Journal of Mathematical Physics

Linear Canonical Transformations and Their Unitary Representations

M. Moshińsky, C. Quesne

Published Aug 1, 1971• 841 citations

Abstract

We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.

DOI: 10.1063/1.1665805

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