay where

they are. Their relative p olar angle changes by :

= dt (t)= ; (4:2)

i;i+1 i;i+1

i.e. we can write

( )=e = exp(i dt ): (4:3)

i i;i+1

The changes in all other relative p olar angles add up to zero, b ecause =0,

j; k 6= i; i + 1, and + =0;j 6= i; i +1. For the inverse , takea

ij i+1;j

clo ckwise rotation = , and

i;i+1

1 +i

( )= e = exp(i dt ) (4:4)

i;i+1

again. Since any lo op can b e built up out of pro ducts of the generators ,we see

that

( )=exp(i dt (t)): (4:5)

Thus, supp osing the dynamics of the anyons is describ ed by a Lagrangian L, the

propagator can b e written

Z Z Z

R R R

t t t

X X

b b b

i dtL i dtL i dtL

ef f ef f

t t t

a a a

K = ( ) D qe = D qe = D qe ;

q (t)2 q (t)2 all paths

(4:6)

which is the path integral of b oson particles with a Lagrangian

L = L

(t): (4:7)

ef f

The last term, the statistical interaction, is a total derivative (and thus a "top o-

logical term") and will not a ect the equations of motion of the system. Its only role

is to provide the correct statistical weight factors in the propagator.

Let us lo ok more closely at the case of free (nonrelativistic) anyons. Then

L =

x : (4:8)

ef f

i=1

The time{derivative of the p olar angle can b e written

______

=(x r + x r ) =(x x )r =(x x)r :

ij i i j j ij i j i ij j i j ij 9

Thus the canonical momentum corresp onding to x is

ef f

_ _

p = = mx r mx + a ; (4:9)

i i i ij i i

@ x

j 6=i

where, by abusing three{dimensional notation,

e (x x )

3 i j

a = : (4:10)

jx x j

i j

j 6=i

(e is a unit vector p erp endicular to the plane where the particles move.)

The Hamiltonian of the system is

N N

X X

H = x p L = (p a ) ; (4:11)

i i ef f i i

i=1 i=1

which is of the same form as the minimally coupled Hamiltonian for N particles

moving in an ab elian gauge eld describ ed by the vector p otential A(x )= a . Indeed,

i i

a has b een given the name the statistical gauge eld.

From Eq. (4.9) we see that in gauge theory language the statistical gauge eld is

a pure gauge, and can thus b e removed by a gauge transformation:

= e (z ;:::;z ;z ;:::;z )= (z ) (z ;:::;z ;z ;:::;z );

Bose 1 N ij Bose 1 N

1 N 1 N

(4:12)

1 2

(z = x + ix ;z = z z )

k ij i j

k k

P P N

i i

ij ij

H = e He =

: (4:13)

i=1

The transformed wave functions are not single{valued as functions on M :

~ ~

(x ;:::;x ;:::;x ;:::;x )=e (x ;:::;x ;:::;x ;:::;x );

1 k l N 1 l k N

since = (dep ending on whichwaywe braid, i.e. interchange x and

kl lk k

x ). Rather, is a prop er function on the universal covering space of M , and

interchanging particles takes us from one fundamental domain to another . Of course,

the gauge transformation (4.12), (4.13) has taken us back to our starting p oint: Eq.

(4.12) is the form of the wave function implied by the propagator (3.3).

5. The Chern{Simons Action and Anyon Statistics

The (ab elian) Chern{Simons (CS) action is

S = d xL =

d x A @ A (5:1)

CS CS

2 10

012

( = = +1). The action (5.1) is invariant under the U (1) gauge transformation

012

A ! A + @ , since L changes only by a total derivative. Let us couple the CS

vector p otential to a current j describing N p oint particles:

j (x)=g v (x x (t)); (5:2)

2 n

n=1

where the 3-velo city v =(1;v), and g is the "CS{ charge". The coupling is

S = d xj (x)A (x) (5:3)

int

(g = diag (1; 1; 1)). Let the particles move nonrelativistically:

