Chapter

The Lorentz group

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/9780199662920.003.0019
The Lorentz group

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The Lorentz group is defined. Special relativity is viewed as the statement that the laws of Physics are invariant under rotations in a four-dimensional space-time. These generalized rotations leave invariant a quadratic form with an indefinite metric, which results in the Lorentz group being non-compact. Its six generators are the ordinary angular momentum J and the boosts N, which are Hermitian in a unitary representation. By identifying the group of proper orthochronous Lorentz transformations with SO0(3,1) the commutation relations of J and N and the expressions for the two Lorentz Casimirs follow. It is shown the covering group of SO0(3,1) is SL(2,C). Matrix elements of N are calculated with the help of the Wigner–Eckart theorem and the principal series and complementary series of infinite-dimensional unitary representations is described. Finite-dimensional non-unitary representations are obtained and used to describe the relativistic wave equations of Klein–Gordon, Dirac, Weyl, Proca and Maxwell. Biographical notes on Minkowski, Klein, Gordon, Dirac and Proca are given.

Keywords: Lorentz group; indefinite metric; non-compact; boost; Lorentz Casimirs; Wigner–Eckart theorem; principal and complementary series; relativistic wave equations

Chapter.  4959 words. 

Subjects: Mathematical and Statistical Physics

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