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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Paraparticles of Infinite Order

James B. Hartle, Robert H. Stolt, John R. Taylor

Published Oct 15, 1970• 21 citations
Publisher

Abstract

We show that with one exception every possible statistical type of particle consistent with the cluster properties of quantum mechanics can be uniquely identified by a single pair of integers ($p, q$). A particle of type ($p, q$) has states corresponding to all Young diagrams whose first $p$ columns (and only these) have arbitrary depth and whose first $q$ rows (and only these) have arbitrary length. Parabosons and parafermions of order $p$ are included in this scheme as types (0, $p$) and ($p$, 0), respectively. If $p$ and $q$ are nonzero, then the particle is of infinite order. The only exception to this classification scheme is a particle with states corresponding to all Young diagrams.

DOI
10.1103/physrevd.2.1759

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arXiv:math-ph/0604045•Journal of Mathematical Physics•Feb 1, 2007•5 citations
We show that with one exception every possible statistical type of particle consistent with the cluster properties of quantum mechanics can be uniquely identified by a single pair of integers (p,qp, qp,q). A particle of type (p,qp, qp,q) has states corresponding to all Young diagrams whose first ppp columns (and only these) have arbitrary depth and whose first qqq rows (and only these) have arbitrary length. Parabosons and parafermions of order ppp are included in this scheme as types (0, ppp) and (ppp, 0), respectively. If ppp and qqq are nonzero, then the particle is of infinite order. The only exception to this classification scheme is a particle with states corresponding to all Young diagrams.
Hartle and Taylor have shown that the cluster law imposes certain restrictions on the allowed symmetries of first-quantized systems of identical particles. We extend their results to show that all particles which are neither bosons nor fermions and which obey the cluster law can be divided into two classes, those of finite order, and those of infinite order. For every positive integer ppp there are two types of finite-order particle, which we call parabosons and parafermions of order ppp. Ordinary bosons and fermions can be fitted into this scheme as particles of order 1. We conjecture that the finite-order particles can be identified with the parafermions and parabosons of the second-quantized, parafield theory. Infinite-order particles would seem to have no analog in the second-quantized theory, as presently formulated.
We show that it is possible to formulate a consistent first-quantized theory of paraparticles, i.e., particles which are neither bosons nor fermions. We examine a number of properties of the theory and show that the formulation of Messiah and Greenberg in terms of generalized rays can be replaced by an equivalent formulation in which states are represented by rays in the usual way. We use this alternative formulation to establish some results of ordinary quantum mechanics. We examine in detail the consistency of the theory with the cluster law and show that paraparticles must have states associated with whole families of different permutation symmetries, according to the following rule: If a given particle has NNN-particle states associated with a given Young diagram, then it must have ($N\ensuremath{-}1$)-, ($N\ensuremath{-}2$)-,..., two-particle states associated with all Young diagrams which can be obtained from the first by successively removing squares. This gives rise to infinitely many different kinds of paraparticle, all with rather complicated properties.
Quantum mechanics broadly classifies the particles into two categories: (1)(1)(1) fermions and (2)(2)(2) bosons. Fermions are half-integer spin particles, obeying Pauli's exclusion principle and Fermi-Dirac statistics. Whereas bosons are integer spin particles, not obeying Pauli's exclusion principle, and obeying Bose-Einstein statistics. However, there are two exceptions to this standard case: first, anyons, which exist in 2-dimensional systems, and secondly, paraparticles, which can exist in any dimension. Paraparticles follow their non-trivial parastatistics, obeying their generalised exclusion principle. In this paper, we provide a detailed review of the foundations of paraparticle statistics established in \cite{wang2025particle}. We extend this work further and then derive an important expression for the heat capacity of paraparticles for a specific case, which would provide a handle for the experimental detection of paraparticles in appropriate systems.
In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator PPP which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of nnn parafermions together with PPP is the orthogonal Lie algebra so(2n+2)so(2n+2)so(2n+2). The Fock spaces correspond to particular irreducible representations of so(2n+2)so(2n+2)so(2n+2), and the action of PPP in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of nnn parabosons together with PPP is the orthosymplectic Lie superalgebra osp(2∣2n)osp(2|2n)osp(2∣2n). In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of osp(2∣2n)osp(2|2n)osp(2∣2n). Both for parafermions and parabosons the spectrum of PPP is closely related to the so-called order of statistics ppp, introduced by Green.
Although usually only two kinds of statistics, namely Bose-Einstein and Fermi-Dirac statistics, are considered in Quantum Mechanics and in Quantum Field Theory, other kinds of statistics, called collectively parastatistics, are conceivable. We critically review theoretical studies of parastatistics to date, point out and clarify several confusions. We first study the "proofs" so far proposed for the symmetrization postulate which excludes parastatistics, emphasizing their ad hoc nature. Then, after exploring in detail the structure of the quantum mechanical theory of paraparticles, we clarify some confusions concerning the compatibility of parastatistics with the so-called cluster property, which has been an issue of controversy for several years. We show, following a suggestion of Greenberg, that the quantum mechanical theory of paraparticles can be formulated in terms of density matrix compatibly with the cluster property. We also discuss such topics as selection rules for systems with variable numbers of paraparticles, the connection between statistics and permutation characters, and the classification of paraparticles. For the quantum field theory of paraparticles, we study discrete representations of the para-commutation relations and illustrate in detail Greenberg and Messiah's theorem concerning Green's ansatzes. Also, fundamental topics such as the spin-statistic theorem, the TCP theorem and the observability of parafields are discussed on the basis of Green's ansatzes. Finally, we point out that the so-called particle permutation operators do not always define multi-dimensional representations of the permutation group both in first and second quantization theories. This questions the validity of the correspondence between the two theories which has recently been proposed.