Path Integrals and Parastatistics Alexios P. Polychronakosy

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SHS-96-4

UUITP-08/96

25 March 1996

Path Integrals and Parastatistics

Alexios P. Polychronakos

Centre for Advanced Study, Norwegian Academy of Science and Letters,

0205 Oslo, Norway

and

Theoretical Physics Dept., Uppsala University

S-751 08 Uppsala, Sweden

ABSTRACT

The propagator and corresp onding path integral for a system of identical par-

ticles ob eying parastatistics are derived. It is found that the statistical weights of

top ological sectors of the path integral for parafermions and parab osons are sim-

ply related through multiplication by the parity of the p ermutation of the nal

p ositions of the particles. Appropriate generalizations of statistics are prop osed

ob eying unitarity and factorizability (strong cluster decomp osition). The realiza-

tion of simple maximal o ccupancy (Gentile) statistics is shown to require ghost

states.

y p [email protected]

Permanent address

Generalizations of the standard concepts of fermions and b osons have b een

extensively considered in the last few decades. These can b e broadly divided into

\phenomenological" (anyons [1-4], exclusion statistics [5]), which are meanttogive

a b etter description or understanding of systems of otherwise ordinary fermions or

b osons, and \fundamental," whichwould b e genuinely new kinds of particles. A

nice review of these approaches can b e found in [6], and a short and concise resume

of some relevant results can b e found in [7].

The rst ever consistent extension of fundamental statistics, given by Green

[8], is parastatistics. In that, the standard b osonic or fermionic elds whichwould

create identical particles are replaced by comp osite elds whose comp onents com-

mute with themselves and anticommute with each other for parab osons, or vice

versa for parafermions. The numb er of comp onents of the elds p de nes the \or-

der" of parastatistics. In general, one can put at most p parafermions in a totally

symmetric wavefunction, and at most p parab osons in a totally antisymmetric one.

The degeneracies of o ccupation of more general multiparticle states are in princi-

ple calculable but rather complicated. Parastatistics in this approach has b een

well-studied [9-12].

The ab ove is a eld theoretic realization of parastatistics. Just as in the case

of fermions or b osons, one can deal with a parastatistical system at a xed particle

numb er in a rst-quantized formalism. In this approach, due mainly to Messiah

and Greenb erg [13,14], the N -b o dy Hilb ert space is decomp osed into irreducible

representations (irreps) of the particle p ermutation group S . Since the particles

are indistinguish abl e, this group should b e viewed as a \gauge" symmetry of the

system, and states transforming in the same representation have to b e identi ed.

Moreover, since all physical op erators are required to commute with the p ermuta-

tion group, each irreducible comp onent is a sup erselection sector. Therefore, one

can pro ject the Hilb ert space to only some of the irreps of S .Further, only one

state in each irrep need b e kept as a representative of the multiplet of physically

equivalent states. The resulting reduced space constitutes a consistent quantiza-

tion of N indistingui shabl e particles. The choice of included irreps constitutes a 2

choice of quantum statistics. In particular, parab osons corresp ond to including

only irreps with up to p rows in their Young tableau, while parafermions to ones

with up to p columns. Clearly the cases p = 1 reduce to ordinary fermions and

b osons.

This description relies on a canonical quantization of the many-b o dy system.

It is of interest to also have a path-integral formulation of a quantum system,

since this complements and completes the conceptual framework and usually o ers

orthogonal intuition in several cases. For ordinary statistics this question was

studied by Laidlaw and DeWitt [15]. In this pap er, we provide such a realization for

parastatistics, or, in general, for any statistics where the Hilb ert space is emb edded

in the tensor pro duct of N one-particle Hilb ert spaces (note that this excludes

anyons and braid statistics).

The starting p oint will b e the co ordinate representation of the full (unpro-

jected) Hilb ert space, spanned by the p osition eigenstates jx ;:::x >jx>

(where x can b e in a space of any dimension). The collection of such states for a

set of distinct x transforms in the N !-dimensional de ning representation of S

P jx>jPx >= jx 1 ;:::x 1 > (1)

P (1) P (N)

where P is a p ermutation (the app earance of P in the ab ove is necessary so

that pro ducts of p ermutations b e represented in the right order). If anyofthe

co ordinates x coincide the ab ove is not the full de ning representation any more.

