Journal article
REPRESENTATIONS OF LIE ALGEBRAS BUILT OVER HILBERT SPACE
Infinite dimensional analysis, quantum probability, and related topics, Vol.14(3), pp.419-442
09/2011
DOI: 10.1142/S0219025711004468
Abstract
Starting with a complex Hilbert space, using inductive limits, we build Lie algebras, and find families of representations. They include those often studied in mathematical physics in order to model quantum statistical mechanics or quantum fields. We explore natural actions on infinite tensor algebras T(H) built with a functorial construction, starting with a fixed Hilbert space H. While our construction applies also when H is infinite-dimensional, the case with N ≔ dim H finite is of special interest as the symmetry group we consider is then a copy of the non-compact Lie group U(N, 1). We give the tensor algebra T(H) the structure of a Hilbert space, i.e. the unrestricted infinite tensor product Fock space [Formula: see text]. The tensor algebra T(H) is naturally represented as acting by bounded operators on [Formula: see text], and U (N, 1) as acting as a unitary representation. From this we built a covariant system, and we explore how the fermion, the boson, and the q on Hilbert spaces are reduced by the representations. In particular we display the decomposition into irreducible representations of the naturally defined U (N, 1) representation.
Details
- Title: Subtitle
- REPRESENTATIONS OF LIE ALGEBRAS BUILT OVER HILBERT SPACE
- Creators
- PALLE E. T JORGENSEN - Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, USA
- Resource Type
- Journal article
- Publication Details
- Infinite dimensional analysis, quantum probability, and related topics, Vol.14(3), pp.419-442
- DOI
- 10.1142/S0219025711004468
- ISSN
- 0219-0257
- eISSN
- 1793-6306
- Language
- English
- Date published
- 09/2011
- Academic Unit
- Mathematics
- Record Identifier
- 9983985867102771
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