S = dt

v : (5:4)

matter

n=1

The total action of the mo del we shall study in this section is then

S = S + S + S : (5:5)

matter CS int

The equations of motion can b e straightforwardly derived. Intro ducing the CS-

eld strength tensor

F = @ A @ A

with comp onents E F (CS{electric eld) and B F (CS-magnetic eld; in

0i 21

2+1 dimensions the magnetic eld is a pseudoscalar), we can write the Lorentz force

equations for the particles:

i i ij j

mv_ = g (E (t; x )+ v B (t; x )) (5:6)

n n

( = = +1). Varying the action with resp ect to A we get the eld equations

j =

F ; (5:7)

i.e.

i ij

E = j ; (5:8)

1 1

B = j : (5:9)

Since the CS action is of rst order in the derivatives of the elds, the eld

equations simply express the elds as functions of the sources and allow the complete

elimination of the elds from the equations of motion. Inserting Eqs (5.8) and (5.9)

in (5.6) we get

ij j j

mv_ =

(v (t) v (t)) (x (t) x (t)): (5:10)

2 n m

m n

m=1 11

We see that the Lorentz force vanishes almost everywhere, so that our system de-

scrib es free particles. The Chern{Simons term is not without consequences, however.

This is most clearly seen in the Hamiltonian picture. By following the standard

canonical pro cedure we derive the Hamiltonian

2 2

H = (p g A(x ;t)) + d xA (x)(B (x)+ (x)): (5:11)

n n 0

n=1

We still have the freedom to cho ose a suitable gauge. The clever choice is:

A =0;@ A =0: (5:12)

0 i

The latter condition allows us to solve Eq. (5.9) (which in the Hamiltonian formula-

tion is to b e imp osed as a constraint, like Gauss' law in the Hamiltonian formulation

of Maxwell electro dynamics) uniquely for the CS vector p otential:

j j j 0j

x x x x g

ij m 2 0 ij 0

A (x)=

d x (x )= : (5:13)

0 2 0 2

2 jx x j 2 jx x j

m=1

Substituting Eq. (5.13) into Eq. (5.11), where the last term is zero in the gauge

(5.12), and dropping the troublesome divergent n = m terms (it has b een argued

that this can b e justi ed through suitable regularization), we arrive at exactly the

statistical interaction (4.11), if we identify the statistical parameter as

: (5:14)

Thus, the Chern{Simons eld minimally coupled to matter particles generates

anyon statistics! This imp ortant observation p oints to a way of constructing a full-

edged eld theory of anyons, a topic to whichwe shall now turn.

6. Nonrelativistic Chern{Simons ({Maxwell) Field Theory

Quantum eld theory remains the preferred vehicle for many-b o dy quantum the-

ory. In the relativistic case, when particle numb ers are not conserved, it is practically

the only available alternative, but even in the nonrelativistic case it is computation-

ally sup erior to its rivals. Thus it is natural to attempt to base a many-anyon theory

on a quantum eld theory of anyons.

However, we do not yet know what such a eld theory should lo ok like. In view

of the results of the preceding section, according to which the minimal interaction of

particles with a Chern{ Simons eld induces anyon statistics for the particles, it is

tempting to start from a theory of a matter eld in interaction with a Chern{Simons

eld.

Sticking to the nonrelativistic case, we might try

y 2

L = i D +

D + A @ A ; (6:1)

0 0

2m 2 12

where is a b oson eld, and D = @ + ig A .To the Lagrangian (6.1) we might

optionally add further terms, like a Maxwell term

L = F F ; (6:2)

or a contact interaction b etween the -particles:

y 2

L = :( ) : : (6:3)

The advantage in adding a Maxwell term lies in making the theory more regular

in that A then b ecomes a physical degree of freedom (the transverse part of A

describ es in that case a massive "photon" of mass m = e ). The contact term (6.3)

again brings in new interesting features. The classical theory based on L + L has

0 1

13;14 15

soliton solutions .Luscher has studied the theory L + L .We shall here see

0 M

12;13

what follows from L alone, mainly following . In order not to get b ogged down

in lots of detail, we will pro ceed briskly ahead with a heuristic account of the main

line of argument and refer to the original pap ers for a more careful treatment (see

16;17

also ).