The set of such states, however, is of measure zero (the co ordinate space assumed

continuous) and thus they can b e safely ignored. (We assume that there are no

interactions singular at coincidence p oints that might dynamically make such states

of nonzero weight.)

Pro jecting the Hilb ert space to an irrep R of S amounts to keeping only linear

combinations of states within this multiplet transforming in R, that is,

ja; x>= C (P)Pjx> ; a=1;:::d ; d = dim(R): (2)

R R

P 3

where the sum is over all elements of the p ermutation group and C (P ) are appro-

priately chosen co ecients. If we denote with R (P ) the matrix elements of the

p ermutation P in the representations R,

P ja; x >= R (P )jb; x > (3)

The de ning representation decomp oses into irreducible comp onents, classi ed by

Young tableaux, each app earing with a certain multiplicity. Should wekeep only

one irrep out of eachmultiplicity or the whole multiplet? To decide it, note that

if instead of the base state jx>for the construction of the states ja; x > we

cho ose a di erent p ermutation P jx>, then although the new states ja; P x>

o o

constructed through (2) still transform in the irrep R, in general they are not

linear combinations of ja; x > but rather span a di erent copyof R. Since we can

continuously move in the con guration space from jx> to P jx>,we conclude

that wemust keep al l irreps R within eachmultiplet. (In other words, although for

each p oint in the Hilb ert space jx> this multiplet is reducible, the b er of these

representations over the Hilb ert space is connected and irreducible.)

To realize explicitly the ab ove, we construct the states

jab; x >= R (P )P jx> (4)

N !

Using the group prop erty of the representation R(P )R(P )=R(P P ), we deduce

1 2 1 2

that under the action of the group S and under change of base p oint x the ab ove

states transform as:

X X

1 1

P jab; x >= R (P )jcb; x > ; jab; P x >= R (P )jac; x > (5)

c c

Thus we see that the rst index in these states lab els the di erent elements of a

single irrep R, while the second index lab els the di erent equivalent irreps in the 4

multiplet. Since b oth indices take d values, we recover the standard result that

each irrep of S is emb edded in the de ning representation a numb er of times

equal to its dimension.

Consider now the matrix element < ab; xjAjcd; y >, where A is anyphysical

op erator, that is, any op erator commuting with all elements P of S . Substituting

y 1

(P )= the de nition (4) and using the unitarityofP (P =P ) and of R (R

R (P )) we obtain, after a change in summation variable,

0 1 0

< ab; xjAjcd; y >= R (P )R (P )R (P ) (6)

be cd

N !

P;P ;e

Using further the orthogonality (Shur's) relation (see, e.g., [16])

R (P )R (P )= (7)

ab cd ad bc

we nally obtain

< ab; xjAjcd; y >= R (P ) (8)

Let us rst cho ose A = 1. Then the ab ove provides the overlap b etween the states

< ab; xjcd; y >= R (P ) (x Py) (9)

For x in the neighb orho o d of y it is P = 1 which contributes to the normalization,

for which R (1) = and we recover the standard continuous normalization

bd bd

between the states.

iH t

Nowwe can cho ose A = e , where H is the hamiltonian, and thus nd the

propagator G(ab; x; cd; y jt)between the states of the system. It is clear from (8)

that the rst index a in the state jab; x > propagates trivially. Since this is the index

that corresp onds to the di erent but physically equivalent states within each irrep 5

R,we conclude that the required pro jection of the Hilb ert space to the physical

subspace amounts to simply omitting this index from all states. (That is, freeze

this index to the same xed value for all states of the theory; no physical quantity

will ever dep end on the choice of this value.) On the other hand, the second

index, corresp onding to di erent equivalent irreps, do es not propagate trivially

and must, as argued b efore, b e kept. We are led therefore to the physical states

jba; x >!ja; x > and the propagator

G (a; x; b; y jt)= R (P ) G(x; P y ; t) (10)

iH t

where G(x; P y ; t)= is the usual many-b o dy propagator. We note

that, due to the transformation prop erty (5), the states ja; P x > are linear com-

binations of states ja; x >. Therefore, pro jecting down to the physical subspace

corresp onding to R amounts to trading the original N ! copies of physically equiv-

alent states jPx > foranumber d of global internal degrees of freedom for the

system, lab eled by the index a.

