The Metaplectic Group

Abstract

The metaplectic group is a unitary representation of the double cover of the sym-plectic 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. 7.1 The metaplectic representation The idea behind the metaplectic representation of the symplectic group is that one can associate to every symplectic matrix a pair of unitary operators on L 2 (R n) differing by a sign. Technically this is achieved by constructing of a unitary representation of the (connected) double covering Sp 2 (2n, R) of Sp(2n, R). This representation (which is not irreducible, see Exercise 138 in Chapter 8) is called the metaplectic group and is denoted by Mp(2n, R). Equivalently, the sequence 0 −→ Z 2 −→ Mp(2n, R) −→ Sp(2n, R) −→ 0 is exact. In many texts the existence of the metaplectic representation is motivated by vague considerations about the uniqueness of the Schrödinger representation and the Heisenberg-Weyl operators T (z) which will be studied in Chapter 8. Following this argument there must exist, for every S ∈ Sp(2n, R) a unitary operator S such that S T (z) S −1 = T (Sz). However this relation certainly does not characterize precisely S since it is still true if we replace it by c S with |c| = 1. At best one obtains in this way a projective representation of the symplectic group. We are following closely de Gosson [67], Chapter 7. 79 M.A. de Gosson,