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arXiv:quant-ph/0610051

A new link between Boson and Fermion algebras: an alternative to supersymmetry

Philip R. Davies

Published Oct 8, 2006• 0 citations
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Abstract

In this paper we describe a new family of algebras which in the case of n = 2 reduces to the Fermion algebra and in the limiting case of n tends to infinity reduces to the Boson algebra. These generalized algebras describe particles which obey a generalized exclusion principle, limiting state occupation to (n-1) and obeying a generalized particle statistics.

DOI
10.48550/arxiv.quant-ph/0610051

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In this paper we describe a new family of algebras which in the case of n = 2 reduces to the Fermion algebra and in the limiting case of n tends to infinity reduces to the Boson algebra. These generalized algebras describe particles which obey a generalized exclusion principle, limiting state occupation to (n-1) and obeying a generalized particle statistics.
A set of operators, the so-called k-fermion operators, that interpolate between boson and fermion operators are introduced through the consideration of an algebra arising from two non-commuting quon algebras. The deformation parameters q and 1/q for these quon algebras are roots of unity with q to the power k being equal to 1. The case k = 2 corresponds to fermions and the case k going to infinity to bosons. Generalized coherent states (connected to the k-fermionic states) and super-coherent states (involving a k-fermionic sector and a purely bosonic sector) are investigated.
A set of operators, the so-called k-fermion operators, that interpolate between boson and fermion operators are introduced through the consideration of an algebra arising from two non-commuting quon algebras. The deformation parameters q and 1/q for these quon algebras are roots of unity with q to the power k being equal to 1. The case k = 2 corresponds to fermions and the case k going to infinity to bosons. Generalized coherent states (connected to the k-fermionic states) and super-coherent states (involving a k-fermionic sector and a purely bosonic sector) are investigated.
Recent experimental results by Metz, Jolie and collaborators have led to renewed interest in the notion of dynamical supersymmetry in fermion many-body systems. By considering the general problem of constructing a boson-fermion realization for Lie algebras with generators expressed in terms of fermion and bifermion operators, also when favored pairs dictate the structure of collective bosons, we demonstrate that dynamical “supersymmetry without bosons” can indeed emerge in a fermion system without any Pauli Violations. Our construction of the general realization utilizes the identification of a physical subspace and allows one to identify the images of single particle transfer operators uniquely, unlike the situation in phenomenological applications of dynamical supersymmetry.
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.
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.