Representation theory of the Lorentz group

Non-technical introduction to representation theory

The present purpose is to illustrate the role of representation theory of groups in mathematics and in physics. Rigor and detail take the back seat, as the main objective is to fix the notion of finite-dimensional and infinite-dimensional representations of the Lorentz group. The reader familiar with these concepts should skip by.

Applications

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.

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.

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. There were speculative theories, (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.

Mathematics

From the point of view that the goal of mathematics is to classify and characterize, the representation theory of the Lorentz group is since 1947 a finished chapter. But in association with the Bargmann–Wigner programme, there are (as of 2006) yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations.

The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the little group of the Poincare group of spacelike vectors in higher spacetime dimension. The corresponding infinite-dimensional unitary representations of the (generalized) Poincaré group are the so-called tachyonic representations. Tachyons appear in the spectrum of bosonic strings and are associated with instability of the vacuum. Even though tachyons may not be realized in nature, these representations must be mathematically understood in order to understand string theory. This is so since tachyon states turn out to appear in superstring theories too in attempts to create realistic models.

One open problem (as of 2006) is the completion of the Bargmann–Wigner programme for the isometry group SO(D – 2, 1) of the de Sitter spacetime dS_{D – 2}. Ideally, one would like to see the physical components of wave functions realized on the hyperboloid dS_{D – 2} of radius μ > 0 embedded in ℝ^{D − 2, 1} and the corresponding O(D − 2, 1) covariant wave equations of the infinite-dimensional unitary representation to be known.

It is common in mathematics to regard the Lorentz group to be, foremost, the Möbius group to which it is isomorphic. The group may be represented in terms of a set of functions defined on the Riemann sphere. These are the Riemann P-functions, which are expressible as hypergeometric functions.

Classical field theory

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, one begins with one or more classical fields, where e.g. the wave functions solving the Dirac equation are considered as classical fields prior to (second) quantization. While second quantization and the Lagrangian formalism associated with it is not a fundamental aspect of QFT, it is the case that so far all quantum field theories can be approached this way, including the standard model. 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.

Relativistic quantum mechanics

For the present purpose one may make the following definition: A relativistic wave function is a set of n functions ψ^α on spacetime which transforms under an arbitrary proper Lorentz transformation Λ as

ψ ′ α ( x ) = D [ Λ ] α β ψ β ( Λ − 1 x ) ,

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 and the Dirac equation 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).

Quantum field theory

In QFT, the demand for relativistic invariance enters, among other ways in that the S-matrix necessarily must be Poincaré invariant. This has the implication that there is one or more infinite-dimensional representation of the Lorentz group acting on Fock space. One way to guarantee the existence of such representations is the existence of a Lagrangian description of the system using the canonical formalism, from which one may deduce a realization of the generators of the Lorentz group.

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. For illustration, consider the definition of some n-component field operator: Given a matrix representation as above, a relativistic field operator is a set of n operator valued functions on spacetime which transforms under proper Lorentz transformations Λ according to

Ψ α ( x ) → Ψ ′ α ( x ′ ) = U [ Λ ] Ψ α ( x ) U [ Λ − 1 ] = D [ Λ − 1 ] α β Ψ β ( Λ x )

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), one finds

Ψ α ( x ) = ∑ σ ∫ d p [ a ( p , σ ) u α ( p , σ ) e i p ⋅ x + a † ( p , σ ) v α ( p , σ ) e − i p ⋅ x ] ,

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

a † ( p , σ ) → a ′ † ( p ′ , σ ) = U [ Λ ] a † ( p , σ ) U [ Λ − 1 ] = a † ( Λ p , ρ ) D ( s ) [ R ( Λ , p ) − 1 ] ρ σ ,

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 is the wave function, also called cofficient function

v α ( p , σ ) e − i p ⋅ x

that carries 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. 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, 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

Speculative theories

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).

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. This results in a relativistically invariant theory in any spacetime dimension. 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. 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 Lorentz Lie algebra. The only possible dimension of spacetime in such theories is 10.

Finite-dimensional representations

Representation theory of groups in general, and Lie groups in particular, is a very rich subject. The full Lorentz group is no exception. 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. Perhaps most importantly, 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, one has to deal with time reversal and space inversion separately.

History

The development of the finite-dimensional representation theory of the Lorentz group mostly follows that of the subject 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 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, see History of infinite-dimensional unitary representations below, due to its exceptional importance in physics. 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.