We are interested in a second quantized theory of anyons. Under the assumption

that Eq. (6.1) do es describ e anyons, our strategy will b e to eliminate the Chern{

Simons eld through an appropriate gauge transformation and to investigate the

prop erties of the transformed eld op erators.

In the pure Chern{Simons case, the equations of motion again connect the eld

strength op erators to the particle eld op erators:

F =

j ; (6:4)

where now

0 y

j = (6:5)

k y k k y

j = ( D (D ) ): (6:6)

In particular, Eq. (6.4) says for =0:

B =F = : (6:7)

In a transverse gauge, rA = 0, this equation can b e solved for the vector p otential

@ g

i ij 2

A (x;t)= ( d yG(x y)(y)): (6:8)

Here G(x) is the two-dimensional Green function

r G(x)= (x);

2 13

i.e.

G(x)= log (jxj);

where is an arbitrary scale. Intro ducing again the p olar angle (x) through

1 2 i(x)

z (x) x + ix = jz je ; (6:9)

the Cauchy{Riemann equations for f (z ) = log z read

log jx y j = (x y):

j i

@x @x

This means that Eq. (6.8) can b e written

A(x;t)= d yr (x y)(y;t): (6:10)

If I am allowed to move the derivation op erator outside the integral (this is not trivial:

is not single{valued; see for a discussion ab out the validity of this step), (6.10)

is a pure gauge A = r, with

(x;t)= d y(x y)(y;t): (6:11)

Returning to Eq. (6.4), now for = i =1;2:

F = @ A @ A = j ; (6:12)

i0 i 0 0 i ik

we can, using the continuity equation

+ r j =0; (6:13)

solve also for A , with the result

g @j

2 ik

A (x;t)= d yG(x y) : (6:14)

Up on integrating by parts and using Eq. (6.13) again, we arriveat

@ g @

A (x;t)= d y(x y)(y;t)= (x;t): (6:15)

@t 2 @t

Thus wehave shown that A = @ , a pure gauge. A gauge transformation will

remove it and bring (6.1) to the form

0 y y 2

~ ~ ~ ~

L = i @ + r ; (6:16)

0 0

2m 14

with the transformed elds

ig (x;t)

(x;t)= e (x;t)

y y ig (x;t)

(x;t)= (x;t)e : (6:17)

The gauge parameter is an op erator, and this will a ect the algebra of the {

op erators. and were ordinary b oson elds satisfying the algebra (2.7), (2.8).

From Eq. (6.11) we deduce

[ (x;t);(y ;t)] =

(y x) (x;t): (6:18)

with this result, it is straightforward to derive the algebra of the transformed op erators

(6.17). We get, using the notation (5.14), e.g.

i ((y x)(xy ))

~ ~ ~ ~

(x;t) (y;t)= e (y ;t) (x;t): (6:19)

This is of the general form (2.13), but wehave to ask is this really what wewant?

All hinges on the argument funtion app earing in (6.19). It is a multivalued

function a priori, but can b e made single{ valued byintro ducing cuts across which

jumps by2. Taking the cut in the direction of the p ositive x -axis, so that

(e + e ) ! 0;(e e ) ! 2 as ! 0, wehave

1 2 1 2

2 2 2 2

(y x) (x y ) = sgn(x y );x 6= y ;

1 1 2 2

= sgn(x y );x = y :

Cutting along the negative x -axis gives the opp osite values for the di erence of the

arguments.

