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.^{[nb 1]} This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established,^{[nb 2]} 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.

Development[edit]

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 are 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.

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,^[1] 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.^[2] 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,^{[nb 3]} in 1947. The corresponding classification for ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$ was published independently by Bargmann and Israel Gelfand together with Mark Naimark in the same year.

Applications[edit]

Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string theory and beyond. The representation theory also provides the theoretical ground for the concept of spin. The theory enters into general relativity in the sense that in small enough regions of spacetime, physics is that of special relativity.^[3]

The finite-dimensional irreducible non-unitary representations together with the irreducible infinite-dimensional unitary representations of the inhomogeneous Lorentz group, the Poincare group, are the representations that have direct physical relevance.^[4][5]

Infinite-dimensional unitary representations of the Lorentz group appear by restriction of the irreducible infinite-dimensional unitary representations of the Poincaré group acting on the Hilbert spaces of relativistic quantum mechanics and quantum field theory. But these are also of mathematical interest and of potential direct physical relevance in other roles than that of a mere restriction.^[6] There were speculative theories,^[7][8] (tensors and spinors have infinite counterparts in the expansors of Dirac and the expinors of Harish-Chandra) consistent with relativity and quantum mechanics, but they have found no proven physical application. Modern speculative theories potentially have similar ingredients per below.

Classical field theory[edit]

While the electromagnetic field together with the gravitational field are the only classical fields providing accurate descriptions of nature, other types of classical fields are important too. In the approach to quantum field theory (QFT) referred to as second quantization, the starting point is one or more classical fields, where e.g. the wave functions solving the Dirac equation are considered as classical fields prior to (second) quantization.^[9] While second quantization and the Lagrangian formalism associated with it is not a fundamental aspect of QFT,^[10] it is the case that so far all quantum field theories can be approached this way, including the standard model.^[11] In these cases, there are classical versions of the field equations following from the Euler–Lagrange equations derived from the Lagrangian using the principle of least action. These field equations must be relativistically invariant, and their solutions (which will qualify as relativistic wave functions according to the definition below) must transform under some representation of the Lorentz group.

The action of the Lorentz group on the space of field configurations (a field configuration is the spacetime history of a particular solution, e.g. the electromagnetic field in all of space over all time is one field configuration) resembles the action on the Hilbert spaces of quantum mechanics, except that the commutator brackets are replaced by field theoretical Poisson brackets.^[9]

Relativistic quantum mechanics[edit]

For the present purposes the following definition is made:^[12] A relativistic wave function is a set of n functions ψ^α on spacetime which transforms under an arbitrary proper Lorentz transformation Λ as

where D[Λ] is an n-dimensional matrix representative of Λ belonging to some direct sum of the (m, n) representations to be introduced below.

The most useful relativistic quantum mechanics one-particle theories (there are no fully consistent such theories) are the Klein–Gordon equation^[13] and the Dirac equation^[14] in their original setting. They are relativistically invariant and their solutions transform under the Lorentz group as Lorentz scalars ((m, n) = (0, 0)) and bispinors respectively ((0, 1/2) ⊕ (1/2, 0)). The electromagnetic field is a relativistic wave function according to this definition, transforming under (1, 0) ⊕ (0, 1).^[15]

The infinite-dimensional representations may be used in the analysis of scattering.^[16]

Quantum field theory[edit]

In quantum field theory, the demand for relativistic invariance enters, among other ways in that the S-matrix necessarily must be Poincaré invariant.^[17] This has the implication that there is one or more infinite-dimensional representation of the Lorentz group acting on Fock space.^{[nb 4]} One way to guarantee the existence of such representations is the existence of a Lagrangian description (with modest requirements imposed, see the reference) of the system using the canonical formalism, from which a realization of the generators of the Lorentz group may be deduced.^[18]

The transformations of field operators illustrate the complementary role played by the finite-dimensional representations of the Lorentz group and the infinite-dimensional unitary representations of the Poincare group, witnessing the deep unity between mathematics and physics.^[19] For illustration, consider the definition an n-component field operator:^[20] A relativistic field operator is a set of n operator valued functions on spacetime which transforms under proper Poincaré transformations (Λ, a) according to^[21][22]

Here U[Λ, a] is the unitary operator representing (Λ, a) on the Hilbert space on which Ψ is defined and D is an n-dimensional representation of the Lorentz group. The transformation rule is the second Wightman axiom of quantum field theory.

