Abstract
There has recently been a resurgence of interest in Born–Jordan quantization, which historically preceded Weyl’s prescription. Both mathematicians and physicists have found that this forgotten quantization scheme is actually not only of great mathematical interest, but also has unexpected application in operator theory, signal processing, and time-frequency analysis. In the present paper we discuss the applications to deformation quantization, which in its traditional form relies on Weyl quantization. Introducing the notion of “Bopp operator” which we have used in previous work, this allows us to obtain interesting new results in the spectral theory of deformation quantization.
Highlights
Deformation quantization is a popular framework for quantum mechanics among mathematical physicists
It was suggested by Moyal [33] and Groenewold [26], and put on a firm mathematical ground by Bayen et al [1,2]; later Kontsevich [29,30] extended the theory to Poisson manifolds
The star product is defined in physics by the suggestive formula a b = a exp i 2
Summary
Deformation quantization is a popular framework for quantum mechanics among mathematical physicists. There are many good reasons to believe that the Born and Jordan quantization scheme is the right one in physics (Kauffmann [28]); in addition, some very recent work of Boggiatto and his collaborators [3,4,5] shows that the Wigner formalism corresponding to Born–Jordan quantization is much more adequate in signal analysis than the traditional Weyl–Wigner approach. It allows to damp the appearance of unwanted “ghost” frequencies in spectrograms; numerical experiments confirm these theoretical facts.
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