In either case, given the lo cations x and y , the phase factor in (6.19) takes a

i i

sp eci c value, either e or e . Note that this is the phase acquired by the wave

function for twoanyons in a state ji :

~ ~

(x; y )= h0j (x) (y)ji (6:20)

under an interchange of the arguments. But this is not enough. The phase of the wave

function should b e able to change both by+ and in resp onse to whichway

we braid in interchanging x and y . In other words, states built by the creation op er-

ators acting on the vacuum, with a sp eci c, single{valued choice of the argument

function (x), do not provide a representation of the braid group B .

It is evident, however, that this is the b est we can achieve with local op erators

~ ~

; . Lo cal information, in the meaning of initial and nal p ositions of particles, is

simply not sucient to co de the braiding, where wehave to sp ecify also whichway

around each other the particles passed in interchanging p ositions.

The only solution to this dilemma (and in fairness it should b e added that not

all exp erts see this as any dilemma) within the present framework, is to give the 15

argument function a de nition that makes a nonlocal op erator. This p oint has

b een emphasized by Semeno .

Let us return to the transformed eld op erators (6.17) with the op erator{valued

gauge function (6.11). We give the following de nition of the argument function

in (6.11): To the p oint x we attach a curve C starting from some reference p oint P

(in nityisaconvenientchoice) and ending at x. Let x move along C from P to x.

0 0

(x; y ) is then de ned as the change in the p olar angle of x , as seen from y ,as x

moves from P to x along C :

(x; y )= dlr (ly): (6:21)

C l

The corresp onding transformed eld op erators now dep end on the curve C as

well:

i d y (x;y)(y)

[C ]= e (x)

y y i d y (x;y)(y)

[C ]= (x)e : (6:22)

A state of two lo calized anyons is now given by

y y i (x;y )

~ ~

[C ] [C ]j0i = e jx; y i; (6:23)

x y

where jx; y i is the symmetric two- b oson state

y y

jx; y i = (x) (y )j0i: (6:24)

The action of an op erator U [ ] implementing the braiding transformation is

taken to mean extending the curves C ;C by pieces describing the braiding (e.g.

x y

adding half a circle, clo ckwise or counterclo ckwise oriented and centered on y ,to

C and then straight line pieces to b oth C and C so that their endp oints are

x x y

interchanged). It is clear from Eq. (6.23) that this gives the two-anyon state just the

correct phase ( ) corresp onding to the braiding . In this waywehave a means of

representing all of B , alb eit with nonlo cal states.

7. Epilogue

Wehave failed miserably in attaining our initial goal of nding a simple, com-

putationally useful formulation of the many-anyon problem, like the second quanti-

zation of b osons and fermions. The solution we ended up with is complicated and

not particularly workable in practical calculations. Some authors even maintain that

Chern{Simons eld theory has nothing to do with anyons, e.g. .

The rather formal manipulations we carried out in the previous section can b e put

on a more rigorous fo oting by formulating the theory on a lattice. This is esp ecially

true of the relativistic theory, which presents diculties of its own. Again, the theory

20;21

with a Maxwell term is b etter b ehaved , the pure Chern{Simons theory presents

22 21

p eculiar pitfalls . (It is interesting to note that Frohlich and Marchetti formulate 16

anyon statistics for asymptotic states in p-space | probably the right thing to do in

relativistic theories!)

But the conclusions remain the same: Anyons are describ ed by nonlo cal op erators,

which are hard to deal with. If we insist on a lo cal formulation, wehave to hide the

statistics in an interaction with a Chern{Simons eld, and pay the price of handling

the extra interaction through whatever means are avilable.

For lack of space, time and comp etence many topics of anyon physics have not

b een treated in these lectures. Among these are: The relation b etween spin and

statistics, the fractional quantum Hall e ect and the statistical mechanics of anyons.

5;11;12

For these, I have to refer you to the reviews already cited , to which a few more

can b e added .

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Anyon Primer, hep-th 9209066. 18