Metaplectic group - Knowlpedia

Definition

The metaplectic group $\operatorname{Mp}(2n,\mathbb R)$ is the connected two-sheeted covering group of the real symplectic group $\operatorname{Sp}(2n,\mathbb R)$ . It fits into the central extension

1\longrightarrow \{\pm1\} \longrightarrow \operatorname{Mp}(2n,\mathbb R) \longrightarrow \operatorname{Sp}(2n,\mathbb R) \longrightarrow 1.

Its Lie algebra is canonically $\mathfrak{sp}(2n,\mathbb R)$ , but the covering is globally nontrivial.

Construction from the fundamental group

The symplectic group is connected and has fundamental group isomorphic to $\mathbb Z$ . Its connected covering groups correspond to subgroups of $\mathbb Z$ ; the subgroup $2\mathbb Z$ gives the metaplectic double cover. It is distinct from the universal covering group, whose kernel over $\operatorname{Sp}(2n,\mathbb R)$ is infinite cyclic.

Why the cover appears

The natural action of linear symplectic transformations on quantum states is only projective at the level of $\operatorname{Sp}(2n,\mathbb R)$ . Passing to $\operatorname{Mp}(2n,\mathbb R)$ resolves the sign ambiguity and gives the genuine metaplectic representation. This is the linear model for replacing a projective unitary representation by a representation of a central covering group.

References

G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989. Publisher record. Relevant: Chapter 4, the metaplectic representation.
M. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Birkhäuser, 2011. DOI record. Relevant: Chapters 7–8, the metaplectic group and its generators.