Intermediate-statistics quantum bracket, coherent state, oscillator, and representation of angular momentum (su(2)) algebra

TL;DR

This paper introduces an intermediate-statistics quantum bracket, constructs related coherent states and oscillators, and explores their algebraic representations, bridging bosonic and fermionic cases with new mathematical tools.

Contribution

It develops a novel intermediate-statistics quantum bracket, constructs corresponding coherent states and oscillators, and extends the representation of angular momentum algebra.

Findings

The intermediate-statistics quantum bracket generalizes existing brackets.

Constructed intermediate-statistics coherent states and oscillators.

Derived the energy spectrum of the intermediate-statistics oscillator.

Abstract

In this paper, we first discuss the general properties of an intermediate-statistics quantum bracket, $[ u,v]_{n}=uv-e^{i2\pi /(n+1)}vu$, which corresponds to intermediate statistics in which the maximum occupation number of one quantum state is an arbitrary integer, $n$. A further study of the operator realization of intermediate statistics is given. We construct the intermediate-statistics coherent state. An intermediate-statistics oscillator is constructed, which returns to bosonic and fermionic oscillators respectively when $n\to \infty $ and $n=1$. The energy spectrum of such an intermediate-statistics oscillator is calculated. Finally, we discuss the intermediate-statistics representation of angular momentum ($su(2)$) algebra. Moreover, a further study of the operator realization of intermediate statistics is given in the Appendix.