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Linear canonical transformations and their unitary representations

par Moshinsky, Marcos;Quesne, Christiane
Référence Journal of mathematical physics, 12, 8, page (1772-1780)
Publication Publié, 1971

Article révisé par les pairs

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Résumé :

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) X script O sign(q). An application to the calculation of matrix elements is given in a following paper.