By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass m and spin s (or helicity), it is deduced that^{[23][nb 5]}

$${\displaystyle \Psi ^{\alpha }(x)=\sum _{\sigma }\int dp\left(a(\mathbf {p} ,\sigma )u^{\alpha }(\mathbf {p} ,\sigma )e^{ip\cdot x}+a^{\dagger }(\mathbf {p} ,\sigma )v^{\alpha }(\mathbf {p} ,\sigma )e^{-ip\cdot x}\right),}$$

(X1)

where a^†, a are interpreted as creation and annihilation operators respectively. The creation operator a^† transforms according to^[23][24]

and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin (m, s) of the particle. The connection between the two are the wave functions, also called coefficient functions

that carry both the indices (x, α) operated on by Lorentz transformations and the indices (p, σ) operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection.^[25] To exhibit the connection, subject both sides of equation (X1) to a Lorentz transformation resulting in for e.g. u,

where D is the non-unitary Lorentz group representative of Λ and D^(s) is a unitary representative of the so-called Wigner rotation R associated to Λ and p that derives from the representation of the Poincaré group, and s is the spin of the particle.

All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the (m, n) representation under which it is supposed to transform,^{[nb 6]} and also that of the wave function, can be derived from group theoretical considerations alone once the frameworks of quantum mechanics and special relativity is given.^{[nb 7]}

Speculative theories[edit]

In theories in which spacetime can have more than D = 4 dimensions, the generalized Lorentz groups O(D − 1; 1) of the appropriate dimension take the place of O(3; 1).^{[nb 8]}

The requirement of Lorentz invariance takes on perhaps its most dramatic effect in string theory. Classical relativistic strings can be handled in the Lagrangian framework by using the Nambu–Goto action.^[26] This results in a relativistically invariant theory in any spacetime dimension.^[27] But as it turns out, the theory of open and closed bosonic strings (the simplest string theory) is impossible to quantize in such a way that the Lorentz group is represented on the space of states (a Hilbert space) unless the dimension of spacetime is 26.^[28] The corresponding result for superstring theory is again deduced demanding Lorentz invariance, but now with supersymmetry. In these theories the Poincaré algebra is replaced by a supersymmetry algebra which is a Z₂-graded Lie algebra extending the Poincaré algebra. The structure of such an algebra is to a large degree fixed by the demands of Lorentz invariance. In particular, the fermionic operators (grade 1) belong to a (0, 1/2) or (1/2, 0) representation space of the (ordinary) Lorentz Lie algebra.^[29] The only possible dimension of spacetime in such theories is 10.^[30]

Finite-dimensional representations[edit]

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.^[31]

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.^{[nb 9][32]} 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.^[33] Lack of simple connectedness gives rise to spin representations of the group.^[34] 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.^[35][36]

History[edit]

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.^[37][38] By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing.^[39][40] 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.^[41][42] 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.^[43][44] The Lorentz group has also historically received special attention in representation theory, see History of infinite-dimensional unitary representations below, due to its exceptional importance in physics. Mathematicians Hermann Weyl^{[41][45][37][46][47]} and Harish-Chandra^[48][49] and physicists Eugene Wigner^[50][51] and Valentine Bargmann^[52][53][54] made substantial contributions both to general representation theory and in particular to the Lorentz group.^[55] 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.^{[56][57][nb 10]}

The Lie algebra[edit]

This section addresses the irreducible complex linear representations of the complexification ${\displaystyle {\mathfrak {so}}(3;1)_{\mathbb {C} }}$ of the Lie algebra ${\displaystyle {\mathfrak {so}}(3;1)}$ of the Lorentz group. A convenient basis for ${\displaystyle {\mathfrak {so}}(3;1)}$ is given by the three generators J_i of rotations and the three generators K_i 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^[58]

The components of A = (A₁, A₂, A₃) and B = (B₁, B₂, B₃) separately satisfy the commutation relations of the Lie algebra ${\displaystyle {\mathfrak {su}}(2)}$ and, moreover, they commute with each other,^[59]

where i, j, k are indices which each take values 1, 2, 3, and ε_ijk is the three-dimensional Levi-Civita symbol. Let ${\displaystyle \mathbf {A} _{\mathbb {C} }}$ and ${\displaystyle \mathbf {B} _{\mathbb {C} }}$ denote the complex linear span of A and B respectively.

