View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server SHS-96-4 UUITP-08/96 25 March 1996 Path Integrals and Parastatistics y 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 N 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 N 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> 1 N (where x can b e in a space of any dimension). The collection of such states for a i set of distinct x transforms in the N !-dimensional de ning representation of S i N P jx>jPx >= jx 1 ;:::x 1 > (1) P (1) P (N) 1 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. i 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 N combinations of states within this multiplet transforming in R, that is, X ja; x>= C (P)Pjx> ; a=1;:::d ; d = dim(R): (2) a R R P 3 where the sum is over all elements of the p ermutation group and C (P ) are appro- a priately chosen co ecients. If we denote with R (P ) the matrix elements of the ab p ermutation P in the representations R, X 1 P ja; x >= R (P )jb; x > (3) ab b 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 o 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 r X d R jab; x >= R (P )P jx> (4) ab N ! P 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 N states transform as: X X 1 1 P jab; x >= R (P )jcb; x > ; jab; P x >= R (P )jac; x > (5) ac cb 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 R each irrep of S is emb edded in the de ning representation a numb er of times N 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 N y 1 (P )= the de nition (4) and using the unitarityofP (P =P ) and of R (R ab 1 R (P )) we obtain, after a change in summation variable, ba X d R 0 1 0 < ab; xjAjcd; y >= R (P )R (P )R (P ) <xjAP jy> (6) ea be cd N ! 0 P;P ;e Using further the orthogonality (Shur's) relation (see, e.g., [16]) X N! 1 R (P )R (P )= (7) ab cd ad bc d R P we nally obtain X < ab; xjAjcd; y >= R (P ) <xjAjPy > (8) ac bd P Let us rst cho ose A = 1. Then the ab ove provides the overlap b etween the states X < ab; xjcd; y >= R (P ) (x Py) (9) ac bd P 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.