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Journal of Physics A Mathematical and General

Comment on quantum representations of non-bijective canonical transformations

Armando Antillón, B. Lesche, T. H. Seligman

Published Aug 1, 1982• 0 citations
Publisher

Abstract

A recent letter by J. Deenen on 'non-bijective canonical transformations in quantum mechanics' (see ibid., vol.14, p.L273, 1981) is analysed and found to be incomplete. The general representation problem of non-bijective canonical transformations is shown to involve infinite sums of the quotient and tensor product spaces defined by Plebanski and Seligman (1982).

DOI
10.1088/0305-4470/15/8/041

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A recent letter by J. Deenen on 'non-bijective canonical transformations in quantum mechanics' (see ibid., vol.14, p.L273, 1981) is analysed and found to be incomplete. The general representation problem of non-bijective canonical transformations is shown to involve infinite sums of the quotient and tensor product spaces defined by Plebanski and Seligman (1982).
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.]
In a previous paper, with a similar title, the authors showed that for bijective (i.e. one-to-one onto) canonical transformations they can obtain their quantum mechanical representations in the form of a kernel in the phase space of Wigner distribution functions and recover the classical expectation for this kernel in the limit to 0. For non-bijective canonical transformations the situation is more complex as the phase space acquires either a Riemann surface structure or requires the introduction of the concept of an ambiguity group. By discussing the non-bijective canonical transformation taking them from an oscillator of frequency kappa -1 , where kappa is an integer, to one of unit frequency, they see how the kernel is generalised to include the indices associated with the irreducible representations of the ambiguity group, i.e. the ambiguity spins. They obtain the kernel explicitly and show that in the limit to 0 they recover the form that they expect in the Riemann surface picture of their phase space. While they illustrate their analysis through the representation in Wigner distribution phase space of a specific non-bijective canonical transformation, the procedure is clearly extendable to the representations of general canonical transformations of this type.
Representations of nonlinear nonbijective canonical transformations in quantum mechanics are discussed. Due to the nonbijectivity the classical phase space has a Riemann sheet structure, and a family of partial isometries translating this structure into quantum mechanics is constructed. If a unitary representation is required, a new variable-the ambiguity spin-has to be introduced in order to recover bijectivity following the approach of Moshinsky (1981) and co-workers. The new degree of freedom is analysed in terms of multiboson operators. The application of this formalism to some non-classical states of light is discussed.