A Multiparametric Quon Algebra
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A Multiparametric Quon Algebra Hery Randriamaro1 Received: 12 June 2019 / Revised: 12 September 2019 / Accepted: 12 May 2020 © Iranian Mathematical Society 2020
Abstract The quon algebra is an approach to particle statistics introduced by Greenberg to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. We generalize these models by introducing a deformation of the quon algebra generated by a collection of operators ai , i ∈ N∗ the set of positive integers, on an infinite dimensional module satisfying the qi, j -mutator relations ai a†j − qi, j a†j ai = δi, j . The realizability of our model is proved by means of the AguiarMahajan bilinear form on the chambers of hyperplane arrangements. We show that, for suitable values of qi, j , the module generated by the particle states obtained by applying combinations of ai ’s and ai† ’s to a vacuum state |0 is an indefinite Hilbert module. Furthermore, we recover the extended Zagier’s conjecture established independently by Meljanac et al., and by Duchamp et al. Keywords Quon algebra · Indefinite hilbert module · Hyperplane arrangement Mathematics Subject Classification 05E15 · 81R10
1 Introduction Denote by C[qi, j ] the polynomial ring C qi, j i, j ∈ N∗ with variables qi, j . Quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons.
Communicated by Jamshid Moori. This research was funded by my mother.
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Hery Randriamaro [email protected] Lot II B 32 bis Faravohitra, 101, Antananarivo, Madagascar
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Bulletin of the Iranian Mathematical Society
Definition quon algebra is meant the free algebra A, equal to By multiparametric 1.1 C[qi, j ] ai i ∈ N∗ , and subject to the anti-involution † exchanging ai with ai† and to the commutation relations ai a†j = qi, j a†j ai + δi, j , where δi, j is the Kronecker delta. The multiparametric quon algebra is a generalization of the deformed quon algebra subject to the restriction qi, j = q¯ j,i independently studied by Bo˙zejko and Speicher [2, § 3], and by Meljanac and Svrtan [6, § 1.1]. Their algebra is in turn a generalization of the deformed quon algebra investigated by Speicher subject to the restriction qi, j = q j,i [8]. Then, his algebra is a generalization of the quon algebra introduced by Greenberg [5] and studied by Zagier [9, § 1] which is subject to the commutation relations ai a†j = q a†j ai + δi, j obeyed by the annihilation and creation operators of the quon particles, and generating a model of infinite statistics. Finally, the quon algebra is a generalization of the classical Bose and Fermi algebras corresponding to the restrictions q = 1 and q = −1, respectively, as well as of the intermediate case q = 0 suggested by Hegstrom and investigated by Greenberg [4]. In a Fock-like representation, the generators of A are the linear operators ai , ai† : V → V on an infinite dimensional C[qi, j ]-vector module V satisfying the commutation relations ai a†j = qi, j a†j
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