Coherent States and Generalized Hermite Polynomials for fractional statistics -- interpolating from fermions to bosons

Abstract

This article develops the algebraic structure that results from the $\theta$-commutator $\alpha \beta -e^{i\theta }\beta \alpha =1$ that provides a continuous interpolation between the Clifford and Heisenberg algebras. We first demonstrate the most general geometrical picture, applicable to all values of $N$. After listing the properties of this Hilbert space, we study the generalized coherent states that result when ${\xi }^N=0$, for $N\ge 2$. We also solve the generalized harmonic oscillator problem and derive generalized versions of the Hermite polynomials for general $N$. Some remarks are made to connect this study to the case of anyons. This study represents the first steps towards developing an anyonic field theory.