Weyl-Wigner representation

In this section we advocate a far more advantageous route to studying conceptual features of the classical-quantum correspondence, and indeed for each mechanics independently, in which phase space distributions are used in both classical and quantum mechanics, that is, classical Liouville dynamics50 in the former and the Wigner-Weyl representation in the latter. This approach provides, as will be demonstrated, powerful conceptual insights into the relationship between classical and quantum mechanics. The essential point of this section is easily stated using similar mathematics in both quantum and classical mechanics results in a similar qualitative picture of the dynamics. 

As discussed subsequently, introduction of the standard p, q representation in classical mechanics, and of the Wigner-Weyl representation in quantum mechanics, defines densities p(p,q) = (p,q p) that both lie in the same Hilbert space. Thus, the essential difference between quantum and classical mechanics... 

The Wigner-Weyl formulation of quantum mechanics provides an appropriate quantum analogue to classical phase space dynamics and consists of a particular representation of quantum mechanics. 

The Wigner-Weyl representation of quantum mechanics is that obtained by the expansion of operators in an orthogonal set of operators A(p,q) the resultant p p.q) lie within a Hilbert space of L2 functions. Specifically,51... 

The algebra of the Wigner-Weyl representation has been extensively reviewed51 and we only call attention to the form that Eqs. (3.5) and (3.6) take in this representation. The Liouville equation becomes... 

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