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Mathematical Proceedings of the Cambridge Philosophical Society

Physical properties of particles obeying generalized statistics

Ian E. McCarthy

Published Jan 1, 1955• 16 citations
Publisher

Abstract

ABSTRACT The Feynman probability amplitudes for interactions between spin ½ fields and fields of integral spin quantized according to the generalized method of Green are investigated. It is shown that the scattering amplitudes are independent of the method of quantization though the amplitudes of the vacuum fluctuations are not. The question of distinguishing particles satisfying the generalized statistics, if they exist in nature, from Fermi or Bose particles is considered. It is shown that the statistical thermodynamics of the particles depends on the statistics, but as this requires a large number of particles in thermodynamical equilibrium, it offers no means of distinction for particles with short lifetimes. In practice, an examination of particles in bound states would identify particles obeying generalized statistics.

DOI
10.1017/s0305004100030012

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ABSTRACT The Feynman probability amplitudes for interactions between spin ½ fields and fields of integral spin quantized according to the generalized method of Green are investigated. It is shown that the scattering amplitudes are independent of the method of quantization though the amplitudes of the vacuum fluctuations are not. The question of distinguishing particles satisfying the generalized statistics, if they exist in nature, from Fermi or Bose particles is considered. It is shown that the statistical thermodynamics of the particles depends on the statistics, but as this requires a large number of particles in thermodynamical equilibrium, it offers no means of distinction for particles with short lifetimes. In practice, an examination of particles in bound states would identify particles obeying generalized statistics.
A method of field quantization is investigated which is more general than the usual methods of quantization in accordance with Bose of Fermi statistics, though these are included in the scheme. The commutation properties and matrix representations of the quantized field amplitudes are determined, and the energy levels of the field are derived in the usual way. It is shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate. With the generalized statistics, the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated. Physical consequences of the assumption that there are particles which obey the generalized statistics are briefly examined.
A new class of identical particles which may exhibit both Bose and Fermi statistics with respective probabilities pb{p}_{b}pb​ and pf{p}_{f}pf​ is introduced. Such an uncertainty may be either an intrinsic property of a particle or can be viewed as an ``experimental uncertainty.'' Statistical equivalence of such particles and particles obeying parastatistics of infinite order is shown. Generalized statistical distributions are derived, and statistical and thermodynamical properties of an ideal gas of the particles are investigated. The physical nature of such particles and the implications of this investigation for the statistics of extremal black holes are discussed.
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.
Abstract It is commonly believed that there are only two types of particle exchange statistics in quantum mechanics, fermions and bosons, with the exception of anyons in two dimensions 1–5 . In principle, a second exception known as parastatistics, which extends outside two dimensions, has been considered 6 but was believed to be physically equivalent to fermions and bosons 7–9 . Here we show that non-trivial parastatistics inequivalent to either fermions or bosons can exist in physical systems. These new types of identical particle obey generalized exclusion principles, leading to exotic free-particle thermodynamics distinct from any system of free fermions and bosons. We formulate our theory by developing a second quantization of paraparticles that naturally includes exactly solvable non-interacting theories and incorporates physical constraints such as locality. We then construct a family of exactly solvable quantum spin models in one and two dimensions, in which free paraparticles emerge as quasiparticle excitations, and their exchange statistics can be physically observed and are notably distinct from fermions and bosons. This demonstrates the possibility of a new type of quasiparticle in condensed matter systems and—more speculatively—the potential for previously unconsidered types of elementary particle.
Generalized statistical distributions for identical particles are introduced for the case where filling a single-particle quantum state by particles depends on filling states of different momenta. The system of one-dimensional bosons with a two-body potential that can be solved by means of the thermodynamic Bethe ansatz is shown to be equivalent thermodynamically to a system of free particles obeying statistical distributions of the above class. The quantum statistics arising in this way are completely determined by the two-particle scattering phases of the corresponding interacting systems. An equation determining the statistical distributions for these statistics is derived.