Abstract
The Heisenberg–Weyl and Grossmann–Royer operators allow us to define in a particular simple way two classical objects from symplectic harmonic analysis, namely the cross-ambiguity and Wigner functions, which are symplectic Fourier transforms of each other. Wigner introduced the eponymic distribution in 1932 as a substitute for a phase space probability density, but he did that in an ad hoc way, a kind of “lucky guess” one could say. It has since then been realized that the Wigner distribution (and its companion, the cross-ambiguity function) have a very natural meaning in Weyl calculus, and that they can be simply defined in terms of the Grossmann–Royer and Heisenberg–Weyl operators of last chapter.
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© 2011 Springer Basel AG
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de Gosson, M.A. (2011). Cross-ambiguity and Wigner Functions. In: Symplectic Methods in Harmonic Analysis and in Mathematical Physics. Pseudo-Differential Operators, vol 7. Springer, Basel. https://doi.org/10.1007/978-3-7643-9992-4_9
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DOI: https://doi.org/10.1007/978-3-7643-9992-4_9
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