Representation theory of the Lorentz group

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]

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

$${\displaystyle \psi '^{\alpha }(x)=D{[\Lambda ]^{\alpha }}_{\beta }\psi ^{\beta }\left(\Lambda ^{-1}x\right),}$$

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]

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]

$${\displaystyle \Psi ^{\alpha }(x)\to \Psi '^{\alpha }(x)=U[\Lambda ,a]\Psi ^{\alpha }(x)U^{-1}\left[\Lambda ,a\right]=D{\left[\Lambda ^{-1}\right]^{\alpha }}_{\beta }\Psi ^{\beta }(\Lambda x+a)}$$

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]

$${\displaystyle a^{\dagger }(\mathbf {p} ,\sigma )\rightarrow a'^{\dagger }\left(\mathbf {p} ,\sigma \right)=U[\Lambda ]a^{\dagger }(\mathbf {p} ,\sigma )U\left[\Lambda ^{-1}\right]=a^{\dagger }(\Lambda \mathbf {p} ,\rho )D^{(s)}{\left[R(\Lambda ,\mathbf {p} )^{-1}\right]^{\rho }}_{\sigma },}$$

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

$${\displaystyle u^{\alpha }(\mathbf {p} ,\sigma )e^{ip\cdot x},\quad v^{\alpha }(\mathbf {p} ,\sigma )e^{-ip\cdot x}}$$

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,

$${\displaystyle {D[\Lambda ]^{\alpha }}_{\alpha '}u^{\alpha '}(\mathbf {p} ,\lambda )={D^{(s)}[R(\Lambda ,\mathbf {p} )]^{\lambda '}}_{\lambda }u^{\alpha }\left(\Lambda \mathbf {p} ,\lambda '\right),}$$

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]

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]

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]

Wilhelm Killing, Independent discoverer of Lie algebras. The simple Lie algebras were first classified by him in 1888.

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]

$${\displaystyle \mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}},\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}.}$$

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

$${\displaystyle \left[A_{i},A_{j}\right]=i\varepsilon _{ijk}A_{k},\quad \left[B_{i},B_{j}\right]=i\varepsilon _{ijk}B_{k},\quad \left[A_{i},B_{j}\right]=0,}$$

where i, j, k are indices which each take values 1, 2, 3, and 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 ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ).}$

Hermann Weyl, inventor of the unitarian trick. There are several concepts and formulas in representation theory named after Weyl, e.g. the Weyl group and the Weyl character formula.

Photo courtesy of ETH-Bibliothek Zürich, Bildarchiv

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 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

$${\displaystyle \varphi _{\mu ,\nu }(X)=\left(\varphi _{\mu }\otimes {\overline {\varphi _{\nu }}}\right)(X)=\varphi _{\mu }(X)\otimes \operatorname {Id} _{\nu +1}+\operatorname {Id} _{\mu +1}\otimes {\overline {\varphi _{\nu }(X)}},\qquad X\in {\mathfrak {sl}}(2,\mathbb {C} )}$$

where ${\textstyle \varphi _{\mu },\mu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }$ 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.

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

$${\displaystyle (m,n)\equiv \pi _{(m,n)}:{\mathfrak {so}}(3;1)\to {\mathfrak {gl}}(V),}$$

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

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

(A2)

where 1_n is the n-dimensional unit matrix and

$${\displaystyle \mathbf {J} ^{(n)}=\left(J_{1}^{(n)},J_{2}^{(n)},J_{3}^{(n)}\right)}$$

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]

$${\displaystyle {\begin{aligned}\left(J_{1}^{(j)}\right)_{a'a}&={\frac {1}{2}}\left({\sqrt {(j-a)(j+a+1)}}\delta _{a',a+1}+{\sqrt {(j+a)(j-a+1)}}\delta _{a',a-1}\right)\\\left(J_{2}^{(j)}\right)_{a'a}&={\frac {1}{2i}}\left({\sqrt {(j-a)(j+a+1)}}\delta _{a',a+1}-{\sqrt {(j+a)(j-a+1)}}\delta _{a',a-1}\right)\\\left(J_{3}^{(j)}\right)_{a'a}&=a\delta _{a',a}\end{aligned}}}$$

where δ denotes the Kronecker delta. In components, with −m ≤ a, a′ ≤ m, −n ≤ b, b′ ≤ n, the representations are given by[65]

$${\displaystyle {\begin{aligned}\left(\pi _{(m,n)}\left(J_{i}\right)\right)_{a'b',ab}&=\delta _{b'b}\left(J_{i}^{(m)}\right)_{a'a}+\delta _{a'a}\left(J_{i}^{(n)}\right)_{b'b}\\\left(\pi _{(m,n)}\left(K_{i}\right)\right)_{a'b',ab}&=-i\left(\delta _{a'a}\left(J_{i}^{(n)}\right)_{b'b}-\delta _{b'b}\left(J_{i}^{(m)}\right)_{a'a}\right)\end{aligned}}}$$

• 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]

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 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.

