A Multiparametric Quon Algebra

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 aia†j−qi,ja†jai=δi,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 qi,j , the module generated by the particle states obtained by applying combinations of ai ’s and a†i ’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.