It is now easy to write down the path integral corresp onding to identical par-

ticles quantized in the R-irrep of S . G(x; P y ; t) can b e expressed as an N -body

path integral in the standard way, with particles starting from p ositions x and

ending in p ositions Py = y 1 . Since all p ermutations of particle p ositions are

P (i)

physically equivalent, (10) instructs us to sum over al l sectors where particles end

up in such p ermuted p ositions, weighted with the factors R (P ) dep ending on the

internal degrees of freedom of the initial and nal states. From (4), (9) we can

write the completeness relation within the physical subspace

d x

ja; x >< a; xj (11) I =

N !

and with the use of (11) it is easy to prove that the ab ove path integral is unitary, 6

that is,

d y

0 0

G(a; x; b; y jt) G(b; y ; c; z jt )=G(a; x; c; z jt + t ) (12)

N !

The extension to parab osons, parafermions or any similar statistics is immedi-

ate. Let S = fR ;:::R g b e the set of allowed irreps of S in the Hilb ert space.

1 n

The internal degree of freedom now takes values A =(R; a), where R 2 S and

a =1;:::d lab els the internal degrees of freedom within each irrep. So, overall,

A takes d + d di erentvalues. The propagator (and corresp onding path

R R

1 n

integral) is obviously

G (A; x; B; yjt)= S(P) G(x; P y ; t) ; where S (P ) = (R ) (P )

S AB AB R ;R A ab

A B

(13)

For parab osons (parafermions) of order p, S is the set of Young tableaux with up

to p rows (columns). We note that the irreps for parafermions are the duals of

those for parab osons (the dual of a tableau is the tableau with rows and columns

interchanged). In an appropriate basis, the representation matrices of dual irreps

R; R are real and satisfy

R (P )=(1) R (P ) (14)

ab ab

where (1) is the parity of the p ermutation. We arrive then at the relation

between the weights for parab osons and parafermions of order p:

S (P ) =(1) S (P ) (15)

pF AB pB AB

This extends a similar relation for ordinary fermions and b osons, for which there

are no internal degrees of freedom and S (P )=1.

From the path integral we can evaluate the partition function, by simply shift-

ing to the euclidean p erio dic propagator G ( )= e and summing over all

E 7

initial and nal states, with the measure implied by (11). Given that

R (P )=trR(P)= (P) (16)

we get the expression in terms of the characters of S

X X

d x

S(P) ; where S (P )= (P) (17) Z (T )=

E R S

P R2S

The interpretation in terms of a p erio dic euclidean path integral is obvious. The

characters (P ) are a set of integers, and thus the \statistical factors" S (P )

weighing each top ological sector of the path integral are (p ositive or negative)

integers. In the case of parab osons of any order p,however, we note that the

statistical weights are positive (or zero) integers. The ones for parafermions can b e

either p ositive or negative, as given by

S (P )=(1) S (P ) ; S (P ) 0 (18)

pF pB pB

We do not have a general formula for S (P ) for arbitrary p.

From the ab ove results we can derive the partition function for a gas of parasta-

tistical particles as well as the allowed o ccupancy of single-particle states. Consider

a collection of non-interacting particles, for which the hamiltonian is separable into

a sum of one-b o dy hamiltonians H = H (x ). Let the energy eigenvalues of

the one-b o dy problem b e and the corresp onding one-b o dy Boltzmann factors

z = e . Consider now a sector of the euclidean path integral characterized by

the p ermutation of nal p oints P . It is clear that this path integral Z decomp oses

into a pro duct of disconnected comp onents, characterized by the fact that the par-

ticle worldlines in each comp onent mix particles only within the same comp onent.

