The Metaplectic Group

Part of the Pseudo-Differential Operators book series (PDO, volume 7)


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.


Unitary Operator Natural Projection Symplectic Group Maslov Index Symplectic Matrix 
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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics, NuHAGUniversity of ViennaViennaAustria

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