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

$${\displaystyle \forall X\in {\mathfrak {gl}}(n,\mathbb {C} ):\quad p(\exp(iX))=\exp(i\pi (X)).}$$

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.

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

Proof

To exhibit the fundamental group of SO(3; 1)⁺, the topology of its covering group ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ is considered. By the polar decomposition theorem, any matrix ${\displaystyle \lambda \in {\text{SL}}(2,\mathbb {C} )}$ may be uniquely expressed as[71]

$${\displaystyle \lambda =ue^{h},}$$

where u is unitary with determinant one, hence in SU(2), and h is Hermitian with trace zero. The trace and determinant conditions imply:[72]

$${\displaystyle {\begin{aligned}h&={\begin{pmatrix}c&a-ib\\a+ib&-c\end{pmatrix}}&&(a,b,c)\in \mathbb {R} ^{3}\\[4pt]u&={\begin{pmatrix}d+ie&f+ig\\-f+ig&d-ie\end{pmatrix}}&&(d,e,f,g)\in \mathbb {R} ^{4}{\text{ subject to }}d^{2}+e^{2}+f^{2}+g^{2}=1.\end{aligned}}}$$

The manifestly continuous one-to-one map is a homeomorphism with continuous inverse given by (the locus of u is identified with ${\displaystyle \mathbb {S} ^{3}\subset \mathbb {R} ^{4}}$)

$${\displaystyle {\begin{cases}\mathbb {R} ^{3}\times \mathbb {S} ^{3}\to {\text{SL}}(2,\mathbb {C} )\\(r,s)\mapsto u(s)e^{h(r)}\end{cases}}}$$

explicitly exhibiting that ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ is simply connected. But ${\displaystyle {\text{SO}}(3;1)\cong {\text{SL}}(2,\mathbb {C} )/\{\pm I\},}$ where ${\displaystyle \{\pm I\}}$ is the center of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$. Identifying λ and −λ amounts to identifying u with −u, which in turn amounts to identifying antipodal points on ${\displaystyle \mathbb {S} ^{3}.}$ Thus topologically,[72]

$${\displaystyle {\text{SO}}(3;1)\cong \mathbb {R} ^{3}\times (\mathbb {S} ^{3}/\mathbb {Z} _{2}),}$$

where last factor is not simply connected: Geometrically, it is seen (for visualization purposes, ${\displaystyle \mathbb {S} ^{3}}$ may be replaced by ${\displaystyle \mathbb {S} ^{2}}$) that a path from u to −u in ${\displaystyle SU(2)\cong \mathbb {S} ^{3}}$ is a loop in ${\displaystyle \mathbb {S} ^{3}/\mathbb {Z} _{2}}$ since u and −u are antipodal points, and that it is not contractible to a point. But a path from u to −u, thence to u again, a loop in ${\displaystyle \mathbb {S} ^{3}}$ and a double loop (considering p(ue^h) = p(−ue^h), where ${\displaystyle p:{\text{SL}}(2,\mathbb {C} )\to {\text{SO}}(3;1)}$ is the covering map) in ${\displaystyle \mathbb {S} ^{3}/\mathbb {Z} _{2}}$ that is contractible to a point (continuously move away from −u "upstairs" in ${\displaystyle \mathbb {S} ^{3}}$ and shrink the path there to the point u).[72] Thus π₁(SO(3; 1)) is a group with two equivalence classes of loops as its elements, or put more simply, SO(3; 1) is doubly connected.

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]

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

$${\displaystyle \left({\frac {1}{2}}\sigma _{1},{\frac {1}{2}}\sigma _{2},{\frac {1}{2}}\sigma _{3},{\frac {i}{2}}\sigma _{1},{\frac {i}{2}}\sigma _{2},{\frac {i}{2}}\sigma _{3}\right)\equiv (j_{1},j_{2},j_{3},k_{1},k_{2},k_{3}),}$$

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)⁺.

More on covering groups in general and the ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ covering of the Lorentz group in particular

A geometric view

E.P. Wigner investigated the Lorentz group in depth and is known for the Bargmann-Wigner equations. The realization of the covering group given here is from his 1939 paper.

Let p_g(t), 0 ≤ t ≤ 1 be a path from 1 ∈ SO(3; 1)⁺ to g ∈ SO(3; 1)⁺, denote its homotopy class by [p_g] and let π_g be the set of all such homotopy classes. Define the set

$${\displaystyle G=\{(g,[p_{g}]):g\in \mathrm {SO} (3;1)^{+},[p_{g}]\in \pi _{g}\}}$$

(C1)

and endow it with the multiplication operation

$${\displaystyle (g_{1},[p_{1}])(g_{2},[p_{2}])=(g_{1}g_{2},[p_{12}]),}$$

(C2)

where ${\displaystyle p_{12}}$ is the path multiplication of ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$:

$${\displaystyle p_{12}(t)=(p_{1}*p_{2})(t)={\begin{cases}p_{1}(2t)&0\leqslant t\leqslant {\tfrac {1}{2}}\\p_{2}(2t-1)&{\tfrac {1}{2}}\leqslant t\leqslant 1\end{cases}}}$$

