A Multiparametric Quon Algebra

Abstract

The quon algebra is an approach to particle statistics introduced by Greenberg in order 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 ${\mathtt{a}}_i$, $i\in {\mathbb{N}}^{\ast }$ the set of positive integers, on an infinite dimensional module satisfying the $q_{i,j}$-mutator relations ${\mathtt{a}}_i{\mathtt{a}}_j^{†}-q_{i,j}{\mathtt{a}}_j^{†}{\mathtt{a}}_i={\delta }_{i,j}$. The realizability of our model is proved by means of the Aguiar-Mahajan bilinear form on the chambers of hyperplane arrangements. We show that, for suitable values of $q_{i,j}$, the module generated by the particle states obtained by applying combinations of ${\mathtt{a}}_i$'s and ${\mathtt{a}}_i^{†}$'s to a vacuum state $|0⟩$ is an indefinite Hilbert module. Furthermore, we refind the extended Zagier's conjecture established independently by Meljanac et al. and by Duchamp et al.

11 pages