Chapter 11 - The Lorentz group

Abstract

We begin by the definition of the Minkowski space-time as a 4D real space with a non-Euclidean inner product defined on it. The Lorentz group is defined as the group of transformations leaving invariant this inner product. We then introduce the important subgroups of the Lorentz group. Then we study the generators of the Lorentz group SO(1,3) and the Lie algebra so(1,3). By exponentiating, we get the general boosts. We then study the so-called imaginary and complex generators. Then we briefly remark on the representations of the Lorentz group. Then we study the transformations of scalar functions and vector fields. Then the group SL(2,C) is introduced and the relation with the Lorentz group is studied. We also remark on the dependence on the velocity of light and the relation to the Galilean group. Finally we study the Poincaré group as the isometry group of the Minkowski space-time, its Lie algebra, and the relation to central extension of the Galilean group.