Exact Phase Space Localized Projectors from Energy Eigenstates

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.