  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 non-commuting
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.
