# Non-bijective canonical transformations and their representations in quantum mechanics

Ontology type: schema:Chapter

### Chapter Info

DATE

1978

AUTHORS

P. Kramer , M. Moshinsky , T. H. Seligman

ABSTRACT

In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are non-bijective, i.e. not one to one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one-dimensional oscillator of frequency K−1 into one of frequency k−1 where k, K are relatively prime integers. For the particular case k = 1, the mapping of the original phase space (x,p) onto the new one $$(\bar x,\bar p)$$is K to 1 and the equivalent points in (x,p) are related by a cyclic group CK of linear canonical transformations. When formulating this problem in Bargmann Hilbert space the canonical transformation can be related with the conformal transformation w = zK which again is K to 1 and where a group CK also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with it suggests that while we can take in the original Bargmann Hilbert space a single component function, in the new Bargmann Hilbert space we must take a K component one. In this way we can map in a one to one fashion the states and operators in the old and new Bargmann Hilbert spaces. When translating these results to ordinary Hilbert space we get in an ambiguous way the quantization of the observables appearing in the equations that determine the representation of the classical canonical transformation relating oscillators of frequencies K−1 and k−1. Furthermore we also get the solutions of these equations, and the resulting representation is unitary. More... »

PAGES

521-521

### Book

TITLE

Group Theoretical Methods in Physics

ISBN

978-3-540-08848-6

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/3-540-08848-2_65

DOI

http://dx.doi.org/10.1007/3-540-08848-2_65

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1015928625

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