Representation theory of the Lorentz group

The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.

The development of the representation theory has historically followed the development of the more general theory of representation theory of semisimple groups, largely due to Élie Cartan and Hermann Weyl, but the Lorentz group has also received special attention due to its importance in physics. Notable contributors are physicist E. P. Wigner and mathematician Valentine Bargmann with their Bargmann–Wigner program, one conclusion of which is, roughly, a classification of all unitary representations of the inhomogeneous Lorentz group amounts to a classification of all possible relativistic wave equations. The classification of the irreducible infinite-dimensional representations of the Lorentz group was established by Paul Dirac's doctoral student in theoretical physics, Harish-Chandra, later turned mathematician, in 1947. Closely related work was published independently by Bargmann and Israel Gelfand together with Mark Naimark in the same year.

The full theory of the finite-dimensional representations of the Lie algebra of the Lorentz group is deduced using the general framework of the representation theory of semisimple Lie algebras. The finite-dimensional representations of the connected component ${\displaystyle {\text{SO}}(3;1)^{+}}$ of the full Lorentz group O(3; 1) are obtained by employing the Lie correspondence and the matrix exponential. The full finite-dimensional representation theory of the universal covering group (and also the spin group, a double cover) ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ of ${\displaystyle {\text{SO}}(3;1)^{+}}$ is obtained, and explicitly given in terms of action on a function space in representations of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ and ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$. The representatives of time reversal and space inversion are given in space inversion and time reversal, completing the finite-dimensional theory for the full Lorentz group. The general properties of the (m, n) representations are outlined. Action on function spaces is considered, with the action on spherical harmonics and the Riemann P-functions appearing as examples. The infinite-dimensional case of irreducible unitary representations is realized for the ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ principal series and the complementary series. Finally, the Plancherel formula for ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ is given, and representations of SO(3, 1) are classified and realized for Lie algebras.