A Differential Characterization of the Metaplectic Kernel

We present a derivation of the integral kernel of a metaplectic representation operator in the Bargmann-Fock model based solely on its defining intertwining property with the Heisenberg representation. Expressing a metaplectic operator as an integral operator transforms the corresponding infinitesimal intertwining identities into a system of first-order partial differential equations satisfied by its kernel. We show that this system determines the kernel (up to a unit scalar) which is precisely the classical Gaussian associated with the symplectic transformation. A similar result is obtained for the Schrödinger model.