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Journal of Mathematical Physics

Sequences of point transformations and linear canonical transformations in classical and quantum mechanics

F. Leyvraz, T. H. Seligman

Published Nov 1, 1989• 18 citations
Publisher

Abstract

Sequences of point transformations and canonical linear transformations are considered in classical and quantum mechanics. It is shown that the unitary representations of such transformations can be obtained, in general, in the sense that in the limit, classical behavior is retrieved. In the particular case of one point transformation combined with two linear transformations, the results found in this way are exact. A new class of differential equations is thereby solved by quadratures.

DOI
10.1063/1.528531

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Sequences of point transformations and canonical linear transformations are considered in classical and quantum mechanics. It is shown that the unitary representations of such transformations can be obtained, in general, in the sense that in the limit, classical behavior is retrieved. In the particular case of one point transformation combined with two linear transformations, the results found in this way are exact. A new class of differential equations is thereby solved by quadratures.
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.
Quantum canonical transformations are defined algebraically outside of a Hilbert space context. This generalizes the quantum canonical transformations of Weyl and Dirac to include non-unitary transformations. The importance of non-unitary transformations for constructing solutions of the Schr\\"odinger equation is discussed. 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 be realized quantum mechanically as a product of these transformations. Each transformation corresponds to a familiar tool used in solving differential equations, and the procedure of solving a differential equation is systematized by the use of the canonical transformations. Several examples are done to illustrate the use of the canonical transformations. [This is an extensively revised version of hep-th-9205080: the first third of the paper is new material; the notation has been simplified, and further discussion has been added to the remainder.]
First a review is presented of the definition and properties of classical and quantum canonical transformations, from the point of view of a current program on the role of canonical transformations in quantum mechanics. The groups and the corresponding infinitesimal algebras are explored. The subgroup of point transformations (i.e. canonical transformations between pairs of conjugate observables, where one of a pair is function only of one of the other pair) is of special interest since it is a group of transformations for which classical and quantum mechanics follow each other. Only the group of inhomogeneous symplectic transformations has similar characteristics. Point transformations are treated in detail and a principal series of unitary representations is constructed. (auth)
The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.
Quantum canonical transformations are defined algebraically outside of a Hilbert space context. This generalizes the quantum canonical transformations of Weyl and Dirac to include non-unitary transformations. The importance of non-unitary transformations for constructing solutions of the Schrödinger equation is discussed. 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 be realized quantum mechanically as a product of these transformations. Each transformation corresponds to a familiar tool used in solving differential equations, and