The Lorentz group

Abstract

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.