This means that, within each comp onent, particles mix under a cyclic p ermutation

(since following the worldline of each particle must successively lead to every other 8

particle in the comp onent). Each element P of S is then decomp osed into a pro d-

uct of commuting cyclic p ermutations. The numb er of particles n participating in

each cyclic p ermutation constitute the cycles of P (obviously n = N ). We are

led, thus, to the fact that for noninteracting particles

Z = Z (19)

n2cy cl es(P )

The path integral Z for a cyclic p ermutation of n particles, on the other hand,

can b e thought of as the path integral of a single particle winding n times around

euclidean time . This means that

Z ( )= Z (n )= z W [z ] (20)

n 1 n i

and the corresp onding expression for Z b ecomes

Z = W [z ] (21)

n i

n2cy cl es(P )

The expression for the full partition function then b ecomes

X X Y

Z = (P ) W [z ] (22)

n i

S R

N !

R2S P

n2cy cl es(P )

We recognize the sum over P in (22) as Frob enius' relation, connecting the sum

over the Schur functions W [z ] to the characters of SU (M ) [z ]. We get the

n i i

nal result

N 1j +`

X X

det(z )

Z = [z ]= (23)

S R

N 1j

det(z )

R2S R2S

where ` is the length of the j -th row of the Young tableau of R. This repro duces

the result of Chaturvedi for the partition function [17] and an earlier result of

Suranyi for p = 2 [18]. We stress that the ab ove result holds only for noninteracting 9

particles. For interacting particles the top ological ly disconnected comp onents Z

of the path integral Z are still dynamically connected and factorization fails. One

has to go back to the full expression (17) for the partition function in that case.

To nd the degeneracy of states, we need to decomp ose [z ] app earing in

(23) in monomials z

X X Y

[z ]= D[p ] z (24)

i i

R2S

fpg

The co ecients D [p ] of these monomials, called Kostka-Foulkes numb ers [17, 19],

are non-negativeintegers which determine the degeneracy of the state with p

particles o ccupying each energy level . To nd these integers in a systematic

way,we use the following trick: consider that the particles are b osons and have

an internal degree of freedom transforming in the fundamental of SU (N ) (in fact,

SU (M ) with M N would also do). Since under total p ermutation of particle

co ordinates and internal degrees of freedom the states must transform trivially,we

conclude that the irrep of the color SU (N ) for each state must b e the same as the

irrep of the co ordinate p ermutation group S (meaning they have the same Young

tableau and thus the same symmetries). A state with p particles in the same

level transforms in the p-fold symmetric irrep of SU (N ). Therefore, a state

with o ccupancies p for each level transforms under the direct pro duct of p -fold

i i

symmetric irreps, one for each level . Decomp osing this pro duct into irreducible

comp onents, we will obtain each representation R of SU (N )a numb er of times

D [p ]. Each such irrep will transform under a similar irrep of S and thus will

R N

corresp ond to a unique physical state in the quantization of N identical particles in

the representation R. Therefore, the degeneracy D [p ] can b e found by summing

the numb er of times that each allowed irrep R 2 S app ears in the direct pro duct

of symmetric irreps p , which can b e found using standard SU (N )Young tableaux

comp osition rules. We also see that, if the internal degree of freedom group is

chosen to b e SU (p) (where p may b e smaller that N ), we will only get irreps with

up to p rows. Therefore, we recover the known result that parab osons of order p 10

can b e viewed as b osons with an internal SU (p) symmetry, where we identify each

irrep of SU (p) as a unique physical state [11,12]. A similar construction can b e

rep eated starting from fermions instead of b osons. We recover a dual expression

for the degeneracies D [p ], where nowwe form the direct pro duct of p -fold fully

i i

antisymmetric irreps of SU (N ), and a similar expression of parafermions of order

p as fermions with an SU (p)internal symmetry.

As was argued in [10,14], parastatistics particles ob ey the cluster decomp osition

principle, in the sense that the density matrix obtained by tracing over a subset of

particles which decouple from the system can b e constructed as a p ossible density

matrix of the reduced system of remaining particles. From (17), however, we see

that the partition function of two dynamically isolated sets of particles N and

N do es not factorize into the pro duct of the two partition functions, since the

statistical weights S (P ) in general do not factorize into S (P )S (P ) when P is

1 2

the pro duct of two commuting elements P and P . Equivalently, this means that

1 2

the o ccupation degeneracy D [p ] do es not factorize into the pro duct of individual

o ccupation degeneracies for each level . This has imp ortantphysical implications.

