Nonlinear canonical transformations and their representations in quantum mechanics

Abstract

In the last few years an extensive literature has developed on linear canonical tranformations and their representation in quantum mechanics. Applications of these results have been made to clustering theory in nuclei, problems of accidental degeneracy, etc. In the present paper we wish to turn our attention to nonlinear canonical transformations. We show that by dealing with appropriate functions fα (α=1,...,2n) of xi, pi (i=1,...,n) rather than with these variables themselves, we can in principle set unambiguously the equations that determine the representation in quantum mechanics of the canonical transformation under study. This result holds when the old and new functions fα have the same spectrum. We discuss specific examples when this last condition is satisfied: nonlinear canonical transformations in the radial variable that were obtained from projection of linear ones in higher-dimensional spaces; canonical transformations that take us from one Hamiltonian to another with the same spectrum, be this one continuous or discrete; canonical transformations that relate two sets of integrals of motion (which include the Hamiltonians) when we are dealing with phase spaces of dimensions higher than 2, etc. We discuss briefly, in the concluding section, the possibility of extending our analysis to canonical tranformations that do not conserve the spectrum of the relevant operators.