The propagator and corresponding path integral for a system of identical particles obeying parastatistics are derived. It is found that the statistical weights of topological sectors of the path integral for parafermions and parabosons are simply related through multiplication by the parity of the permutation of the final positions of the particles. Appropriate generalizations of statistics are proposed obeying unitarity and factorizability (strong cluster decomposition). The realization of simple maximal occupancy (Gentile) statistics is shown to require ghost states.
The propagator and corresponding path integral for a system of identical particles obeying parastatistics are derived. It is found that the statistical weights of topological sectors of the path integral for parafermions and parabosons are simply related through multiplication by the parity of the permutation of the final positions of the particles. Appropriate generalizations of statistics are proposed obeying unitarity and factorizability (strong cluster decomposition). The realization of simple maximal occupancy (Gentile) statistics is shown to require ghost states.
This chapter follows on from the previous chapter in quantum statistical mechanics but specialising on systems with identical particles. Using Gibbs prescritpion on generic states from Chapter 3, the occupation number representation is introduced. Constraints imposed on statistics by irreducible representations of the permutation group are discussed. These group-theoretic considerations are used to justify the use of Gentile’s parastatistics. Fermions and bosons are introduced as special cases of Gentile’s statistics, corresponding to the trivial representation of the permutation group for bosons, and the sign representation of the permutation group for fermions. Basic applications to fermions and bosons is given, including the Fermi–Dirac and Bose–Einstein statistics. A detailed expose of why photons are said to have zero chemical potential is also proposed.
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.
ABS>Use of a quantum field theory obeying parastatistics to treat free particles is further examined. Relativistic quantum field theories independent of locality are noted to hold for paralocal fields, and the TCP theorem is generalized to paralocal fields. The following generalization of the connection of spin and statistics is demonstrated: given the choice between parabose and parafermi statistics, integral spin flelds must obey parabose, and odd halfintegral spin fields must obey parafermi statistics (D.C.W.)
A new generalization of quantum statistics is described, which is different from parastatistics, though it includes Fermi statistics and parafermi statistics of order two. It can be applied to the quantization of nonlinear field theories, without violating the correspondence principle. Like parastatistics, it allows the occupation of a given dynamical state by more than one particle of half-odd-integral spin. A special feature is that quark-like particles are naturally associated in modules, which have many of the characteristics of baryons and mesons. The group theoretical properties of the new statistics, and the implied classification of states, are briefly examined.
Quantum mechanics broadly classifies the particles into two categories: (1) fermions and (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.
Possible generalizations of the statistics of identical particles, parastatistics, and the corresponding theories of parafields are reviewed. The connection between parastatistics and cases of degeneracy of particles with respect to an internal degree of freedom is discussed, and also the restrictions that are imposed on internal symmetries if they are given a parafield description. In particular, the difference between the hypotheses of ''three-color'' quarks and ''paraquarks'' is discussed.