One has the isomorphisms^{[60][nb 11]}

$${\displaystyle {\begin{aligned}{\mathfrak {so}}(3;1)\hookrightarrow {\mathfrak {so}}(3;1)_{\mathbb {C} }&\cong \mathbf {A} _{\mathbb {C} }\oplus \mathbf {B} _{\mathbb {C} }\cong {\mathfrak {su}}(2)_{\mathbb {C} }\oplus {\mathfrak {su}}(2)_{\mathbb {C} }\\[5pt]&\cong {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )\\[5pt]&\cong {\mathfrak {sl}}(2,\mathbb {C} )\oplus i{\mathfrak {sl}}(2,\mathbb {C} )={\mathfrak {sl}}(2,\mathbb {C} )_{\mathbb {C} }\hookleftarrow {\mathfrak {sl}}(2,\mathbb {C} ),\end{aligned}}}$$

(A1)

where ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ is the complexification of ${\displaystyle {\mathfrak {su}}(2)\cong \mathbf {A} \cong \mathbf {B} .}$

The utility of these isomorphisms comes from the fact that all irreducible representations of ${\displaystyle {\mathfrak {su}}(2)}$, and hence all irreducible complex linear representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ are known. The irreducible complex linear representation of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ).}$

The unitarian trick[edit]

The Lie algebra ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$ is the Lie algebra of ${\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} ).}$ It contains the compact subgroup SU(2) × SU(2) with Lie algebra ${\displaystyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2).}$ The latter is a compact real form of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} ).}$ 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 ${\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} ).}$

By compactness, the Peter–Weyl theorem applies to SU(2) × SU(2),^[61] 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).^[62]

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 ${\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} )}$
• Smooth representations of SU(2) × SU(2)
• Real linear representations of ${\displaystyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)}$
• Complex linear representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$

Tensor products of representations appear at the Lie algebra level as either of^{[nb 12]}

$${\displaystyle {\begin{aligned}\pi _{1}\otimes \pi _{2}(X)&=\pi _{1}(X)\otimes \mathrm {Id} _{V}+\mathrm {Id} _{U}\otimes \pi _{2}(X)&&X\in {\mathfrak {g}}\\\pi _{1}\otimes \pi _{2}(X,Y)&=\pi _{1}(X)\otimes \mathrm {Id} _{V}+\mathrm {Id} _{U}\otimes \pi _{2}(Y)&&(X,Y)\in {\mathfrak {g}}\oplus {\mathfrak {g}}\end{aligned}}}$$

(A0)

where Id is the identity operator. Here, the latter interpretation, which follows from (G6), is intended. The highest weight representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ 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 ${\displaystyle {\mathfrak {so}}(3;1).}$) The tensor products of two such complex linear factors then form the irreducible complex linear representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} ).}$

Finally, the ${\displaystyle \mathbb {R} }$-linear representations of the real forms of the far left, ${\displaystyle {\mathfrak {so}}(3;1)}$, and the far right, ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$^{[nb 13]} in (A1) are obtained from the ${\displaystyle \mathbb {C} }$-linear representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$ characterized in the previous paragraph.

The (μ, ν)-representations of sl(2, C)[edit]

The complex linear representations of the complexification of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),{\mathfrak {sl}}(2,\mathbb {C} )_{\mathbb {C} },}$ obtained via isomorphisms in (A1), stand in one-to-one correspondence with the real linear representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ).}$^[63] The set of all real linear irreducible representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ are thus indexed by a pair (μ, ν). The complex linear ones, corresponding precisely to the complexification of the real linear ${\displaystyle {\mathfrak {su}}(2)}$ representations, are of the form (μ, 0), while the conjugate linear ones are the (0, ν).^[63] All others are real linear only. The linearity properties follow from the canonical injection, the far right in (A1), of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ 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 ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ are

where ${\textstyle \varphi _{\mu },\mu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }$ are the complex linear irreducible representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ and ${\displaystyle {\overline {\varphi _{\nu }}},\nu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }$ 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 ${\displaystyle {\mathfrak {so}}(3,1)}$ Lie algebra.) Here the tensor product is interpreted in the former sense of (A0). These representations are concretely realized below.

