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A description of quantum mechanics in phase space a common ambit with classical mechanics, through the Wigner map from Hilbert space. May be used to address Quantum Mechanics in phase space, the star product binary operation controlling composition of observables, and Wigner, Husimi, and other distribution functions in phase space.

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Classical limit of Moyal bracket in integral representation

It is well-known that the Poisson bracket can be recovered out of the Moyal bracket under the limit when $\hbar$ goes to zero $$\lim_{\hbar\rightarrow 0} \lbrace f,g\rbrace_M=\lbrace f,g \rbrace_P.$$ …