If the two sets of particles are totally isolated, it do es not make sense to evaluate the

partition function of the total system, since the statistical distribution can never

relax to the one predicted by that partition function. The individual partition

functions of the subsystems are the relevant ones. If, however, the two sets are

only weakly coupled, then initially each set will distribute according to its reduced

partition function, but after some relaxation time (dep ending on the strength of

the coupling b etween the two sets) they will relax to the joint distribution function,

which, we stress, will not even approximately equal the pro duct of the individual

ones. Thus, cluster decomp osition holds in an absolute sense but fails in a more

realistic sense. In contrast, fermions and b osons resp ect cluster decomp osition in

b oth senses.

The obvious generalization of quantum statistics, based always on the assump-

tion that the many-b o dy Hilb ert state is emb edded into the tensor pro duct of

many one-b o dy Hilb ert spaces, is to generalize the set of allowed irreps S beyond 11

the one relevant to parastatistics. Wemay,however, further include more than one

state for each included irrep of S . This seems unmotivated, in view of the fact

that such states are physically indistinct, but it is certainly consistent. It could

mean, for instance, that the particles have some hidden internal degrees of freedom

accounting for the extra degeneracy, which are invisible to the present hamiltonian

but may b ecome dynamically relevant later. The most general situation, then, is

that we include C states from each irrep R. The generalization of all previous

formulae for this case is quite immediate, S (P ) and C b eing related by

X X

S (P )= C (P) ; C = S(P) (P) (25)

R R R R

R P

The case of distinguishable particles (\in nite statistics" [12,20,21]), in particular,

is repro duced by accepting all states in each irrep, that is, C = d . Since R

R R

app ears exactly d times in the de ning representation of S , S (P )abovebe-

R N

comes the trace of P in that representation. But all P 6= 1 are o -diagonal in

the de ning representation, so we get S (P )= N! , recovering the standard

inf P;1

distinguishabl e particles result.

We summarize here by p ointing out that the most general statistics of the typ e

examined here is parametrized byany of three p ossible sets of numb ers. The rst

is, as just stated, the numb er of states C accepted for each irrep R of S . Since

R N

the irreps of S are parametrized by the partitions of N (lengths of rows or the

Young tableau), there are as many C as there are partitions of N . The second set

is the statistical weights S (P ) app earing in the partition function (euclidean path

integral). Clearly these weights are invariant under conjugation of P ! QP Q ,

since this simply amounts to a relab eling of the particle worldlines. Thus S (P )

dep ends only on the conjugacy class of P , that is, the cycles of P . The p ossible sets

of cycles are the same as the partitions of N ; so, again, the S (P ) are numb ered by

partitions of N . Finally,we could use the degeneracy of a many-b o dy o ccupancy

state D [p ] as our de nition. There are as manyways to distribute particles in

one-b o dy states as there are partitions of N , so this set also has the same number

of elements as the previous two. 12

What are the restrictions or criteria to b e imp osed on the ab ove parameters?

The rst one is unitarity, that is, the existence of a well-de ned Hilb ert space with

p ositive metric. This requires that C b e non-negative (no negative norm states)

integers (no \fractional dimension" states). The other will b e what we call \strong

cluster decomp osition principle," that the partition function of isolated systems

factorize. This is a physical criterion, rather than a consistency requirement. To

summarize:

Unitarity: C non negativeintegers

Y Y

Strong cluster decomp osition : S (P )= S (n)or D[p]= D(p )

i i

n2cy cl es(P )

The strong cluster decomp osition, in particular, implies the existence of a grand

partition function, obtained (in the case of noninteracting particles) by exp onen-

tiating the sum of all connected path integrals (P a cyclic p ermutation of degree

n) with weights S (n)=n (1=n is the symmetry factor of this path integral, corre-

sp onding to cyclic relab elings of the particles). The grand partition function will

further factorize into a pro duct of partition functions for each level .Thus, S (n)

are cluster co ecients connected to D (n) in the standard way

1 1

X X

S (n)

p n

D (p)z = exp (26) z

p=0 n=1

The ab ove formula, in fact, provides the easiest way to relate D [p ] and S (P )in

the general case (no strong cluster prop erty): simply expand the right-hand side

of (26) in p owers of z and substitute every term S (n ) S(n ) with S (n ;:::n ).

