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.The way in which one represents the spacetime symmetries may take many shapes depending on the theory at hand. While not being the present topic, some details will be provided in footnotes labeled "nb", and in the section applications. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, "If it turned out that a system could not be described by a quantum field theory, it would be a sensation; if it turned out it did not obey the rules of quantum mechanics and relativity, it would be a cataclysm." 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 1945, Harish-Chandra came to see Dirac in Cambridge. Harish-Chandra became convinced that theoretical physics was not the field he should be in. He had found an error in a proof by Dirac in his work on the Lorentz group. Dirac said, "I am not interested in proofs but only interested in what nature does." Harish-Chandra later wrote, "This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics, and I soon decided to move over to mathematics." Dirac did, however, suggest the topic of Harish-Chandra's thesis, the classification of the irreducible infinite-dimensional representations of the Lorentz group. See 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 of the full Lorentz group 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) of is obtained, and explicitly given in terms of action on a function space in representations of and . 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 principal series and the complementary series. Finally, the Plancherel formula for is given, and representations of are classified and realized for Lie algebras.
Finite-dimensional representations Representation theory of groups in general, and Lie groups in particular, is a very rich subject. The Lorentz group has some properties that makes it "agreeable" and others that make it "not very agreeable" within the context of representation theory; the group is simple and thus semisimple, but is not connected, and none of its components are simply connected. Furthermore, the Lorentz group is not compact.These facts can be found in most introductory mathematics and physics texts. See e.g. , and .
For finite-dimensional representations, the presence of semisimplicity means that the Lorentz group can be dealt with the same way as other semisimple groups using a well-developed theory. In addition, all representations are built from the irreducible ones, since the Lie algebra possesses the complete reducibility property.One says that a group has the complete reducibility property if every representation decomposes as a direct sum of irreducible representations. But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, cannot be dealt with in all the aspects as in the simple framework that applies to simply connected, compact groups. Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional unitary representations exist. Lack of simple connectedness gives rise to spin representations of the group. The non-connectedness means that, for representations of the full Lorentz group, time reversal and reversal of spatial orientation have to be dealt with separately.
History The development of the finite-dimensional representation theory of the Lorentz group mostly follows that of representation theory in general. Lie theory originated with Sophus Lie in 1873., 1890, 1893. Primary source. By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing. Primary source. In 1913 the theorem of highest weight for representations of simple Lie algebras, the path that will be followed here, was completed by Élie Cartan. Primary source. Richard Brauer was during the period of 1935–38 largely responsible for the development of the Weyl-Brauer matrices describing how spin representations of the Lorentz Lie algebra can be embedded in Clifford algebras. Primary source. The Lorentz group has also historically received special attention in representation theory due to its exceptional importance in physics (see History of infinite-dimensional unitary representations below). Mathematicians Hermann Weyl Primary source. Primary source. and Harish-Chandra Primary source. and physicists Eugene Wigner Primary source. and Valentine Bargmann Primary source.Bargmann was also a mathematician. He worked as Albert Einsteins assistant at the Institute for Advanced Study in Princeton (). made substantial contributions both to general representation theory and in particular to the Lorentz group. Primary source. Physicist Paul Dirac was perhaps the first to manifestly knit everything together in a practical application of major lasting importance with the Dirac equation in 1928. Primary source.Dirac suggested the topic of as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in (), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations ().
Lie algebra
The irreducible representations of the Lie algebra of the Lorentz group can be derived by factoring that Lie algebra into a direct product of two subalgebras. Each subalgebra is isomorphic to , and the irreducible representations of are labeled by nonnegative half-integers. Consequently, the irreducible representations of the Lorentz group's Lie algebra are labeled by ordered pairs of nonnegative half-integers.
This section addresses the irreducible complex linear representations of the complexification of the Lie algebra of the Lorentz group. A convenient basis for is given by the three generators of rotations and the three generators of boosts. They are explicitly given in conventions and Lie algebra bases.
The Lie algebra is complexified, and the basis is changed to the components of its two ideals
The components of and separately satisfy the commutation relations of the Lie algebra and, moreover, they commute with each other,
where are indices which each take values , and is the three-dimensional Levi-Civita symbol. Let and denote the complex linear span of and respectively.
One has the isomorphisms The rather mysterious looking third isomorphism is proved in chapter 2, paragraph 4.
where is the complexification of
The utility of these isomorphisms comes from the fact that all irreducible representations of , and hence all irreducible complex linear representations of are known. The irreducible complex linear representation of is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of
Unitarian trick
The Lie algebra is the Lie algebra of It contains the compact subgroup with Lie algebra The latter is a compact real form of Thus from the first statement of the unitarian trick, representations of are in one-to-one correspondence with holomorphic representations of
By compactness, the Peter–Weyl theorem applies to , and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of are precisely the tensor products of irreducible unitary representations of .This is an application of
By appeal to simple connectedness, the second statement of the unitarian trick is applied. The objects in the following list are in one-to-one correspondence: Holomorphic representations of Smooth representations of Real linear representations of Complex linear representations of
Tensor products of representations appear at the Lie algebra level as either ofTensor products of representations, of can, when both factors come from the same Lie algebra either be thought of as a representation of or .
where is the identity operator. Here, the latter interpretation, which follows from , is intended. The highest weight representations of are indexed by for . (The highest weights are actually , but the notation here is adapted to that of ) The tensor products of two such complex linear factors then form the irreducible complex linear representations of
Finally, the -linear representations of the real forms of the far left, , and the far right, When complexifying a complex Lie algebra, it should be thought of as a real Lie algebra of real dimension twice its complex dimension. Likewise, a real form may actually also be complex as is the case here. in are obtained from the -linear representations of characterized in the previous paragraph.
(μ, ν)-representations of sl(2, C) The complex linear representations of the complexification of obtained via isomorphisms in , stand in one-to-one correspondence with the real linear representations of The set of all real linear irreducible representations of are thus indexed by a pair . The complex linear ones, corresponding precisely to the complexification of the real linear representations, are of the form , while the conjugate linear ones are the . All others are real linear only. The linearity properties follow from the canonical injection, the far right in , of into its complexification. Representations on the form or are given by real matrices (the latter are not irreducible). Explicitly, the real linear -representations of are
where are the complex linear irreducible representations of and their complex conjugate representations. (The labeling is usually in the mathematics literature , but half-integers are chosen here to conform with the labeling for the Lie algebra.) Here the tensor product is interpreted in the former sense of . These representations are concretely realized below.