Strategy

Classification of the finite-dimensional irreducible representations generally consists of two steps. The first step is to hypothesize the existence of representations. One assumes heuristically that all representations that a priori could exist, do exist. One investigates the properties of these hypothetical representations, primarily using the Lie algebra. The goal of this study is twofold. First, some of these hypothetical representations may not exist. The goal in this situation is to show that their existence would imply a falsehood such as 0 = 1. If this can be done, then the initial hypothesis that the representation existed must be false, and one can therefore exclude these hypothetical representations from later studies. Second, one can better understand the representations that do exist. These representations must have enough structure to manifest the symmetries of the group action, but describing this structure may not be easy. Before a classification has been completed, it is unclear which representations fall into the first class and which fall into the second.

If this first step of the classification is successful, it results in a tentative classification of the possible representations. This is often a short list. Each list entry is a single representation or a family of related representations, and ideally, the entry gives requirements so specific that they can be met by at most a single representation. The second step consists of explicit construction of the representations on this list. If successful, it justifies the existence hypotheses made in the first step. The results of investigations performed in the first step provide hints about how to construct the representations, i.e. construction of a vector space V and a specified Lie algebra action on V, since most of the properties they must have are then known.

For finite-dimensional irreducible representations of finite-dimensional semisimple Lie algebras the general result is Cartan's theorem of highest weight. It provides a classification of the irreducible representations in terms of the weights of the Lie algebra.

For some semisimple Lie algebras, especially non-compact ones, it is easier to proceed indirectly via Weyl's unitarian trick instead of applying Cartan's theorem directly. In the present case of so(3; 1) one sets up a chain of isomorphisms between Lie algebras and other correspondences preserving irreducible representations, so that the representations may be obtained from representations of SU(2) ⊗ SU(2). See equation (A1) and references around it. It is essential here that SU(2) is compact, since then the irreducible representations of SU(2) ⊗ SU(2) are simply tensor products of irreducible representations of SU(2), that can all be obtained from the irreducible representations of su(2).

Then the classification part. Cartan's theorem is applied to su(2) (together with knowledge of its highest weights) and one obtains a classification of the representations of so(3; 1) via (A1). An explicit construction of the representations of SL(2, ℂ) is then given (which is not much more difficult to obtain than the more basic su(2) representations), thus completing the task with the (m, n) representations of so(3; 1) as the final result.

Representative matrices may be obtained by choice of basis in the representation space. An explicit formula for matrix elements is presented and some common representations are listed.

The Lie correspondence is subsequently employed for obtaining group representations of the connected component of the Lorentz group, SO(3, 1)⁺. This is effected by taking the matrix exponential of the matrices of the Lie algebra representation, a topic which is investigated in some depth. A subtlety arises due to the (in physics parlance) doubly connected nature of SO(3, 1)⁺. This results in the projective representations or two-value representations that are actually spin representations of the covering group SL(2, ℂ).

The Lie correspondence gives results only for the connected component of the groups, and thus the components of the full Lorentz that contain the operations of time reversal and space inversion are treated separately, mostly from physical considerations, by defining representatives for the space inversion and time reversal matrices.

The Lie algebra

According to the general representation theory of Lie groups, one first looks for the representations of the complexification, so(3; 1)_C of the Lie algebra so(3; 1) of the Lorentz group. A convenient basis for 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.

Now complexify the Lie algebra, and then change basis to the components of

A = J + i K 2 , B = J − i K 2 .

One may verify that the components of A = (A₁, A₂, A₃) and B = (B₁, B₂, B₃) separately satisfy the commutation relations of the Lie algebra su(2) and moreover that 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 ,

where i, j, k are indices which each take values 1, 2, 3, and ε_ijk is the three-dimensional Levi-Civita symbol. Let A_C and B_C denote the complex linear span of A and B respectively.

One has the isomorphisms

where sl(2, C) is the complexification of su(2) ≈ A ≈ B.

The utility of these isomorphisms comes from the fact that all irreducible representations of su(2) are known. Every irreducible representation of su(2) is isomorphic to one of the highest weight representations. Moreover, there is a one-to-one correspondence between linear representations of su(2) and complex linear representations of sl(2, C).

