Finite‐ and Infinite‐Dimensional Representations of the Lorentz Group

Finite‐ and infinite‐dimensional representations of the Lorentz group are discussed and various topics in which this group is currently in use are mentioned. The infinitesimal approach of finding representations is reviewed and all finite‐dimensional spinor representations of the Lorentz group are obtained. Infinite‐dimensional representations are then discussed, including the principal, complementary, and complete series of representations. A generalized Fourier transformation is introduced which enables one to use the global approach to representation theory so as to express infinite‐dimensional representations in terms of matrices. This method is shown to lead to a generalization of the spinor form of finite‐dimensional representation to the infinite‐dimensional case. However, whereas the usual spinor representations are nonunitary, the obtained new form describes both unitary and non‐unitary representations, depending on the choice of certain parameters appearing in the representation formula.