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Equivalence by Chernoff and evolution equations for density matrix and Wigner function for linear quantization

Leonid Andreevich Borisov, Yu. N. Orlov, Vsevolod Zhanovich Sakbaev

Published Jan 1, 2015• 0 citations
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Abstract We consider the inversion problem for linear quantization defined by an integral transformation relating the matrix of a quantum operator to its classical symbol. For an arbitrary linear quantization, we construct evolution equations for the density matrix and the Wigner function. It is shown that the Weyl quantization is the only one for which the evolution equation of the Wigner function is free of a quasi-probability source, which distinguishes this quantization as the only physically adequate one in the class under consideration. As an example, we give an exact stationary solution for the Wigner function of a harmonic oscillator with an arbitrary linear quantization, and construct a sequence of quantizations that approximate the Weyl quantization and tend to it in the weak sense so that the Wigner function remains positive definite.