Metaplectic group

"Weil representation" redirects here. For representations of the Weil group, see Galois module § Representations of the Weil group.

In mathematics and mathematical physics, the metaplectic group is the group that describes how the basic symmetries of classical mechanics act in quantum mechanics. More precisely, the symplectic group consists of the linear changes of position and momentum that preserve the form of Hamiltonian mechanics; equivalently, it is the group of canonical transformations that are linear in position and momentum. When one tries to make those same transformations act on wavefunctions, one is naturally led not to the symplectic group itself but to a closely related two-fold cover of it, called the metaplectic group. For a symplectic space of dimension , it is usually denoted by Mp2n.

The metaplectic group is thus a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence.

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