Metaplectic representation

Definition

The metaplectic representation is the strongly continuous unitary representation

\mu:\operatorname{Mp}(2n,\mathbb R) \longrightarrow \mathcal U\bigl(L^2(\mathbb R^n)\bigr)

that implements the action of the metaplectic group on the Schrödinger representation of the Heisenberg group. Equivalently, its operators intertwine Weyl operators according to the underlying symplectic transformation. It descends only to a projective unitary representation of $\operatorname{Sp}(2n,\mathbb R)$ .

Concrete generators

Metaplectic operators are generated by unitary changes of variables, multiplication by quadratic phase functions, and a normalized Fourier transform. Different factorizations of a symplectic matrix can differ by sign at the operator level; the point of $\operatorname{Mp}(2n,\mathbb R)$ is to retain that sign as part of the group element.

Derived representation

The derived representation preserves the Schwartz space and gives

d\mu:\mathfrak{sp}(2n,\mathbb R) \longrightarrow \operatorname{End}(\mathcal S(\mathbb R^n)).

After multiplication by the convention-dependent factor involving $i\hbar$ , these infinitesimal operators are the Weyl quantizations of quadratic Hamiltonians. This realizes the exact quantization of polynomial observables of degree at most two that survives the Groenewold–Van Hove obstruction.

Terminology

“Oscillator representation,” “Segal–Shale–Weil representation,” and “Weil representation” are closely related names. Over the real numbers, authors may use them for the representation of the metaplectic group, for its even and odd irreducible summands, or for a larger representation involving the Heisenberg group. The group and normalization should be checked when comparing sources.

References

G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989. Publisher record. Relevant: §§4.1–4.3, construction and infinitesimal representation.
M. de Gosson, Symplectic Geometry and Quantum Mechanics, Birkhäuser, 2006. DOI record. Relevant: Chapters 6–7, the metaplectic group and representation.