The unitarian trick

In (A1), all isomorphisms are C-linear (the last is just a defining equality). The most important part of the manipulations below is that the R-linear (irreducible) representations of a (real or complex) Lie algebra are in one-to-one correspondence with C-linear (irreducible) representation of its complexification. With this in mind, it is seen that the R-linear representations of the real forms of the far left, so(3; 1), and the far right, sl(2, C), in (A1) can be obtained from the C-linear representations of sl(2, C) ⊕ sl(2, C).

The manipulations to obtain representations of a non-compact algebra (here so(3; 1)), and subsequently the non-compact group itself, from qualitative knowledge about unitary representations of a compact group (here SU(2)) is a variant of Weyl's so-called unitarian trick. The trick specialized to SL(2, C) can be summarized concisely.

Let V be a finite-dimensional complex vector space. The following statements are equivalent, in the sense that if one of them holds, then there is a uniquely determined (modulo choice of basis for V) corresponding representation (either via given Lie algebra isomorphisms, or via complexification of Lie algebras per above, or via restriction to real forms, or via the exponential mapping (to be introduced), or, finally, via a standard mechanism (also to be introduced) for obtaining Lie algebra representations given group representations) of the appropriate type for the other groups and Lie algebras:

There is a representation of SL(2, R) on V.

There is a representation of SU(2) on V.

There is a holomorphic representation of SL(2, C) on V.

There is a representation of sl(2, R) on V.

There is a representation of su(2) on V.

There is a complex linear representation of sl(2, C) on V.

If one representation is irreducible, then all of them are. In this list, direct products (groups) or direct sums (Lie algebras) may be introduced (if done consistently). The essence of the trick is that the starting point in the above list is immaterial. Both qualitative knowledge (like existence theorems for one item on the list) and concrete realizations for one item on the list will translate and propagate, respectively, to the others.

Now, the representations of sl(2, C) ⊕ sl(2, C), which is the Lie algebra of SL(2, C) × SL(2, C), are supposed to be irreducible. This means that they must be tensor products of complex linear representations of sl(2, C), as can be seen by restriction to the subgroup SU(2) × SU(2) ⊂ SL(2, C) × SL(2, C), a compact group to which the Peter–Weyl theorem applies. The irreducible unitary representations of SU(2) × SU(2) are precisely the tensor products of irreducible unitary representations of SU(2). These stand in one-to-one correspondence with the holomorphic representations of SL(2, C) × SL(2, C) and these, in turn, are in one-to-one correspondence with the complex linear representations of sl(2, C) ⊕ sl(2, C) because SL(2, C) × SL(2, C) is simply connected.

For sl(2, C), there exists the highest weight representations (obtainable, via the trick, from the corresponding su(2)-representations), here indexed by μ for μ = 0, 1, … . The tensor products of two complex linear factors then form the irreducible complex linear representations of sl(2, C) ⊕ sl(2, C). For reference, if (π₁, U) and (π₂, V) are representations of a Lie algebra g, then their tensor product (π₁ ⊗ π₂, U ⊗ V) is given by either of

where Id is the identity operator. Here, the latter interpretation is intended. The not necessarily complex linear representations of sl(2, C) come using another variant of the unitarian trick as is shown in the last Lie algebra isomorphism in (A1).

The representations

The representations for all Lie algebras and groups involved in the unitarian trick can now be obtained. The real linear representations for sl(2, C) and so(3; 1) follow here assuming the complex linear representations of sl(2, C) are known. Explicit realizations and group representations are given later.

sl(2, C)

The complex linear representations of the complexification of sl(2, C), sl(2, C)_C, obtained via isomorphisms in (A1), stand in one-to-one correspondence with the real linear representations of sl(2, C). The set of all, at least real linear, irreducible representations of sl(2, C) are thus indexed by a pair (μ, ν). The complex linear ones, corresponding precisely to the complexification of the real linear 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 sl(2, C) into its complexification. Representations on the form (ν, ν) or (μ, ν) ⊕ (ν, μ) are given by real matrices (the latter is not irreducible). Explicitly, the real linear (μ, ν)-representations of sl(2, C) are

where Φ_μ, μ = 0,1, … are the complex linear irreducible representations of sl(2, C) and Φ_ν, ν = 0,1, … their complex conjugate representations. Here the tensor product is interpreted in the former sense of (A0). These representations are concretely realized below.

so(3; 1)

Via the displayed isomorphisms in (A1) and knowledge of the complex linear irreducible representations of sl(2, C) ⊕ sl(2, C), upon solving for J and K, all irreducible representations of so(3; 1)_C, and, by restriction, those of so(3; 1) are known. It's worth noting that the representations of 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 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/2 and n = ν/2, conventionally written as one of

( m , n ) ≡ D ( m , n ) ≡ π m , n .