The (m, n)-representations of so(3; 1)[edit]

Via the displayed isomorphisms in (A1) and knowledge of the complex linear irreducible representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$ upon solving for J and K, all irreducible representations of ${\displaystyle {\mathfrak {so}}(3;1)_{\mathbb {C} },}$ and, by restriction, those of ${\displaystyle {\mathfrak {so}}(3;1)}$ are obtained. The representations of ${\displaystyle {\mathfrak {so}}(3;1)}$ 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.^[60] Since ${\displaystyle {\mathfrak {so}}(3;1)}$ is semisimple,^[60] 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

where V is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given by^{[nb 14]}

$${\displaystyle \pi _{(m,n)}(J_{i})=J_{i}^{(m)}\otimes 1_{(2n+1)}+1_{(2m+1)}\otimes J_{i}^{(n)}}$$ $${\displaystyle \pi _{(m,n)}(K_{i})=-i\left(J_{i}^{(m)}\otimes 1_{(2n+1)}-1_{(2m+1)}\otimes J_{i}^{(n)}\right),}$$

(A2)

where 1_n is the n-dimensional unit matrix and

are the (2n + 1)-dimensional irreducible representations of ${\displaystyle {\mathfrak {so}}(3)\cong {\mathfrak {su}}(2)}$ also termed spin matrices or angular momentum matrices. These are explicitly given as^[64] where δ denotes the Kronecker delta. In components, with −m ≤ a, a′ ≤ m, −n ≤ b, b′ ≤ n, the representations are given by^[65]

Common representations[edit]

Irreducible representations for small (m, n). Dimension in parentheses.

	m = 0	1/2	1	3/2
n = 0	Scalar (1)	Left-handed Weyl spinor (2)	Self-dual 2-form (3)	(4)
1/2	Right-handed Weyl spinor (2)	4-vector (4)	(6)	(8)
1	Anti-self-dual 2-form (3)	(6)	Traceless symmetric tensor (9)	(12)
3/2	(4)	(8)	(12)	(16)

• The (0, 0) representation is the one-dimensional trivial representation and is carried by relativistic scalar field theories.
• Fermionic supersymmetry generators transform under one of the (0, 1/2) or (1/2, 0) representations (Weyl spinors).^[29]
• The four-momentum of a particle (either massless or massive) transforms under the (1/2, 1/2) representation, a four-vector.
• A physical example of a (1,1) traceless symmetric tensor field is the traceless^{[nb 15]} part of the energy–momentum tensor T^μν.^{[66][nb 16]}

Off-diagonal direct sums[edit]

Since for any irreducible representation for which m ≠ n it is essential to operate over the field of complex numbers, the direct sum of representations (m, n) and (n, m) have particular relevance to physics, since it permits to use linear operators over real numbers.

• (1/2, 0) ⊕ (0, 1/2) is the bispinor representation. See also Dirac spinor and Weyl spinors and bispinors below.
• (1, 1/2) ⊕ (1/2, 1) is the Rarita–Schwinger field representation.
• (3/2, 0) ⊕ (0, 3/2) would be the symmetry of the hypothesized gravitino.^{[nb 17]} It can be obtained from the (1, 1/2) ⊕ (1/2, 1) representation.
• (1, 0) ⊕ (0, 1) is the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation.

The group[edit]

The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence.^[67] The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras.^[68] The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted ${\displaystyle \exp :{\mathfrak {g}}\to G.}$

If ${\displaystyle \pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)}$ for some vector space V is a representation, a representation Π of the connected component of G is defined by

$${\displaystyle {\begin{aligned}\Pi (g=e^{iX})&\equiv e^{i\pi (X)},&&X\in {\mathfrak {g}},\quad g=e^{iX}\in \mathrm {im} (\exp ),\\\Pi (g=g_{1}g_{2}\cdots g_{n})&\equiv \Pi (g_{1})\Pi (g_{2})\cdots \Pi (g_{n}),&&g\notin \mathrm {im} (\exp ),\quad g_{1},g_{2},\ldots ,g_{n}\in \mathrm {im} (\exp ).\end{aligned}}}$$

(G2)

This definition applies whether the resulting representation is projective or not.

Surjectiveness of exponential map for SO(3, 1)[edit]

From a practical point of view, it is important whether the first formula in (G2) can be used for all elements of the group. It holds for all ${\displaystyle X\in {\mathfrak {g}}}$, however, in the general case, e.g. for ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$, not all g ∈ G are in the image of exp.

