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 SO ( 3 ; 1 ) + {\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) SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} of SO ( 3 ; 1 ) + {\displaystyle {\text{SO}}(3;1)^{+}} is obtained, and explicitly given in terms of action on a function space in representations of SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} and s l ( 2 , C ) {\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 SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} principal series and the complementary series. Finally, the Plancherel formula for SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} is given, and representations of SO(3, 1) 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. 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. 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. By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing. 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. 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. 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 and Harish-Chandra and physicists Eugene Wigner and Valentine Bargmann made substantial contributions both to general representation theory and in particular to the Lorentz group. 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.
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 s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} , and the irreducible representations of s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} are labeled by nonnegative half-integers. Consequently, the irreducible representations of the Lorentz group's Lie algebra are labeled by ordered pairs ( m , n ) {\displaystyle (m,n)} of nonnegative half-integers. This section addresses the irreducible complex linear representations of the complexification s o ( 3 ; 1 ) C {\displaystyle {\mathfrak {so}}(3;1)_{\mathbb {C} }} of the Lie algebra s o ( 3 ; 1 ) {\displaystyle {\mathfrak {so}}(3;1)} of the Lorentz group. A convenient basis for s o ( 3 ; 1 ) {\displaystyle {\mathfrak {so}}(3;1)} is given by the three generators Ji of rotations and the three generators Ki 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
A = J + i K 2 , B = J − i K 2 . {\displaystyle \mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}},\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}.}
The components of A = (A1, A2, A3) and B = (B1, B2, B3) separately satisfy the commutation relations of the Lie algebra s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} and, moreover, they commute with each other,
[ A i , A j ] = i ε i j k A k , [ B i , B j ] = i ε i j k B k , [ A i , B j ] = 0 , {\displaystyle \left[A_{i},A_{j}\right]=i\varepsilon _{ijk}A_{k},\quad \left[B_{i},B_{j}\right]=i\varepsilon _{ijk}B_{k},\quad \left[A_{i},B_{j}\right]=0,}
where i, j, k are indices which each take values 1, 2, 3, and εijk is the three-dimensional Levi-Civita symbol. Let A C {\displaystyle \mathbf {A} _{\mathbb {C} }} and B C {\displaystyle \mathbf {B} _{\mathbb {C} }} denote the complex linear span of A and B respectively. One has the isomorphisms
where s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} is the complexification of s u ( 2 ) ≅ A ≅ B . {\displaystyle {\mathfrak {su}}(2)\cong \mathbf {A} \cong \mathbf {B} .}
The utility of these isomorphisms comes from the fact that all irreducible representations of s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} , and hence all irreducible complex linear representations of s l ( 2 , C ) , {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),} are known. The irreducible complex linear representation of s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of s l ( 2 , C ) . {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ).}
Unitarian trick
The Lie algebra s l ( 2 , C ) ⊕ s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )} is the Lie algebra of SL ( 2 , C ) × SL ( 2 , C ) . {\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} ).} It contains the compact subgroup SU(2) × SU(2) with Lie algebra s u ( 2 ) ⊕ s u ( 2 ) . {\displaystyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2).} The latter is a compact real form of s l ( 2 , C ) ⊕ s l ( 2 , C ) . {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} ).} Thus from the first statement of the unitarian trick, representations of SU(2) × SU(2) are in one-to-one correspondence with holomorphic representations of SL ( 2 , C ) × SL ( 2 , C ) . {\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} ).}
By compactness, the Peter–Weyl theorem applies to SU(2) × SU(2), and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of SU(2) × SU(2) are precisely the tensor products of irreducible unitary representations of SU(2). 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 SL ( 2 , C ) × SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} )}
Smooth representations of SU(2) × SU(2) Real linear representations of s u ( 2 ) ⊕ s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)}
Complex linear representations of s l ( 2 , C ) ⊕ s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}
Tensor products of representations appear at the Lie algebra level as either of
where Id is the identity operator. Here, the latter interpretation, which follows from (G6), is intended. The highest weight representations of s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} are indexed by μ for μ = 0, 1/2, 1, .... (The highest weights are actually 2μ = 0, 1, 2, ..., but the notation here is adapted to that of s o ( 3 ; 1 ) . {\displaystyle {\mathfrak {so}}(3;1).} ) The tensor products of two such complex linear factors then form the irreducible complex linear representations of s l ( 2 , C ) ⊕ s l ( 2 , C ) . {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} ).}
Finally, the R {\displaystyle \mathbb {R} } -linear representations of the real forms of the far left, s o ( 3 ; 1 ) {\displaystyle {\mathfrak {so}}(3;1)} , and the far right, s l ( 2 , C ) , {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),} in (A1) are obtained from the C {\displaystyle \mathbb {C} } -linear representations of s l ( 2 , C ) ⊕ s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )} characterized in the previous paragraph.
