Commuting Position and Momentum Operators, Exact Decoherence and Emergent Classicality
 

J.J.Halliwell
 
The non-commutativity of position and momentum operators lies at the very heart of quantum theory, yet is completely absent in classical mechanics. Therefore, any account of the emergence of classical behaviour from quantum theory should explain how the position and momentum operators become effectively commuting under appropriate conditions. It is normally claimed that they become ``approximately'' commuting at sufficiently coarse-grained scales.  However, it is of interest to see if a more precise statement may be made. One possible approach to this is to construct, in quantum theory, 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. This idea was first considered by von Neumann in 1932, although no details are available, and more recent results suggest that there may be some difficulties with his results. The construction of the commuting operators X,P, requires the construction of orthonormal sets of phase space localized states, and the Balian-Low theorem puts serious restrictions on the from these construction may take. Here these difficulties are avoided by restricting attention to operators acting on density matrices which are reasonably decohered (i.e., spread out in phase space). The results may be valuable in the discussion of the relationship between exact and approximate decoherence in the decoherent histories approach to quantum theory.