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arXiv:1906.02503 · Applied and numerical harmonic analysis

Linear Perturbations of the Wigner Transform and the Weyl Quantization

Dominik Bayer, Elena Cordero, Karlheinz Gröchenig, S. Ivan Trapasso

Published Jan 1, 2020• 26 citations
arXivHTMLPDF
DOI
10.1007/978-3-030-36138-9_5

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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.
Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.