The notation D^(m,n) is usually reserved for the group representations. Let π_(m, n) : so(3; 1) → gl(V), where V is a vector space, denote the irreducible representations of so(3; 1) according to this classification. These are, up to a similarity transformation, uniquely given by

where the J⁽ⁿ⁾ = (J⁽ⁿ⁾₁, J⁽ⁿ⁾₂, J⁽ⁿ⁾₃) are the (2n + 1)-dimensional irreducible spin n representations of so(3) ≈ su(2) and 1_n is the n-dimensional unit matrix.

Explicit formula for matrix elements

Let π_(m, n) : so(3; 1) → gl(V), where V is a vector space, denote the irreducible representations of so(3; 1) according to the (m,  n) classification. In components, with −m ≤ a, a′ ≤ m, −n ≤ b, b′ ≤ n, the representations are given by

where δ is the Kronecker delta and the J_i⁽ⁿ⁾ are the (2n + 1)-dimensional irreducible representations of so(3), also termed spin matrices or angular momentum matrices. These are explicitly given as

Common representations

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) has a particular relevance to physics, since it permits to use linear operators over real numbers.

(0, 0) is the Lorentz scalar representation. This representation is carried by relativistic scalar field theories.

(1/2, 0) is the left-handed Weyl spinor and (0, 1/2) is the right-handed Weyl spinor representation. Fermionic supersymmetry generators transform under one of these representations.

(1/2, 0) ⊕ (0, 1/2) is the bispinor representation. (See also Dirac spinor and Weyl spinors and bispinors below.)

(1/2, 1/2) is the four-vector representation. The four-momentum of a particle (either massless or massive) transforms under this representation.

(1, 0) is the self-dual 2-form field representation and (0, 1) is the anti-self-dual 2-form field representation.

(1, 0) ⊕ (0, 1) is the adjoint representation and the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation.

(1, 1/2) ⊕ (1/2, 1) is the Rarita–Schwinger field representation.

(1, 1) is the spin 2 representation of a traceless symmetric tensor field. A physical example is the traceless part of the energy-momentum tensor T^μν.

(3/2, 0) ⊕ (0, 3/2) would be the symmetry of the hypothesized gravitino. It can be obtained from the (1, 1/2) ⊕ (1/2, 1)-representation.

The group

The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence. The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras. The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted exp:g → G. It is one-to-one in a neighborhood of the identity.

The Lie correspondence

The Lie correspondence and some results based on it needed here and below are stated for reference. If G denotes a Lie group and g a Lie algebra, let Γ(g) denote the group generated by exp(g), the image of the Lie algebra under the exponential mapping, and let L(G) denote the Lie algebra of G. The Lie correspondence reads in modern language as follows:

There is a one-to-one correspondence between connected and simply connected Lie groups G and Lie algebras g under which g corresponds to L(G) and G to Γ(g). Equivalently, Γ(L(G)) = G and L(Γ(g)) = g. (Lie)

A linear Lie group is one that has at least one faithful finite-dimensional representation. The following are some corollaries that will be used in the sequel:

A connected linear Lie group G is abelian if and only if g is abelian. (Lie i)

A connected subgroup H with Lie algebra h of a connected linear Lie group G is normal if and only if h ⊂ g is an ideal. (Lie ii)

If G, H are linear Lie groups with Lie algebras g, h and Π:G → H is a group homomorphism, then π:g → h, its pushforward at the identity, is a Lie algebra homomorphism and Π(e^iX) = e^iπ(X) for every X ∈ g. (Lie iii)

Lie algebra representations from group representations

Using the above theorem it is always possible to pass from a representation of a Lie group G to a representation of its Lie algebra g. If Π : G → GL(V) is a group representation for some vector space V, then its pushforward (differential) at the identity, or Lie map, π : g → End V is a Lie algebra representation. It is explicitly computed using

This, of course, holds for the Lorentz group in particular, but not all Lie algebra representations arise this way because their corresponding group representations may not exist as proper representations, i.e. they are projective, see below.

