Representation theory of the Lorentz group
The Lorentz group, a Lie group on which special relativity is based, has a wide variety of representations. Many of these representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics in the description of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory.This representation theory also provides the theoretical ground for the concept of spin, which, for a particle, can be either integer or half-integer in the unit of the reduced Planck constant ℏ. Quantum mechanical wave functions representing particles with half-integer spin are called spinors. The classical electromagnetic field has spin as well. It transforms under a representation with spin one. It enters into general relativity in that in small enough regions of spacetime, physics is that of special relativity.The group may also be represented in terms of a set of functions defined on the Riemann sphere. These are the Riemann P-functions, which are expressible as hypergeometric functions. The identity component SO(3,1)+ of the Lorentz group is isomorphic to the Möbius group, and hence any representation of the Lorentz group is necessarily a representation of the Möbius group and vice versa.The subgroup SO(3) with its representation theory form a simpler theory, but the two are related and both are prominent in theoretical physics as descriptions of spin, angular momentum, and other phenomena related to rotation.The adopted Lie algebra basis and conventions used are presented here.