# Emergent Classicality via Commuting Position and Momentum Operators

## Prof. Jonathan Haliwell, Imperial College, London

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 form 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.

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