Group representations from Lie algebra representations

Given a so(3; 1) representation, one may try to construct a representation of SO(3; 1)⁺, the identity component of the Lorentz group, by using the exponential mapping. Since SO(3; 1)⁺ is a matrix Lie group, the exponential mapping is simply the matrix exponential. If X is an element of so(3; 1) in the standard representation, then

is a Lorentz transformation by general properties of Lie algebras. Motivated by this and the Lie correspondence theorem stated above, let π : so(3; 1) → gl(V) for some vector space V be a representation and tentatively define a representation Π of SO(3; 1)⁺ by first setting

The subscript U indicates a small open set containing the identity. Its precise meaning is defined below. There are at least two potential problems with this definition. The first is that it is not obvious that this yields a group homomorphism, or even a well defined map at all (local existence). The second problem is that for a given g ∈ U ⊂ SO(3; 1)⁺ there may not be exactly one X ∈ so(3; 1) such that g = e^iX (local uniqueness). The soundness of the tentative definition (G2) is shown in several steps below:

1. Π_U is a local homomorphism.
2. Π(g) defined along a path using properties of Π_U is a global homomorphism.
3. The exponential mapping exp:so(3; 1) → SO(3; 1)⁺ is surjective.
4. Π(g) defined along a path coincides with Π_U(g) with U = SO(3; 1)⁺.

Local existence and uniqueness

A theorem based on the inverse function theorem states that the map exp : so(3; 1) → SO(3; 1)⁺ is one-to-one for X small enough (A). This makes the map well-defined. The qualitative form of the Baker–Campbell–Hausdorff formula then guarantees that it is a group homomorphism, still for X small enough (B). Let U ⊂ SO(3; 1)⁺ denote image under the exponential mapping of the open set in so(3; 1) where conditions (A) and (B) both hold. Let g, h ∈ U, g = e^X, h = e^Y, then

This shows that the map Π_U is a well-defined group homomorphism on U.

Global existence and uniqueness

Technically, formula (G2) is used to define Π near the identity. For other elements g ∉ U one chooses a path from the identity to g and defines Π along that path by partitioning it finely enough so that formula (G2) can be used again on the resulting factors in the partition. In detail, one sets

where the g_i are on the path and the factors on the far right are uniquely defined by (G2) provided that all g_i g_i+1⁻¹ ∈ U and, for all conceivable pairs h,k of points on the path between g_i and g_i+1, hk⁻¹ ∈ U as well. For each i take, by the inverse function theorem, the unique X_i such that exp(X_i) = g_ig_i−1⁻¹ and obtain

By compactness of the path there is an n large enough so that Π(g) is well defined, possibly depending on the partition and/or the path, whether g is close to the identity or not.

Partition independence

It turns out that the result is always independent of the partitioning of the path. To demonstrate the independence of a chosen path, one employs the Baker–Campbell–Hausdorff formula. It shows that Π_U is a group homomorphism for elements in U.

To see this, first fix a partitioning used in (G3). Then insert a new point h somewhere on the path, say

g = ⋯ ( g i + 1 h − 1 ) ( h g i − 1 ) ⋯ , ⋯ Π U ( g i + 1 h − 1 ) Π U ( h g i − 1 ) ⋯ .

But

⋯ Π U ( g i + 1 h − 1 ) Π U ( h g i − 1 ) ⋯ = ⋯ Π U ( g i + 1 h − 1 h g i − 1 ) ⋯ = ⋯ Π U ( g i + 1 g i − 1 ) ⋯

as a consequence of the Baker–Campbell–Hausdorff formula and the conditions on the original partitioning. Thus, adding a point on the path has no effect on the definition of Π(g).

Then, for any two given partitions of a given path, they have common refinement, their union. This refinement can be reached from any of the two partitionings by, one-by-one, adding points from the other partition. No individual addition changes the definition of Π(g), hence, since there are finitely many points in each partition, the value of Π(g) must have been the same for the two partitionings to begin with.

Path independence

For simply connected groups, the construction will be independent of the path as well, yielding a well defined representation. In that case formula (G2) can unambiguously be used directly. Simply connected spaces have the property that any two paths can be continuously deformed into each other. Any such deformation is called a homotopy and is usually chosen as a continuous function H from the unit square {s,t ∈ R: 0 ≤ s, t ≤ 1} into the group. For s = 0 the image is one of the paths, for s = 1 the other, for intermediate s, an intermediate path results, but endpoints are kept fixed.

