On predicting correlations from Wigner functions

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Abstract
A proposal for obtaining predictions of correlations between positions and momenta from the peaks of the Wigner function is studied. An implementation of this proposal based on approximate Wigner functions of a special form obtained from semiclassical wave functions is shown to be unreliable because the exact Wigner functions do not have this form. This approach has been used to argue that classical general relativity is predicted from the semiclassical wave function of the Universe, and this must now be reconsidered. A new measure of correlation, involving projection onto coherent states, is proposed for predicting classical correlations from a general Wigner function. In the harmonic oscillator, it predicts a correlation of position and momentum given by the Hamiltonian equated to the classical energy.