×

A deformed quon algebra. (English) Zbl 1464.05357

Summary: The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators \(a_{i,k}, (i, k) \in \mathbb{N}^\ast \times [m]\), on an infinite dimensional vector space satisfying the deformed \(q\)-mutator relations \(a_{j, l} a_{i,k}^\dagger = qa_{i,k}^\dagger a_{j,l} + q^{\beta_{k,l}} \delta_{i,j}\) We prove the realizability of our model by showing that, for suitable values of \(q\), the vector space generated by the particle states obtained by applying combinations of \(a_{i,k}\)’s and \(a_{i,k}^\dagger\)’s to a vacuum state \(|0\rangle\) is a Hilbert space. The proof particularly needs the investigation of the new statistic cinv and representations of the colored permutation group.

MSC:

05E16 Combinatorial aspects of groups and algebras
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
15A15 Determinants, permanents, traces, other special matrix functions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] O.W. Greenberg: Example of Infinite Statistics. Physical Review Letters 64 (7) (1990) 705. · Zbl 1050.81571
[2] O.W. Greenberg: Particles with small Violations of Fermi or Bose Statistics. Physical Review D 43 (12) (1991) 4111.
[3] D. Zagier: Realizability of a Model in Infinite Statistics. Communications in Mathematical Physics 147 (1) (1992) 199-210. · Zbl 0789.47042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.