Localization and the canonical commutation relations

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Let Wn (R) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of Wn (R) onto its image in a localization, or a quotient thereof, of U(so(2, q)), the universal enveloping algebra of so(2, q), forn depending upon q [1]. Here we treat the so(2, 1) case in complete detail. We establish an isomorphism of skew fields, specifically, D(so(2, 1)) ~ D(1,1)(R) where D1,1(R) is the fraction field of W1.1(R) ~ W1(R) ⊗ R(y) with R(y) being the ring of polynomials in the indeterminate y and D(so(2, 1)) is a certain extension of the skew field of fractions of U(so(2, 1)), which is described below. We give applications of this result to representations. In particular we are able to construct representations of W1 (R) out of representations of so (2, 1). Thus, we are able, for this lowest dimensional case, to obtain the canonical commutation relations and representations of them out of so(2, 1) symmetry. Using similar results in higher dimensions [1] we are able to construct representations of Wn(R) out of representations of so (2, q).

Original languageEnglish (US)
Title of host publicationLie Theory and Its Applications in Physics
EditorsVladimir Dobrev
PublisherSpringer New York LLC
Pages423-430
Number of pages8
ISBN (Print)9789811026355
DOIs
StatePublished - Jan 1 2016
EventProceedings of the 11th International Workshop on Lie Theory and Its Applications in Physics, 2015 - Varna, Bulgaria
Duration: Jun 15 2015Jun 21 2015

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume191
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

OtherProceedings of the 11th International Workshop on Lie Theory and Its Applications in Physics, 2015
CountryBulgaria
CityVarna
Period6/15/156/21/15

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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  • Cite this

    Moylan, P. (2016). Localization and the canonical commutation relations. In V. Dobrev (Ed.), Lie Theory and Its Applications in Physics (pp. 423-430). (Springer Proceedings in Mathematics and Statistics; Vol. 191). Springer New York LLC. https://doi.org/10.1007/978-981-10-2636-2_30