Exact Phase Space Localized Projectors from Energy Eigenstates

TL;DR

This paper introduces a method to construct exact phase space localized projectors from energy eigenstates, enabling precise decoherence and classical predictability in quantum systems, especially for the harmonic oscillator.

Contribution

It presents a novel approach to create exact phase space projectors using Hamiltonian eigenstates, improving upon approximate methods for analyzing quantum-classical transition.

Findings

Exact phase space projectors can be constructed from energy eigenstates.

For the harmonic oscillator, these projectors lead to exactly decoherent histories.

Classical evolution probabilities can be made exact in certain quantum systems.

Abstract

In investigations of the emergence of classicality from quantum theory, a useful step is the construction of quantum operators corresponding to the classical notion that the system resides in a region of phase space. The simplest such constructions, using coherent states, yield operators which are approximate projection operators -- their eigenvalues are approximately equal to 1 or 0. Such projections may be shown to have close to classical behaviour under time evolution and these results have been used to prove some useful results about emergent classicality in the decoherent histories approach to quantum theory. Here, we show how to use the eigenstates of a suitably chosen Hamiltonian to construct exact projection operators which are localized on regions of phase. We elucidate the properties of such operators and explore their time evolution. For the special case of the harmonic oscillator, the time evolution is particularly simple, and we find sets of phase space localized histories which are exactly decoherent for any initial state and have probability 1 for classical evolution. These results show how approximate decoherence of histories and classical predictability for phase space histories may be made exact in certain cases.