Quantum Groups, Coherent States, Squeezing and Lattice Quantum Mechanics

TL;DR

This paper develops a formalism connecting quantum groups, coherent states, and lattice quantum mechanics using the Fock--Bargmann representation and finite difference operators, emphasizing the role of $q$-deformation in discretized systems.

Contribution

It introduces a novel approach to incorporate the $q$-Weyl--Heisenberg algebra into analytic function theory and lattice quantum mechanics, highlighting the importance of $q$-deformation in discretized physical systems.

Findings

Coherent states are expressed via theta functions on the von Neumann lattice.

Finite difference operators are key in realizing the $q$-WH algebra.

A formalism for quantum mechanics on lattice systems is established.

Abstract

By resorting to the Fock--Bargmann representation, we incorporate the quantum Weyl--Heisenberg ($q$-WH) algebra into the theory of entire analytic functions. The main tool is the realization of the $q$--WH algebra in terms of finite difference operators. The physical relevance of our study relies on the fact that coherent states (CS) are indeed formulated in the space of entire analytic functions where they can be rigorously expressed in terms of theta functions on the von Neumann lattice. The r\^ole played by the finite difference operators and the relevance of the lattice structure in the completeness of the CS system suggest that the $q$--deformation of the WH algebra is an essential tool in the physics of discretized (periodic) systems. In this latter context we define a quantum mechanics formalism for lattice systems.