But ${\displaystyle \exp :{\mathfrak {so}}(3;1)\to {\text{SO}}(3;1)^{+}}$ is surjective. One way to show this is to make use of the isomorphism ${\displaystyle {\text{SO}}(3;1)^{+}\cong {\text{PGL}}(2,\mathbb {C} ),}$ the latter being the Möbius group. It is a quotient of ${\displaystyle {\text{GL}}(n,\mathbb {C} )}$ (see the linked article). The quotient map is denoted with ${\displaystyle p:{\text{GL}}(n,\mathbb {C} )\to {\text{PGL}}(2,\mathbb {C} ).}$ The map ${\displaystyle \exp :{\mathfrak {gl}}(n,\mathbb {C} )\to {\text{GL}}(n,\mathbb {C} )}$ is onto.^[69] Apply (Lie) with π being the differential of p at the identity. Then

Since the left hand side is surjective (both exp and p are), the right hand side is surjective and hence ${\displaystyle \exp :{\mathfrak {pgl}}(2,\mathbb {C} )\to {\text{PGL}}(2,\mathbb {C} )}$ is surjective.^[70] Finally, recycle the argument once more, but now with the known isomorphism between SO(3; 1)⁺ and ${\displaystyle {\text{PGL}}(2,\mathbb {C} )}$ to find that exp is onto for the connected component of the Lorentz group.

Fundamental group[edit]

The Lorentz group is doubly connected, i. e. π₁(SO(3; 1)) is a group with two equivalence classes of loops as its elements.

Projective representations[edit]

Since π₁(SO(3; 1)⁺) has two elements, some representations of the Lie algebra will yield projective representations.^{[73][nb 18]} Once it is known whether a representation is projective, formula (G2) applies to all group elements and all representations, including the projective ones — with the understanding that the representative of a group element will depend on which element in the Lie algebra (the X in (G2)) is used to represent the group element in the standard representation.

For the Lorentz group, the (m, n)-representation is projective when m + n is a half-integer. See § Spinors.

For a projective representation Π of SO(3; 1)⁺, it holds that^[72]

$${\displaystyle \left[\Pi (\Lambda _{1})\Pi (\Lambda _{2})\Pi ^{-1}(\Lambda _{1}\Lambda _{2})\right]^{2}=1\Rightarrow \Pi (\Lambda _{1}\Lambda _{2})=\pm \Pi (\Lambda _{1})\Pi (\Lambda _{2}),\quad \Lambda _{1},\Lambda _{2}\in \mathrm {SO} (3;1),}$$

(G5)

since any loop in SO(3; 1)⁺ traversed twice, due to the double connectedness, is contractible to a point, so that its homotopy class is that of a constant map. It follows that Π is a double-valued function. It is not possible to consistently choose a sign to obtain a continuous representation of all of SO(3; 1)⁺, but this is possible locally around any point.^[33]

The covering group SL(2, C)[edit]

Consider ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ as a real Lie algebra with basis

where the sigmas are the Pauli matrices. From the relations

$${\displaystyle [\sigma _{i},\sigma _{j}]=2i\epsilon _{ijk}\sigma _{k}}$$

(J1)

is obtained

$${\displaystyle [j_{i},j_{j}]=i\epsilon _{ijk}j_{k},\quad [j_{i},k_{j}]=i\epsilon _{ijk}k_{k},\quad [k_{i},k_{j}]=-i\epsilon _{ijk}j_{k},}$$

(J2)

which are exactly on the form of the 3-dimensional version of the commutation relations for ${\displaystyle {\mathfrak {so}}(3;1)}$ (see conventions and Lie algebra bases below). Thus, the map J_i ↔ j_i, K_i ↔ k_i, extended by linearity is an isomorphism. Since ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ is simply connected, it is the universal covering group of SO(3; 1)⁺.

Non-surjectiveness of exponential mapping for SL(2, C)[edit]

The exponential mapping ${\displaystyle \exp :{\mathfrak {sl}}(2,\mathbb {C} )\to {\text{SL}}(2,\mathbb {C} )}$ is not onto.^[76] The matrix

$${\displaystyle q={\begin{pmatrix}-1&1\\0&-1\\\end{pmatrix}}}$$

(S6)

is in ${\displaystyle {\text{SL}}(2,\mathbb {C} ),}$ but there is no ${\displaystyle Q\in {\mathfrak {sl}}(2,\mathbb {C} )}$ such that q = exp(Q).^{[nb 22]}

In general, if g is an element of a connected Lie group G with Lie algebra ${\displaystyle {\mathfrak {g}},}$ then, by (Lie),

$${\displaystyle g=\exp(X_{1})\cdots \exp(X_{n}),\qquad X_{1},\ldots X_{n}\in {\mathfrak {g}}.}$$