One deforms the path, a little bit at a time, using the previous result, the independence of partitioning. Each consecutive deformation is so small that two consecutive deformed paths can be partitioned using the same partition points. Thus two consecutive deformed paths yield the same value for Π(g). But any two pairs of consecutive deformations need not have the same choice partition points, so the actual path laid out in the group as one progresses through the deformation does indeed change.

Using compactness arguments, in a finite number of steps, the original (s = 0) path is deformed into the other (s = 1) without affecting the value of Π(g).

Global homomorphism

The map Π_U is, by the Baker-Campbell-Hausdorff formula, a local homomorphism. To show that Π is a global homomorphism, consider two elements g, h ∈ SO(3; 1)⁺. Lay out paths p_g, p_h from the identity to them and define a path p_gh going along p_g(2t) for 0 ≤ t ≤ 1/2 and along p_g · p_h(2t - 1) for 1/2 ≤ t ≤ 1. This is a path from the identity to gh. Select adequate partitionings for p_g, p_h. This corresponds to a choice of "times" t₀, t₁, ...t_m and s₀, s₁, ...s_n. Divide the first set with 2 and divide the second set with 2 and add 1/2 and so obtain a new (adequate) set of "times" to be used for p_gh. Direct computation shows that, with these partitionings (and hence all partitionings), Π(gh) = Π(g)Π(h).

Surjectiveness of exponential mapping

From a practical point of view, it is important that formula (G2) can be used for all elements of the group. The Lie correspondence theorem above guarantees that (G2) holds for all X ∈ so(3; 1), but provides no guarantee that all g ∈ SO(3; 1)⁺ are in the image of exp:so(3; 1) → SO(3; 1)⁺. For general Lie groups, this is not the case, especially not for non-compact groups, as for example for SL(2, C), the universal covering group of SO(3; 1)⁺. It will be treated in this respect below.

But exp: so(3; 1) → SO(3; 1)⁺ is surjective. One way to see this is to make use of the isomorphism SO(3; 1)⁺ ≈ PGL(2, C), the latter being the Möbius group. It is a quotient of GL(n, C) (see the linked article). Let p:GL(n, C) → PGL(2, C) denote the quotient map. Now exp:gl(n, C) → GL(n, C) is onto. Apply the Lie correspondence theorem with π being the differential at the identity of p. Then for all X ∈ gl(n, C) p(e^iX) = e^iπ(X). Since the left hand side is surjective (both exp and p are), the right hand side is surjective and hence exp:pgl(2, C) → PGL(2, C) is surjective. Finally, recycle the argument once more, but now with the known isomorphism between SO(3; 1)⁺ and PGL(2, C) to find that exp is onto for the connected component of the Lorentz group.

Consistency

From the way Π(g) has been defined for elements far from the identity, it not immediately clear that formula (G2) holds for all elements of SO(3; 1)⁺, i.e. that one can take U = G in (G2). But, in summary,

Π is a uniquely constructed homomorphism.

Using (G6) with Π as defined here, then one ends up with the π one started with since Π was defined that way near the identity, and (G6) depends only on an arbitrarily small neighborhood of the identity.

exp: so(3; 1) → SO(3; 1)⁺ is surjective.

Hence (G2) holds everywhere. One finally unconditionally writes

Fundamental group

The above construction relies on simple connectedness. The result needs modifications for non-simply connected groups per below. To exhibit the fundamental group of SO(3; 1)⁺, one may consider first the topology of its covering group SL(2, ℂ). By the polar decomposition theorem, any matrix λ ∈ SL(2, C) may be uniquely expressed as

λ = u e h , det u = 1 , tr ⁡ v = 0 ,

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

h = ( c a − i b a + i b c − c ) , u = ( d + i e f + i g − f + i g d − i e ) , d 2 + e 2 + f 2 + g 2 = 1 ,

with (a, b, c) ∈ ℝ³ unconstrained and (d, e, f, g) ∈ ℝ⁴ constrained to the 3-sphere S³. It follows that the manifestly continuous one-to-one map ℝ³ × S³ → SL(2, ℂ); (r, s) ↦ u(s)e^h(r) is a homeomorphism (hence preserves the fundamental group). Since ℝⁿ is simply connected for all n and Sⁿ is simply connected for n > 1 and since simple connectedness is preserved under cartesian products, it follows that SL(2, ℂ) is simply connected. Now, SO(3; 1) ≈ SL(2, ℂ)/{I, −I}, where {I, −I} is the center of SL(2, ℂ). Identifying λ and −λ amounts to identifying u with −u, which in turn amounts to identifying antipodal points on S³. Thus topologically,

