Commuting position and momentum operators, exact decoherence, and emergent classicality

Inspired by an old idea of von Neumann, we seek a pair of commuting operators $X,P$ which are, in a specific sense, “close” to the canonical noncommuting position and momentum operators $x,p$. The construction of such operators is related to the problem of finding complete sets of orthonormal phase-space-localized states, a problem severely constrained by the Balian-Low theorem. Here these constraints are avoided by restricting attention to situations in which the density matrix is reasonably decohered (i.e., spread out in phase space). Commuting position and momentum operators are argued to be of use in discussions of emergent classicality from quantum mechanics. In particular, they may be used to give a discussion of the relationship between exact and approximate decoherence in the decoherent-histories approach to quantum theory.