(S7)

The matrix q can be written

$${\displaystyle {\begin{aligned}&\exp(-X)\exp(i\pi H)\\{}={}&\exp \left({\begin{pmatrix}0&-1\\0&0\\\end{pmatrix}}\right)\exp \left(i\pi {\begin{pmatrix}1&0\\0&-1\\\end{pmatrix}}\right)\\[6pt]{}={}&{\begin{pmatrix}1&-1\\0&1\\\end{pmatrix}}{\begin{pmatrix}-1&0\\0&-1\\\end{pmatrix}}\\[6pt]{}={}&{\begin{pmatrix}-1&1\\0&-1\\\end{pmatrix}}\\{}={}&q.\end{aligned}}}$$

(S8)

Realization of representations of SL(2, C) and sl(2, C) and their Lie algebras[edit]

The complex linear representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ and ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ are more straightforward to obtain than the ${\displaystyle {\mathfrak {so}}(3;1)^{+}}$ representations. They can be (and usually are) written down from scratch. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are related to the complex linear Lie algebra representations by exponentiation. The real linear representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ are exactly the (μ, ν)-representations. They can be exponentiated too. The (μ, 0)-representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here).

The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of i and there is no factor of i in the exponential mapping compared to the physics convention used elsewhere. Let the basis of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ be^[77]

$${\displaystyle H={\begin{pmatrix}1&0\\0&-1\\\end{pmatrix}},\quad X={\begin{pmatrix}0&1\\0&0\\\end{pmatrix}},\quad Y={\begin{pmatrix}0&0\\1&0\\\end{pmatrix}}.}$$

(S1)

This choice of basis, and the notation, is standard in the mathematical literature.

Complex linear representations[edit]

The irreducible holomorphic (n + 1)-dimensional representations ${\displaystyle {\text{SL}}(2,\mathbb {C} ),n\geqslant 2,}$ can be realized on the space of homogeneous polynomial of degree n in 2 variables ${\displaystyle \mathbf {P} _{n}^{2},}$^[78][79] the elements of which are

The action of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ is given by^[80][81]

$${\displaystyle (\phi _{n}(g)P){\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=\left[\phi _{n}{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}P\right]{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=P\left({\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}^{-1}{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}\right),\qquad P\in \mathbf {P} _{n}^{2}.}$$

(S2)

The associated ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$-action is, using (G6) and the definition above, for the basis elements of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$^[82]

$${\displaystyle \phi _{n}(H)=-z_{1}{\frac {\partial }{\partial z_{1}}}+z_{2}{\frac {\partial }{\partial z_{2}}},\quad \phi _{n}(X)=-z_{2}{\frac {\partial }{\partial z_{1}}},\quad \phi _{n}(Y)=-z_{1}{\frac {\partial }{\partial z_{2}}}.}$$

(S5)

With a choice of basis for ${\displaystyle P\in \mathbf {P} _{n}^{2}}$, these representations become matrix Lie algebras.

Real linear representations[edit]

The (μ, ν)-representations are realized on a space of polynomials ${\displaystyle \mathbf {P} _{\mu ,\nu }^{2}}$ in ${\displaystyle z_{1},{\overline {z_{1}}},z_{2},{\overline {z_{2}}},}$ homogeneous of degree μ in ${\displaystyle z_{1},z_{2}}$ and homogeneous of degree ν in ${\displaystyle {\overline {z_{1}}},{\overline {z_{2}}}.}$^[79] The representations are given by^[83]

$${\displaystyle (\phi _{\mu ,\nu }(g)P){\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=\left[\phi _{\mu ,\nu }{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}P\right]{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=P\left({\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}^{-1}{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}\right),\quad P\in \mathbf {P} _{\mu ,\nu }^{2}.}$$

(S6)

By employing (G6) again it is found that

$${\displaystyle {\begin{aligned}\phi _{\mu ,\nu }(E)P=&-{\frac {\partial P}{\partial z_{1}}}\left(E_{11}z_{1}+E_{12}z_{2}\right)-{\frac {\partial P}{\partial z_{2}}}\left(E_{21}z_{1}+E_{22}z_{2}\right)\\&-{\frac {\partial P}{\partial {\overline {z_{1}}}}}\left({\overline {E_{11}}}{\overline {z_{1}}}+{\overline {E_{12}}}{\overline {z_{2}}}\right)-{\frac {\partial P}{\partial {\overline {z_{2}}}}}\left({\overline {E_{21}}}{\overline {z_{1}}}+{\overline {E_{22}}}{\overline {z_{2}}}\right)\end{aligned}},\quad E\in {\mathfrak {sl}}(2,\mathbf {C} ).}$$

