The Metaplectic Group
The metaplectic group is a unitary representation of the double cover of the symplectic group; it plays an essential role in Weyl pseudodifferential calculus, because it appears as a characteristic group of symmetries for Weyl operators. In fact – and this fact seems to be largely ignored in the literature – this property (called “symplectic covariance”) actually is characteristic (in a sense that will be made precise) of Weyl calculus. Metaplectic operators of course have many other applications; they allow us, for instance, to give explicit solutions to the time-dependent Schrödinger equation with quadratic Hamiltonian, as will be shown later, but they are also used with profit in optics, engineering, and last but not least, in quantum mechanics.
KeywordsUnitary Operator Natural Projection Symplectic Group Maslov Index Symplectic Matrix
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