Theory of the Poincaré Group

Wigner introduced group theory to physics in 1926 and since then it has been an essential tool in almost all branches of theoretical physics. The study of space-time coordinate transformations and the construction of explicit representations of non-compact groups, particularly fundamental space-time symmetries in the four-dimensional Minkowski space is now of great interest. The Poincaré group occupies an important place here. The purpose of this book is to discuss the representations of the Poincaré group which is non-compact and which is neither simple nor semisimple. We study first the four-by-four Lorentz transformation matrices and the resulting Lie algebra. We then study the orbits and little groups of the Lorentz group, in preparation for constructing representations of the Poincaré group, the little group decomposition of the Poincaré group, and the Casimir operators for each little group. The problem of constructing unitary representations of Lorentz transformations is discussed as well as the finite-dimensional representations, which are non-unitary and commonly used in quantum field theory. We also discuss space and time reflections applicable to the representations frequently used in physics, with a particular emphasis on little groups.