Weyl quantization and metaplectic representation

Abstract

The formal expansion defining the twisted exponential of an element of the Lie algebra ℋ _n □ (⊕ⁿ Sp(2, ℝ)) can be summed and this result is used to explicitly obtain the classical function u _t corresponding to an evolution operator associated to a quantum Hamiltonian belonging to the above mentioned Lie algebra.

Then, by applying the Weyl quantization procedure to u _t we get a representation of the group W _n □ (⊕ⁿ Sp(2, ℝ)) in terms of integral operators, the kernels of which are expressed by means of the classical action. The family u _t being only locally defined, it must be considered as a distribution on the classical phase space in order to get the metaplectic representation.