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Wikipedia / Metaplectic 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, 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.

Motivation from quantum mechanics

One way to understand the metaplectic group is as the group needed to make linear canonical transformations act (linearly) on wave functions.

, and the linear transformations preserving the basic structure of Hamiltonian mechanics form the symplectic group, this classical canonical transformation is the identity.

does one recover the identity operator on wave functions. This shows that the natural quantum-mechanical object is not the symplectic group itself, but a double cover of it. That double cover is the metaplectic group.

This phenomenon is closely analogous to the role of the spin group as a double cover of the rotation group: in each case, a symmetry that is classically trivial after one full turn may still act nontrivially on quantum states. The metaplectic group is the version of this idea associated with symplectic geometry and canonical transformations rather than with Euclidean rotations.

Definition

The fundamental group of the symplectic Lie group Sp2n(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2n(R) and called the metaplectic group.

The metaplectic group Mp2n(R) is not a matrix group: it has no faithful finite-dimensional representationss. Therefore, the question of its explicit realization is nontrivial. It has faithful irreducible infinite-dimensional representations, such as the Weil representation described below.

It can be proved that if F is any local field other than C, then the symplectic group Sp2n(F) admits a unique perfect central extension with the kernel Z/2Z, the cyclic group of order 2, which is called the metaplectic group over F. It serves as an algebraic replacement of the topological notion of a cover used when . The approach through the notion of central extension is useful even in the case of real metaplectic group, because it allows a description of the group operation via a certain cocycle.

Explicit construction for n = 1

In the case , the symplectic group coincides with the special linear group SL2(R). This group biholomorphically acts on the complex upper half-plane by fractional-linear transformations, such as the Möbius transformation,

where

is a real 2-by-2 matrix with the unit determinant and z is in the upper half-plane, and this action can be used to explicitly construct the metaplectic cover of SL2(R).

The elements of the metaplectic group Mp2(R) are the pairs (g, and is a holomorphic function on the upper half-plane. The multiplication law is defined by:

. The map

is a surjection from Mp2(R) to SL2(R) which does not admit a continuous section. Hence, we have constructed a non-trivial 2-fold cover of the latter group.

Construction of the Weil representation

The existence of the Weil representation can be proven abstractly, as follows. The Heisenberg group has an irreducible unitary representation on a Hilbert space, that is,

with the center acting as multiplication by a given nonzero constant. The Stone–von Neumann theorem is another such representation, there exists an automorphism

  1. .

—more precisely, the action is only well-defined up to multiplication by a nonzero constant.

The automorphisms of the Heisenberg group (fixing its center) form the symplectic group. This is a projective representation, a homomorphism from the symplectic group to the projective. The general theory of projective representations gives an action of some central extension. This central extension can be taken to be a double cover, which is the metaplectic group.

Concretely, in the case of Mp2(R. In the Schrödinger model, the Heisenberg group acts by translations and by multiplication by characters. The corresponding Weil representation of the metaplectic group is generated by the Fourier transform, dilation operators, and multiplication by quadratic phase factors.

Generalizations

by any locally compact abelian group G whose Pontryagin dual (the group of characters) is isomorphic to G. The Hilbert space H is then the space of all L2 functions on G. The (analogue of) the Heisenberg group is generated by translations by elements of G, and multiplication by elements of the dual group (considered as functions from G to the unit circle). There is an analogue of the symplectic group acting on the Heisenberg group, and this action lifts to a projective representation on H. The corresponding central extension of the symplectic group is called the metaplectic group.

Some important examples of this construction are given by:

  • G is a vector space over the reals of dimension n. This gives a metaplectic group that is a double cover of the symplectic group Sp2n(R).

  • More generally G can be a vector space over any local field F of dimension n). This gives a metaplectic group that is a double cover of the symplectic group Sp2n(F).

  • G is a vector space over the adeles of a number field (or global field). This case is used in the representation-theoretic approach to automorphic forms.

  • G is a finite abelian group. The corresponding metaplectic group is then also finite. This case occurs in the theory of theta functions attached to lattices, where G is typically the discriminant group of an even lattice, equipped with its natural quadratic form.

  • A modern point of view on the existence of the linear (not projective) Weil representation over a finite field, namely, that it admits a canonical Hilbert space realization, was proposed by David Kazhdan. Using the notion of canonical intertwining operators suggested by Joseph Bernstein, such a realization was constructed by Gurevich-Hadani.

Generators

See also

Notes

References