1 1

k k

This gives D (p). To nd D [p ]=D(p;:::p ) simply evaluate D (p ) D(p )

i 1 1

k k

using the ab ove formula and again consolidate each pro duct S (n ) S(n )into a

single S (n ;:::n ).

If we assume that S (1) = D (1) = 1, then it is easy to verify that the only

solution of the ab ovetwo criteria is ordinary fermions and b osons. The situation

is di erent, however, when S (1) = D (1) = q>1 (this would mean, e.g., that the 13

particles come a priori in q di erent \ avors"). The p ossibili ties are manifold. All

these generalized statistics share the following generic features:

The degeneracy of the state where n particles o ccupy di erent levels is q .

n n

(Indeed, D (1; 1;::: 1) = D (1) = q .)

If state A can b e obtained from state B by `lumping' together particles that

previously o ccupied di erent levels, then D (A) D (B ). (E.g., D (3) D (2; 1)

D (1; 1; 1).)

This second prop erty is actually related to the (weak) cluster decomp osition as

formulated in [14], whichisobviously covered by the strong cluster prop erty.

The ab ove p ossibiliti es include the obvious sp ecial cases of q b osonic avors

and q fermionic ones (q + q = q ), for which S (n)=q (1) q , along with

2 1 2 1 2

many other. As an example, we give the rst few degeneracies for many-particle

level o ccupation for all statistics with q =2:

D(1) = 2;D(2) = 4;D(3) = 8

D (1) = 2;D(2) = 3;D(3) = 6; 5; 4(B + B )

D (1) = 2;D(2) = 2;D(3) = 4; 3; 2(B + F ); 1; 0

D (1) = 2;D(2) = 1;D(3) = 2; 1; 0(F + F )

D (1) = 2;D(2) = 0;D(3) = 0

The sp eci c choices denoted by B + B , B + F and F + F are the ones cor-

resp onding to two b osonic, one b osonic and one fermionic, and two fermionic a-

vors resp ectively. The topmost statistics could b e termed \sup erb osons" and the

b ottom one \sup erfermions" of order 2. We also remark here that the \(p; q )-

statistics" intro duced in [14] can b e realized as particles with p b osonic and q

fermionic avors, where we identify eachmultiplet transforming irreducibly under

the sup ergroup SU (p; q ) as a unique physical state.

Finally,we direct our attention to the rst known attempt to generalize the

ordinary Fermi or Bose statistical mechanics, by Gentile [22]. The rule is simply 14

that up to p particles can b e put in each single-particle level. This corresp onds

to D (n) = 1 for n p, and D (n) = 0 otherwise. This has b een criticized [7]

on the grounds that xing the allowed o ccupations for each single-particle state

is not a statementinvariant under change of single-particle basis. It is clear that,

in the language of this pap er, any statistics satisfying the unitarity requirementis

consistent and basis-indep endent. Therefore, Gentile statistics must violate uni-

tarity. Indeed, it is easy to check that all weights C for such statistics are integers

(this is generic for all statistics with integer D (n)), but not necessarily p ositive.

In the sp eci c case of p = 2, e.g., where up to double o ccupancy of each level is

allowed, the degeneracies of each irrep of S (parametrized, as usual, by the length

of Young tableau rows) up to N = 5 are

C = C = C = C =1;C = C = C = 1; else C =0 (27)

2 21 22 221 111 1111 2111

We see that representations 111, 1111, 2111 corresp ond to ghost (negative norm)

states and their e ect is to subtract (rather than add) degrees of freedom. We

also remark that the path integral realization of exclusion statistics exhibits b oth

negative and fractional statistical weights, signaling breakdown of unitarity [23].

This is inconsequential in that case, since exclusion statistics is valid only as a

macroscopic (statistical) description of some (interacting) systems of particles.

