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arXiv:quant-ph/9903069 · Kluwer Academic Publishers eBooks

Small Violations of Statistics

Oscar Wallace Greenberg

Published Dec 19, 2005• 3 citations
arXivHTMLPDF

Abstract

There are two motivations to consider statistics that are neither Bose nor Fermi: (1) to extend the framework of quantum theory and of quantum field theory, and (2) to provide a quantitative measure of possible violations of statistics. After reviewing tests of statistics for various particles, and types of statistics that are neither Bose nor Fermi, I discuss quons, particles characterized by the parameter q, which permit a smooth interpolation between Bose and Fermi statistics; q=1 gives bosons, q=−1 gives fermions. The new result of this talk is work by Robert C. Hilborn and myself that gives a heuristic argument for an extension of conservation of statistics to quons with trilinear couplings of the form $$ \bar f$$ fb , where f is fermion-like and b is boson-like. We showed that q f 2=q b. In particular, we related the bound on qg for photons to the bound on q e for electrons, allowing the very precise bound for electrons to be carried over to photons. An extension of our argument suggests that all particles are fermions or bosons to high precision.

DOI
10.1007/0-306-47094-2_17

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The quon algebra describes particles, “quons, ” that are neither fermions nor bosons, using a label q that parametrizes a smooth interpolation between bosons (q = 1) and fermions (q = −1). Understanding the relation of quons on the one side and bosons or fermions on the other can shed light on the different properties of these two kinds of operators and the statistics which they carry. In particular, local bilinear observables can be constructed from bosons and fermions, but not from quons. In this paper we construct bosons and fermions from quon operators. For bosons, our construction works for −1 ≤ q ≤ 1. The case q = −1 is paradoxical, since that case makes a boson out of fermions, which would seem to be impossible. None the less, when the limit q → −1 is taken from above, the construction works. For fermions, the analogous construction works for −1 ≤ q ≤ 1, which includes the paradoxical case q = 1.

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In a recent paper, we proposed a ``paronic'' field-theory framework for possible small deviations from the Pauli exclusion principle. This theory cannot be represented in a positive-metric (Hilbert) space. Nonetheless, the issue of possible small violations of the exclusion principle can be addressed in the framework of quantum mechanics, without being connected with a local quantum field theory. In this paper, we discuss the phenomenology of small violations of both Fermi and Bose statistics. We consider the implications of such violations in atomic, nuclear, particle, and condensed-matter physics and in astrophysics and cosmology. We also discuss experiments that can detect small violations of Fermi and Bose statistics or place stringent bounds on their validity.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 1, 1989•113 citations
There are two motivations to consider statistics that are neither Bose nor Fermi: (1) to extend the framework of quantum theory and of quantum field theory, and (2) to provide a quantitative measure of possible violations of statistics. After reviewing tests of statistics for various particles, and types of statistics that are neither Bose nor Fermi, I discuss quons, particles characterized by the parameter q, which permit a smooth interpolation between Bose and Fermi statistics; q=1 gives bosons, q=−1 gives fermions. The new result of this talk is work by Robert C. Hilborn and myself that gives a heuristic argument for an extension of conservation of statistics to quons with trilinear couplings of the form fˉ \bar ffˉ​ fb , where f is fermion-like and b is boson-like. We showed that q f 2=q b. In particular, we related the bound on qg for photons to the bound on q e for electrons, allowing the very precise bound for electrons to be carried over to photons. An extension of our argument suggests that all particles are fermions or bosons to high precision.
I discuss the statistics of "quons" (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the qqq-deformed commutation relation (the quon algebra or qqq-mutator) which interpolates between fermions and bosons. Topics discussed include representations of the quon algebra, proof of the TCP\mathrm{TCP}TCP theorem and clustering, violation of the usual locality properties, and experimental constraints on violations of the Pauli exclusion principle (i.e., Fermi statistics) and of Bose statistics.
After a brief mention of Bose and Fermi oscillators and of particles which obey other types of statistics, including intermediate statistics, parastatistics, paronic statistics, anyon statistics and infinite statistics, I discuss the statistics of ``quons'' (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the qqq-deformed commutation relation (the quon algebra or q-mutator) which interpolates between fermions and bosons. I emphasize that the operator for interaction with an external source must be an effective Bose operator in all cases. To accomplish this for parabose, parafermi and quon operators, I introduce parabose, parafermi and quon Grassmann numbers, respectively. I also discuss interactions of non-relativistic quons, quantization of quon fields with antiparticles, calculation of vacuum matrix elements of relativistic quon fields, demonstration of the TCP theorem, cluster decomposition, and Wick's theorem for relativistic quon fields, and the failure of local commutativity of observables for relativistic quon fields. I conclude with the bound on the parameter qqq for electrons due to the Ramberg-Snow experiment.
The quon algebra describes particles, “quons, ” that are neither fermions nor bosons using a label q that parametrizes a smooth interpolation between bosons (q = +1) and fermions (q = −1). We derive “conservation of statistics” relations for quons in relativistic theories, and show that in relativistic theories quons must be either bosons or fermions. There are three reasons to study theories that allow violations of statistics, i.e., that allow particles that are neither bosons nor fermions. One, which may seem frivolous, is to stretch the framework of quantum physics and to find out what possibilities open up when one does so. The second, quite concrete, is to respond to experimental interest in high-precision tests of the symmetrization postulate (that all identical particles occur in one-dimensional representations of the symmetric
The quon algebra describes particles, ``quons,'' that are neither fermions nor bosons. The parameter qqq attached to a quon labels a smooth interpolation between bosons ( q=+1q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+1q=+1) and fermions ( $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1$). Wigner and Ehrenfest and Oppenheimer showed that a composite system of identical bosons and fermions is a fermion if it contains an odd number of fermions and is a boson otherwise. We generalize this and show that qcomposite=qconstituentn2{q}_{\mathrm{composite}}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}q}_{\mathrm{constituent}}^{{n}^{2}}qcomposite​=qconstituentn2​ for a system of nnn identical quons. Using this generalization, we find bounds on possible violations of the Pauli exclusion principle for nucleons and quarks based on such bounds for nuclei.
The quon algebra describes particles, “quons, ” that are neither fermions nor bosons, using a label q that parametrizes a smooth interpolation between bosons (q = 1) and fermions (q = −1). Understanding the relation of quons on the one side and bosons or fermions on the other can shed light on the different properties of these two kinds of operators and the statistics which they carry. In particular, local bilinear observables can be constructed from bosons and fermions, but not from quons. In this paper we construct bosons and fermions from quon operators. For bosons, our construction works for −1 ≤ q ≤ 1. The case q = −1 is paradoxical, since that case makes a boson out of fermions, which would seem to be impossible. None the less, when the limit q → −1 is taken from above, the construction works. For fermions, the analogous construction works for −1 ≤ q ≤ 1, which includes the paradoxical case q = 1.
In a recent paper, we proposed a ``paronic'' field-theory framework for possible small deviations from the Pauli exclusion principle. This theory cannot be represented in a positive-metric (Hilbert) space. Nonetheless, the issue of possible small violations of the exclusion principle can be addressed in the framework of quantum mechanics, without being connected with a local quantum field theory. In this paper, we discuss the phenomenology of small violations of both Fermi and Bose statistics. We consider the implications of such violations in atomic, nuclear, particle, and condensed-matter physics and in astrophysics and cosmology. We also discuss experiments that can detect small violations of Fermi and Bose statistics or place stringent bounds on their validity.