Heisenberg–Weyl and Grossmann–Royer Operators
The Heisenberg–Weyl operators (also sometimes called simply “Heisenberg operators”) are in a sense the easiest way to access quantum mechanics, because their definition can be understood in terms of a simple Hamiltonian dynamics: they are the time-one evolution operator for the quantized displacement Hamiltonian. One can actually also define these operators in terms of the phase function of a Lagrangian manifold without invoking any quantization at all; we will not use this approach here and refer the interested reader to Chapter 5 in de Gosson ). Together with their cousins, the Grossmann–Royer operators, the Heisenberg–Weyl operators play a key role in the theory of Weyl pseudo-differential operators, and moreover allow us to simplify many statements and proofs. In particular they allow a neat definition of the cross-ambiguity and Wigner transforms as we will see in Chapter 9. We will also briefly discuss the notion of Weyl–Heisenberg frame, also called Gabor frame in time-frequency analysis.
KeywordsHeisenberg Group Riesz Basis Tight Frame Gabor Frame Frame Operator
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