The irreducible unitary representations of the extended Poincaré group in (1+1) dimensions

We prove that the extended Poincaré group in (1+1) dimensions P̄ is non-nilpotent solvable exponential, and therefore that it belongs to type I. We determine its first and second cohomology groups in order to work out a classification of the two-dimensional relativistic elementary systems. Moreover,...

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Bibliographic Details
Published in:Journal of mathematical physics 2004-03, Vol.45 (3), p.1156-1167
Main Authors: de Mello, R. O., Rivelles, V. O.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical methods in physics
Mathematics
Physics
Sciences and techniques of general use
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Summary:We prove that the extended Poincaré group in (1+1) dimensions P̄ is non-nilpotent solvable exponential, and therefore that it belongs to type I. We determine its first and second cohomology groups in order to work out a classification of the two-dimensional relativistic elementary systems. Moreover, all irreducible unitary representations of P̄ are constructed by the orbit method. The most physically interesting class of irreducible representations corresponds to the anomaly-free relativistic particle in (1+1) dimensions, which cannot be fully quantized. However, we show that the corresponding coadjoint orbit of P̄ determines a covariant maximal polynomial quantization by unbounded operators, which is enough to ensure that the associated quantum dynamical problem can be consistently solved, thus providing a physical interpretation for this particular class of representations.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.1644901