Representations of canonical commutation relations describing infinite coherent states

TL;DR

This paper explores the mathematical structure and physical implications of infinite-volume coherent states of photon fields, revealing their random phase nature, associated Hilbert space representations, and their decoherence effects on small quantum systems.

Contribution

It introduces a new framework for representing infinite coherent states with random phases using stochastic integrals and analyzes their impact on quantum decoherence.

Findings

Infinite volume coherent states have random phases modeled by stochastic integrals.

Initial phase distributions tend to become uniform under free field dynamics.

Decoherence caused by infinite coherent states is faster than thermal states, indicating classical behavior.

Abstract

We investigate the infinite volume limit of quantized photon fields in multimode coherent states. We show that for states containing a continuum of coherent modes, it is mathematically and physically natural to consider their phases to be random and identically distributed. The infinite volume states give rise to Hilbert space representations of the canonical commutation relations which we construct concretely. In the case of random phases, the representations are random as well and can be expressed with the help of It\^o stochastic integrals. We analyze the dynamics of the infinite state alone and the open system dynamics of small systems coupled to it. We show that under the free field dynamics, initial phase distributions are driven to the uniform distribution. We demonstrate that coherences in small quantum systems, interacting with the infinite coherent state, exhibit Gaussian time decay. The decoherence is qualitatively faster than the one caused by infinite thermal states, which is known to be exponentially rapid only. This emphasizes the classical character of coherent states.