Chapter

The Poincaré and Liouville groups

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662920.003.0020
The Poincaré and Liouville groups

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The Poincaré and Liouville groups are defined. The Poincaré group is the universal cover of the group of transformations that leave invariant the distance between two points in space-time. This means that it is the semidirect product of the group of translations and the universal cover of the proper orthochronous Lorentz transformations. There are two Poincaré Casimirs: the square of the momentum four-vector and the square of the Pauli–Lubanski four-vector. The irreducible representations of the Poincaré group come in three classes corresponding to the momentum four-vector being, time-like, light-like and space-like. This can be rephrased in terms of the little group. All known massive elementary particles belong to class one, all known massless particles belong to class two. The Poincaré group is non-compact so the unitary representations are infinite-dimensional except for a subset of representations in class 2, which are one-dimensional. This feat is accomplished by representing several generators by zero. Extending the Poincaré group by dilations results in the Weyl group, and extending it further by special conformal transformations results in the Liouville group. In conclusion, the Virasoro and Kac–Moody algebras are briefly mentioned. Biographical notes on Poincaré, Pauli, Lubanski, Kac and Moody are given.

Keywords: Poincaré group; Liouville group; Kac–Moody algebras; semidirect product; Poincaré Casimir; Pauli–Lubanski four-vector; little group; dilations; special conformal transformations

Chapter.  5709 words. 

Subjects: Mathematical and Statistical Physics

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