GENERATING FUNCTIONS FOR AFFINE SYMPLECTIC GROUP

The generating function notion is used to give a representation of the inhomogeneous symplectic group as group of affine canonical transformations. Then the classical action for linear mechanical systems, the Hamiltonians of which belong to the algebra h sp(2 n, R), is deduced; it is explicitely constructed for all the Hamiltonians belonging to some particular subalgebras of h sp(2 n, R). The metaplectic representation of W Sp(2 n, R) on L2(R) and the solutions of the Schrödinger equation for linear systems are also obtained in terms of generating functions. The Maslov index is explicitly constructed for the quantum corresponding sets of Hamiltonians considered in the classical case. © 1978 Springer-Verlag.