With this multiplication, G becomes a group isomorphic to ${\displaystyle {\text{SL}}(2,\mathbb {C} ),}$[74] the universal covering group of SO(3; 1)⁺. Since each π_g has two elements, by the above construction, there is a 2:1 covering map p : G → SO(3; 1)⁺. According to covering group theory, the Lie algebras ${\displaystyle {\mathfrak {so}}(3;1),{\mathfrak {sl}}(2,\mathbb {C} )}$ and ${\displaystyle {\mathfrak {g}}}$ of G are all isomorphic. The covering map p : G → SO(3; 1)⁺ is simply given by p(g, [p_g]) = g.

An algebraic view

For an algebraic view of the universal covering group, let ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ act on the set of all Hermitian 2×2 matrices ${\displaystyle {\mathfrak {h}}}$ by the operation[72]

$${\displaystyle {\begin{cases}\mathbf {P} (A):{\mathfrak {h}}\to {\mathfrak {h}}\\X\mapsto A^{\dagger }XA\end{cases}}\qquad A\in \mathrm {SL} (2,\mathbb {C} )}$$

(C3)

The action on ${\displaystyle {\mathfrak {h}}}$ is linear. An element of ${\displaystyle {\mathfrak {h}}}$ may be written in the form

$${\displaystyle X={\begin{pmatrix}\xi _{4}+\xi _{3}&\xi _{1}+i\xi _{2}\\\xi _{1}-i\xi _{2}&\xi _{4}-\xi _{3}\\\end{pmatrix}}\qquad \xi _{1},\ldots ,\xi _{4}\in \mathbb {R} .}$$

(C4)

The map P is a group homomorphism into ${\displaystyle {\text{GL}}({\mathfrak {h}})\subset {\text{End}}({\mathfrak {h}}).}$ Thus ${\displaystyle \mathbf {P}$ :{\text{SL}}(2,\mathbb {C} )\to {\text{GL}}({\mathfrak {h}})} is a 4-dimensional representation of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$. Its kernel must in particular take the identity matrix to itself, A^†IA = A^†A = I and therefore A^† = A⁻¹. Thus AX = XA for A in the kernel so, by Schur's lemma,[nb 19] A is a multiple of the identity, which must be ±I since det A = 1.[75] The space ${\displaystyle {\mathfrak {h}}}$ is mapped to Minkowski space M⁴, via

$${\displaystyle X=(\xi _{1},\xi _{2},\xi _{3},\xi _{4})\leftrightarrow {\overrightarrow {(\xi _{1},\xi _{2},\xi _{3},\xi _{4})}}=(x,y,z,t)={\overrightarrow {X}}.}$$

(C5)

The action of P(A) on ${\displaystyle {\mathfrak {h}}}$ preserves determinants. The induced representation p of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ on ${\displaystyle \mathbb {R} ^{4},}$ via the above isomorphism, given by

$${\displaystyle \mathbf {p} (A){\overrightarrow {X}}={\overrightarrow {AXA^{\dagger }}}}$$

(C6)

preserves the Lorentz inner product since

$${\displaystyle -\det X=\xi _{1}^{2}+\xi _{2}^{2}+\xi _{3}^{2}-\xi _{4}^{2}=x^{2}+y^{2}+z^{2}-t^{2}.}$$

This means that p(A) belongs to the full Lorentz group SO(3; 1). By the main theorem of connectedness, since ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ is connected, its image under p in SO(3; 1) is connected, and hence is contained in SO(3; 1)⁺.

It can be shown that the Lie map of ${\displaystyle \mathbf {p}$ :{\text{SL}}(2,\mathbb {C} )\to {\text{SO}}(3;1)^{+},} is a Lie algebra isomorphism: ${\displaystyle \pi$ :{\mathfrak {sl}}(2,\mathbb {C} )\to {\mathfrak {so}}(3;1).} [nb 20] The map P is also onto.[nb 21]

Thus ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$, since it is simply connected, is the universal covering group of SO(3; 1)⁺, isomorphic to the group G of above.

This diagram shows the web of maps discussed in the text. Here V is a finite-dimensional vector space carrying representations of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),{\mathfrak {so}}(3;1),}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ and ${\displaystyle {\text{SO}}(3;1)^{+}.}$ ${\displaystyle \exp }$ is the exponential mapping, p is the covering map from ${\displaystyle {\text{SL}}(2,\mathbb {C} ),}$ onto SO(3; 1)⁺ and σ is the Lie algebra isomorphism induced by it. The maps Π, π and the two Φ are representations. the picture is only partially true when Π is projective.

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)

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.

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

$${\displaystyle P{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=c_{n}z_{1}^{n}+c_{n-1}z_{1}^{n-1}z_{2}+\cdots +c_{0}z_{2}^{n},\quad c_{0},c_{1},\ldots ,c_{n}\in \mathbb {Z} .}$$

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.

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