Canonical Transformations in Quantum Mechanics

TL;DR

This paper explores how elementary canonical transformations can be implemented in quantum mechanics, showing their role in generating the full algebra and solving differential equations through sequences of transformations, linking quantum integrability to these sequences.

Contribution

It demonstrates that elementary canonical transformations have quantum implementations and can generate the entire canonical algebra, providing a new perspective on quantum integrability.

Findings

Elementary canonical transformations have quantum finite implementations.

Sequences of elementary transformations can trivialize super-Hamiltonians.

Quantum integrability is linked to the existence of transformation sequences.

Abstract

Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can, in principle, be realized quantum mechanically as a product of these transformations. It is found that the intertwining of two super-Hamiltonians is equivalent to there being a canonical transformation between them. A consequence is that the procedure for solving a differential equation can be viewed as a sequence of elementary canonical transformations trivializing the super-Hamiltonian associated to the equation. It is proposed that the quantum integrability of a system is equivalent to the existence of such a sequence.