(μ, ν)-representations of sl(2, C) The complex linear representations of the complexification of s l ( 2 , C ) , s l ( 2 , C ) C , {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),{\mathfrak {sl}}(2,\mathbb {C} )_{\mathbb {C} },} obtained via isomorphisms in (A1), stand in one-to-one correspondence with the real linear representations of s l ( 2 , C ) . {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ).} The set of all real linear irreducible representations of s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} are thus indexed by a pair (μ, ν). The complex linear ones, corresponding precisely to the complexification of the real linear s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} representations, are of the form (μ, 0), while the conjugate linear ones are the (0, ν). All others are real linear only. The linearity properties follow from the canonical injection, the far right in (A1), of s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} into its complexification. Representations on the form (ν, ν) or (μ, ν) ⊕ (ν, μ) are given by real matrices (the latter are not irreducible). Explicitly, the real linear (μ, ν)-representations of s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} are
φ μ , ν ( X ) = ( φ μ ⊗ φ ν ¯ ) ( X ) = φ μ ( X ) ⊗ Id ν + 1 + Id μ + 1 ⊗ φ ν ( X ) ¯ , X ∈ s l ( 2 , C ) {\displaystyle \varphi _{\mu ,\nu }(X)=\left(\varphi _{\mu }\otimes {\overline {\varphi _{\nu }}}\right)(X)=\varphi _{\mu }(X)\otimes \operatorname {Id} _{\nu +1}+\operatorname {Id} _{\mu +1}\otimes {\overline {\varphi _{\nu }(X)}},\qquad X\in {\mathfrak {sl}}(2,\mathbb {C} )}
where φ μ , μ = 0 , 1 2 , 1 , 3 2 , … {\textstyle \varphi _{\mu },\mu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots } are the complex linear irreducible representations of s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} and φ ν ¯ , ν = 0 , 1 2 , 1 , 3 2 , … {\displaystyle {\overline {\varphi _{\nu }}},\nu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots } their complex conjugate representations. (The labeling is usually in the mathematics literature 0, 1, 2, ..., but half-integers are chosen here to conform with the labeling for the s o ( 3 , 1 ) {\displaystyle {\mathfrak {so}}(3,1)} Lie algebra.) Here the tensor product is interpreted in the former sense of (A0). These representations are concretely realized below.
(m, n)-representations of so(3; 1) Via the displayed isomorphisms in (A1) and knowledge of the complex linear irreducible representations of s l ( 2 , C ) ⊕ s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )} upon solving for J and K, all irreducible representations of s o ( 3 ; 1 ) C , {\displaystyle {\mathfrak {so}}(3;1)_{\mathbb {C} },} and, by restriction, those of s o ( 3 ; 1 ) {\displaystyle {\mathfrak {so}}(3;1)} are obtained. The representations of s o ( 3 ; 1 ) {\displaystyle {\mathfrak {so}}(3;1)} obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible. Since s o ( 3 ; 1 ) {\displaystyle {\mathfrak {so}}(3;1)} is semisimple, all its representations can be built up as direct sums of the irreducible ones. Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers m = μ and n = ν, conventionally written as one of
( m , n ) ≡ π ( m , n ) : s o ( 3 ; 1 ) → g l ( V ) , {\displaystyle (m,n)\equiv \pi _{(m,n)}:{\mathfrak {so}}(3;1)\to {\mathfrak {gl}}(V),}
where V is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given by
where 1n is the n-dimensional unit matrix and
J ( n ) = ( J 1 ( n ) , J 2 ( n ) , J 3 ( n ) ) {\displaystyle \mathbf {J} ^{(n)}=\left(J_{1}^{(n)},J_{2}^{(n)},J_{3}^{(n)}\right)}
are the (2n + 1)-dimensional irreducible representations of s o ( 3 ) ≅ s u ( 2 ) {\displaystyle {\mathfrak {so}}(3)\cong {\mathfrak {su}}(2)} also termed spin matrices or angular momentum matrices. These are explicitly given as
( J 1 ( j ) ) a ′ a = 1 2 ( ( j − a ) ( j + a + 1 ) δ a ′ , a + 1 + ( j + a ) ( j − a + 1 ) δ a ′ , a − 1 ) ( J 2 ( j ) ) a ′ a = 1 2 i ( ( j − a ) ( j + a + 1 ) δ a ′ , a + 1 − ( j + a ) ( j − a + 1 ) δ a ′ , a − 1 ) ( J 3 ( j ) ) a ′ a = a δ a ′ , a {\displaystyle {\begin{aligned}\left(J_{1}^{(j)}\right)_{a'a}&={\frac {1}{2}}\left({\sqrt {(j-a)(j+a+1)}}\delta _{a',a+1}+{\sqrt {(j+a)(j-a+1)}}\delta _{a',a-1}\right)\\\left(J_{2}^{(j)}\right)_{a'a}&={\frac {1}{2i}}\left({\sqrt {(j-a)(j+a+1)}}\delta _{a',a+1}-{\sqrt {(j+a)(j-a+1)}}\delta _{a',a-1}\right)\\\left(J_{3}^{(j)}\right)_{a'a}&=a\delta _{a',a}\end{aligned}}}
where δ denotes the Kronecker delta. In components, with −m ≤ a, a′ ≤ m, −n ≤ b, b′ ≤ n, the representations are given by
( π ( m , n ) ( J i ) ) a ′ b ′ , a b = δ b ′ b ( J i ( m ) ) a ′ a + δ a ′ a ( J i ( n ) ) b ′ b ( π ( m , n ) ( K i ) ) a ′ b ′ , a b = − i ( δ b ′ b ( J