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Annalen der Physik

Canonical Transformations in Quantum Mechanics

Joachim Schröter, M. Weström

Published Jan 1, 1971• 3 citations
Publisher

Abstract

Abstract The kinematics and dynamics of classical and quantum mechanical systems are discussed. In a mathematical strict way canonical transformations are introduced and the time development of quantum mechanical systems as well as the Schrödinger and Heisenberg picture are considered.
DOI
10.1002/andp.19714810108

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ppp-Mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how non-linear canonical transformations affect ppp-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of ppp-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.
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Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.