A Multiparametric Quon Algebra

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^{\dag} - q{i,j}, $\mathtt{a}$_j^{\dag} $\mathtt{a}$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 ${\mathtt{a}}_i^{\mathrm{†}}$'s to a vacuum state $\mathrm{\mid }0⟩$ is an indefinite Hilbert module. Furthermore, we refind the extended Zagier’s conjecture established independently by Meljanac et al. and by Duchamp et al.