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 2 n {\displaystyle 2n} , 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. In classical mechanics, the phase space of a system with one degree of freedom has coordinates ( q , p ) {\displaystyle (q,p)} , and the linear transformations preserving the basic structure of Hamiltonian mechanics form the symplectic group. For example, the Hamiltonian of the one-dimensional harmonic oscillator,
H ( q , p ) = 1 2 ( q 2 + p 2 ) , {\displaystyle H(q,p)={\tfrac {1}{2}}(q^{2}+p^{2}),}
generates rotations of phase space:
( q , p ) ↦ ( q cos t + p sin t , − q sin t + p cos t ) . {\displaystyle (q,p)\mapsto (q\cos t+p\sin t,\,-q\sin t+p\cos t).}
After a full turn, at t = 2 π {\displaystyle t=2\pi } , this classical canonical transformation is the identity. In quantum mechanics, however, the corresponding time evolution acts on wave functions by a unitary operator. For the harmonic oscillator, with Hamiltonian
H ^ = 1 2 ( P 2 + Q 2 ) , {\displaystyle {\hat {H}}={\tfrac {1}{2}}(P^{2}+Q^{2}),}
the evolution is
U ( t ) = e − i t H ^ . {\displaystyle U(t)=e^{-it{\hat {H}}}.}
If ψ 0 ( x ) = π − 1 / 4 e − x 2 / 2 {\displaystyle \psi _{0}(x)=\pi ^{-1/4}e^{-x^{2}/2}} is the ground-state wave function, then
U ( t ) ψ 0 = e − i t / 2 ψ 0 , {\displaystyle U(t)\psi _{0}=e^{-it/2}\psi _{0},}
because 2 i ∂ t ψ − ∂ x 2 ψ + x 2 ψ = 0 {\displaystyle 2i\partial _{t}\psi -\partial _{x}^{2}\psi +x^{2}\psi =0} for ψ = e − x 2 / 2 − i t / 2 {\displaystyle \psi =e^{-x^{2}/2-it/2}} . In particular,
U ( 2 π ) ψ 0 = − ψ 0 . {\displaystyle U(2\pi )\psi _{0}=-\psi _{0}.}
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