A Deformed Quon Algebra

TL;DR

This paper introduces a generalized deformed quon algebra with new $q$-mutator relations, expanding particle statistics models by proving their realizability within a Hilbert space framework.

Contribution

It develops a novel deformation of the quon algebra using $q$-mutator relations and demonstrates its mathematical realizability in quantum state spaces.

Findings

Established the realizability of the deformed quon algebra in a Hilbert space.

Introduced a new statistic called $ exttt{cinv}$ for analyzing the algebra.

Connected the algebra's structure to representations of the colored permutation group.

Abstract

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 $a_{i,k}$, $(i,k) \in \mathbb{N}^* \times [m]$, on an infinite dimensional vector space satisfying the deformed $q$-mutator relations $a_{j,l} a_{i,k}^{\dag} = q a_{i,k}^{\dag} a_{j,l} + q^{\beta_{-k,l}} \delta_{i,j}$. 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 $a_{i,k}^{\dag}$'s to a vacuum state $|0\rangle$ is a Hilbert space. The proof particularly needs the investigation of the new statistic $\mathtt{cinv}$ and representations of the colored permutation group.