Canonical transformations and non-unitary evolution

Abstract

We test the idea that transformations which, at the classical level, can be interpreted as evolutions are represented within quantum mechanics by unitary operators. To this end, we consider non-trivial canonical transformations which leave invariant the form of the Hamilton function of a system. We demonstrate that infinite families of such transformations exist for a variety of familiar conservative systems of one degree of freedom. We show how the precise form of integral equations for the stationary state wavefunctions implied by the existence of these canonical transformations can be pinned down by exploiting the algebra of the transformations and a symmetry of their generating functions. We recover several integral equations found in the literature on standard special functions of mathematical physics. We find that when one of the classical canonical transformations we consider is non-linear, its quantum implementation is non-unitary. We end with some comments on the implications of our findings for semiclassical studies and a brief discussion relevant to string theory of the generalization to scalar field theories in 1 + 1 dimensions.