(S7)

In particular for the basis elements,

$${\displaystyle {\begin{aligned}\phi _{\mu ,\nu }(H)&=-z_{1}{\frac {\partial }{\partial z_{1}}}+z_{2}{\frac {\partial }{\partial z_{2}}}-{\overline {z_{1}}}{\frac {\partial }{\partial {\overline {z_{1}}}}}+{\overline {z_{2}}}{\frac {\partial }{\partial {\overline {z_{2}}}}}\\\phi _{\mu ,\nu }(X)&=-z_{2}{\frac {\partial }{\partial z_{1}}}-{\overline {z_{2}}}{\frac {\partial }{\partial {\overline {z_{1}}}}}\\\phi _{\mu ,\nu }(Y)&=-z_{1}{\frac {\partial }{\partial z_{2}}}-{\overline {z_{1}}}{\frac {\partial }{\partial {\overline {z_{2}}}}}\end{aligned}}}$$

(S8)

Properties of the (m, n) representations[edit]

The (m, n) representations, defined above via (A1) (as restrictions to the real form ${\displaystyle {\mathfrak {sl}}(3,1)}$) of tensor products of irreducible complex linear representations π_{m = μ} and π_{n = ν} of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ are irreducible, and they are the only irreducible representations.^[61]

• Irreducibility follows from the unitarian trick^[84] and that a representation Π of SU(2) × SU(2) is irreducible if and only if Π = Π_μ ⊗ Π_ν,^{[nb 23]} where Π_μ, Π_ν are irreducible representations of SU(2).
• Uniqueness follows from that the Π_m are the only irreducible representations of SU(2), which is one of the conclusions of the theorem of the highest weight.^[85]

Dimension[edit]

The (m, n) representations are (2m + 1)(2n + 1)-dimensional.^[86] This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ and ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$. For a Lie general algebra ${\displaystyle {\mathfrak {g}}}$ the Weyl dimension formula,^[87]

applies, where R⁺ is the set of positive roots, ρ is the highest weight, and δ is half the sum of the positive roots. The inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ is that of the Lie algebra ${\displaystyle {\mathfrak {g}},}$ invariant under the action of the Weyl group on ${\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}},}$ the Cartan subalgebra. The roots (really elements of ${\displaystyle {\mathfrak {h}}^{*}}$ are via this inner product identified with elements of ${\displaystyle {\mathfrak {h}}.}$ For ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ the formula reduces to dim π_μ = 2μ + 1 = 2m + 1, where the present notation must be taken into account. The highest weight is 2μ.^[88] By taking tensor products, the result follows.

Faithfulness[edit]

If a representation Π of a Lie group G is not faithful, then N = ker Π is a nontrivial normal subgroup.^[89] There are three relevant cases.

1. N is non-discrete and abelian.
2. N is non-discrete and non-abelian.
3. N is discrete. In this case N ⊂ Z, where Z is the center of G.^{[nb 24]}

In the case of SO(3; 1)⁺, the first case is excluded since SO(3; 1)⁺ is semi-simple.^{[nb 25]} The second case (and the first case) is excluded because SO(3; 1)⁺ is simple.^{[nb 26]} For the third case, SO(3; 1)⁺ is isomorphic to the quotient ${\displaystyle {\text{SL}}(2,\mathbb {C} )/\{\pm I\}.}$ But ${\displaystyle \{\pm I\}}$ is the center of ${\displaystyle {\text{SL}}(2,\mathbb {C} ).}$ It follows that the center of SO(3; 1)⁺ is trivial, and this excludes the third case. The conclusion is that every representation Π : SO(3; 1)⁺ → GL(V) and every projective representation Π : SO(3; 1)⁺ → PGL(W) for V, W finite-dimensional vector spaces are faithful.

By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial non-discrete normal subgroups replaced by (one-dimensional) nontrivial ideals in the Lie algebra,^[90] and the center of SO(3; 1)⁺ replaced by the center of ${\displaystyle {\mathfrak {sl}}(3;1)^{+}}$The center of any semisimple Lie algebra is trivial^[91] and ${\displaystyle {\mathfrak {so}}(3;1)}$ is semi-simple and simple, and hence has no non-trivial ideals.

