Coherent States of the q--Canonical Commutation Relations

TL;DR

This paper investigates the structure of representations of q-canonical commutation relations, establishing conditions for their existence, classifying their types, and exploring their algebraic properties, including a q-analogue of the Cuntz algebra.

Contribution

It characterizes the representations of q-canonical relations, introduces a q-analogue of the Cuntz algebra, and analyzes the structure of these representations for different q values.

Findings

Representations exist iff the vector norm is ≤ 1.

For norms < 1, representations are unitarily equivalent to the Fock representation.

The universal C*-algebra has a largest proper ideal, leading to a q-analogue of the Cuntz algebra.

Abstract

For the $q$-deformed canonical commutation relations $a(f)a^\dagger(g) = (1-q)\,\langle f,g\rangle{\bf1}+q\,a^\dagger(g)a(f)$ for $f,g$ in some Hilbert space ${\cal H}$ we consider representations generated from a vector $\Omega$ satisfying $a(f)\Omega=\langle f,\phi\rangle\Omega$, where $\phi\in{\cal H}$. We show that such a representation exists if and only if $\Vert\phi\Vert\leq1$. Moreover, for $\Vert\phi\Vert<1$ these representations are unitarily equivalent to the Fock representation (obtained for $\phi=0$). On the other hand representations obtained for different unit vectors $\phi$ are disjoint. We show that the universal C*-algebra for the relations has a largest proper, closed, two-sided ideal. The quotient by this ideal is a natural $q$-analogue of the Cuntz algebra (obtained for $q=0$). We discuss the Conjecture that, for $d<\infty$, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting cases $q=\pm1$ we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.