In conclusion, wehave presented the many-b o dy propagators and corresp ond-

ing path integrals of particles ob eying parastatistics or any other typ e of statistics

based on irreps of the p ermutation group. We argued that there are many p os-

sible unitary generalizations ob eying the strong cluster decomp osition principle,

although they all require more than one avor of particles. Several other direc-

tions of investigation and op en questions suggest themselves. To name a few, the

statistical mechanics of such generalized statistics particles should b e examined.

Also, it should b e checked if they can b e realized as particles with sp eci c hid-

den internal symmetries and an appropriate pro jection of the Hilb ert space, in a

fashion similar to parastatistics. Indep endently,itwould b e interesting to see if 15

Gentile statistics can b e consistently realized byintro ducing `b enign' ghosts which

account for the negative norm states while decoupling from all physical pro cesses,

just as in gauge theories. Finally, a similar analysis could b e attempted for gener-

alized particle statistics in 2 + 1 dimensions. In fact, a similar treatment, based on

the p ermutation group, has b een used to obtain p erturbative results for anyonic

particles [24-26]. It would b e interesting to examine whether non-ab elian irreps of

the braid group, instead of the p ermutation group, could b e considered.

Acknowledgements: Iwould like to thank J. Myrheim for discussions.

REFERENCES

1. J.M. Leinaas and J. Myrheim, Nuovo Cimento 37B, 1 (1977).

2. G.A. Goldin, R. Meniko and D.H. Sharp, J. Math. Phys. 21, 650 (1980);

22, 1664 (1981); Phys. Rev. D28, 830 (1983).

3. F. Wilczek, Phys. Rev. Lett. 48, 1114 (1982); 49, 957 (1982).

4. Y.S. Wu, Phys. Rev. Lett. 52, 2103 (1984).

5. F.D.M. Haldane, Phys. Rev. Lett. 67 (1991) 937.

6. J. Myrheim, Anyons (Notes for the Course on Geometric Phases, ICTP,

Trieste 6-17 Sept. 1993).

7. O.W. Greenb erg et al., hep-ph/9306225 .

8. H.S. Green, Phys. Rev. 90, 270 (1953).

9. O. Steinmann, Nuovo Cimento 44, A755 (1966).

10. P.V. Landsho and H.P. Stapp, Ann. of Phys. 45, 72 (1967).

11. Y. Ohnuki and S. Kamefuchi, Phys. Rev. 170, 1279 (1968); Ann. of Phys.

51, 337 (1969).

12. S. Doplicher, R. Haag and J. Rob erts, Comm. Math. Phys. 23, 199 (1971);

35, 49 (1974). 16

13. A.M.L. Messiah and O.W. Greenb erg, Phys. Rev. B136, 248 (1964); B138,

1155 (1965).

14. J.B. Hartle and J.R. Taylor, Phys. Rev. 178, 2043 (1969); R.H. Stolt

and J.R. Taylor, Phys. Rev. D1, 2226 (1970); Nucl. Phys. B19, 1 (1970);

J.B. Hartle, R.H. Stolt and J.C. Taylor, Phys. Rev. D2, 1759 (1970).

15. M.G.G. Laidlaw and C.M. DeWitt, Phys. Rev. D3, 1375 (1971).

16. M. Hammermesh, Group Theory, Addison-Wesley Eds. (1962).

17. S. Chaturvedi, U. Hyderabad preprint, hepth/9509150.

18. P. Suranyi, Phys. Rev. Lett. 65, 2329 (1990).

19. C. Kostka, Crel les Journal 93, 98 (1882); H.O. Foulkes, Permutations,

Eds. Gauthier-Villars, Paris (1974).

20. A.B. Govorkov, Theor. Math. Phys. 54, 234 (1983).

21. O.W. Greenb erg, Phys. Rev. Lett. 64, 705 (1990).

22. G. Gentile, Nuovo Cimento 17, 493 (1940).

23. A.P.Polychronakos, Phys. Lett. B365, 202 (1996).

24. J. Myrheim and K. Olaussen, Phys. Lett. B299, 267 (1993).

25. J. Desb ois, C. Heinemann and S. Ouvry, Phys. Rev. D51, 942 (1995).

26. A.D. de Veigy, Nucl. Phys. B458 [FS], 533, 1996). 17