A related fact is that if the corresponding representation of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ representation is not faithful, but is 2:1.

Non-unitarity[edit]

The (m, n) Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary.^{[nb 27]} This is due to the non-compactness of the Lorentz group. In fact, a connected simple non-compact Lie group cannot have any nontrivial unitary finite-dimensional representations.^[33] There is a topological proof of this.^[92] Let u : G → GL(V), where V is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group G. Then u(G) ⊂ U(V) ⊂ GL(V) where U(V) is the compact subgroup of GL(V) consisting of unitary transformations of V. The kernel of u is a normal subgroup of G. Since G is simple, ker u is either all of G, in which case u is trivial, or ker u is trivial, in which case u is faithful. In the latter case u is a diffeomorphism onto its image,^[93] u(G) ≅ G and u(G) is a Lie group. This would mean that u(G) is an embedded non-compact Lie subgroup of the compact group U(V). This is impossible with the subspace topology on u(G) ⊂ U(V) since all embedded Lie subgroups of a Lie group are closed^[94] If u(G) were closed, it would be compact,^{[nb 28]} and then G would be compact,^{[nb 29]} contrary to assumption.^{[nb 30]}

In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of A and B used in the construction are Hermitian. This means that J is Hermitian, but K is anti-Hermitian.^[95] The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.^[96]

Restriction to SO(3)[edit]

The (m, n) representation is, however, unitary when restricted to the rotation subgroup SO(3), but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an (m, n) representation have SO(3)-invariant subspaces of highest weight (spin) m + n, m + n − 1, ..., | m − n|,^[97] where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) j is (2j + 1)-dimensional. So for example, the (1/2, 1/2) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.

Since the angular momentum operator is given by J = A + B, the highest spin in quantum mechanics of the rotation sub-representation will be (m + n)ℏ and the "usual" rules of addition of angular momenta and the formalism of 3-j symbols, 6-j symbols, etc. applies.^[98]

Spinors[edit]

It is the SO(3)-invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the (m, n) representation has spin if m + n is half-integer. The simplest are (1/2, 0) and (0, 1/2), the Weyl-spinors of dimension 2. Then, for example, (0, 3/2) and (1, 1/2) are a spin representations of dimensions 2⋅3/2 + 1 = 4 and (2 + 1)(2⋅1/2 + 1) = 6 respectively. According to the above paragraph, there are subspaces with spin both 3/2 and 1/2 in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under SO(3). It cannot be ruled out in general, however, that representations with multiple SO(3) subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out unphysical components, leaving only a single spin.^[99]

Construction of pure spin n/2 representations for any n (under SO(3)) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints.^[100]

Dual representations[edit]

The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation:

1. The set of weights of the dual representation of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation.^[101]
2. Two irreducible representations are isomorphic if and only if they have the same highest weight.^{[nb 31]}
3. For each semisimple Lie algebra there exists a unique element w₀ of the Weyl group such that if μ is a dominant integral weight, then w₀ ⋅ (−μ) is again a dominant integral weight.^[102]
4. If ${\displaystyle \pi _{\mu _{0}}}$ is an irreducible representation with highest weight μ₀, then ${\displaystyle \pi _{\mu _{0}}^{*}}$ has highest weight w₀ ⋅ (−μ).^[102]

Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If −I is an element of the Weyl group of a semisimple Lie algebra, then w₀ = −I. In the case of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ the Weyl group is W = {I, −I}.^[103] It follows that each π_μ, μ = 0, 1, ... is isomorphic to its dual ${\displaystyle \pi _{\mu }^{*}.}$ The root system of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$ is shown in the figure to the right.^{[nb 32]} The Weyl group is generated by ${\displaystyle \{w_{\gamma }\}}$ where ${\displaystyle w_{\gamma }}$ is reflection in the plane orthogonal to γ as γ ranges over all roots.^{[nb 33]} Inspection shows that w_α ⋅ w_β = −I so −I ∈ W. Using the fact that if π, σ are Lie algebra representations and π ≅ σ, then Π ≅ Σ,^[104] the conclusion for SO(3; 1)⁺ is

Complex conjugate representations[edit]

If π is a representation of a Lie algebra, then ${\displaystyle {\overline {\pi }}}$ is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication.^[105] In general, every irreducible representation π of ${\displaystyle \mathfrak{s}\mathfrak{l}}$