S O ( 3 ; 1 ) ≈ R 3 × S 3 / Z 2 ,

where last factor is not simply connected: Geometrically, it is easy to see (for visualization purposes, replace S³ by S²) that a path from u to −u in SU(2) ≈ S³ is a loop in S³/Z₂ 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 S³ and a double loop (considering p(ue^h) = p(−ue^h), where p is the covering map SL(2, ℂ) → SL(3; 1)) in S³/Z₂ that is contractible to a point (continuously move away from −u "upstairs" in S³ and shrink the path there to the point u). Thus π₁(SO(3; 1)) is a two-element group with two equivalence classes of loops as its elements – or put more simply, SO(3; 1) is doubly connected.

Projective representations

For a group that is connected but not simply connected, such as SO(3; 1)⁺, the result may depend on the homotopy class of the chosen path. The result, when using (G2), will then depend on which X in the Lie algebra is used to obtain the representative matrix for g.

Since π₁(SO(3; 1)⁺) per above has two elements, not all representations of the Lie algebra will yield representations of the group, but some will instead yield projective representations. Once these conclusions have been reached, and once one knows whether a representation is projective, there is no need to be concerned about paths and partitions. Formula (G2) applies to all group elements and all representations, including the projective ones.

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

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

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. One cannot consistently chose a sign to obtain a continuous representation of all of SO(3; 1)⁺, but this is possible locally around any point.

The covering group

Consider sl(2, C) as a real Lie algebra with basis

( 1 2 σ 1 , 1 2 σ 2 , 1 2 σ 3 , i 2 σ 1 , i 2 σ 2 , i 2 σ 3 ) ≡ ( j 1 , j 2 , j 3 , k 1 , k 2 , k 3 ) ,

where the sigmas are the Pauli matrices. From the relations

one obtains

which are exactly on the form of the 3-dimensional version of the commutation relations for so(3; 1) (see conventions and Lie algebra bases below). Thus, one may map J_i ↔ j_i, K_i ↔ k_i, and extend by linearity to obtain an isomorphism. Since SL(2, C) is simply connected, it is the universal covering group of SO(3; 1)⁺.

A geometric view

Let π_g denote the set of path homotopy classes [p_g] of paths p_g(t), 0 ≤ t ≤ 1, from 1 ∈ SO(3; 1)⁺ to g ∈ SO(3; 1)⁺ and define the set

and endow it with the multiplication operation

The dot on the far right denotes path multiplication.

With this multiplication, G is a group and G ≈ SL(2, C), the universal covering group of SO(3; 1)⁺. By the above construction, there is, since each π_g has two elements, a 2:1 covering map p : G → SO(3; 1)⁺ and an isomorphism G ≈ SL(2, C). According to covering group theory, the Lie algebras so(3; 1), sl(2, C) and 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 SL(2, C) act on the set of all Hermitian 2×2 matrices h by the operation

Since X ∈ h is Hermitian, A^†XA is again Hermitian because (A^†XA)^† = A^†X^†A^†† = A^†XA, and also A^†(αX + βY)A = αA^†XA + βA^†YA, so the action is linear as well. An element of h may generally be written in the form

for ξ_i real, showing that h is a 4-dimensional real vector space. Moreover, (AB)^†X(AB) = B^†A^†XAB meaning that P is a group homomorphism into GL(h) ⊂ End h. Thus P : SL(2, C) → GL (h) is a 4-dimensional representation of SL(2, C). Its kernel must in particular take the identity matrix to itself, A^†IA = A^†A = I ⇒ A^† = A⁻¹. Thus AX = XA for A in the kernel so, by Schur's lemma, A is a multiple of the identity, which must be ±I since det A = 1. Now map h to spacetime R⁴ endowed with the Lorentz metric, Minkowski space, via

The action of P(A) on h preserves determinants since det(A^†XA) = (det A)(det A^†)(det X) = det X. The induced representation p of SL(2, C) on R⁴, via the above isomorphism, given by

will preserve the Lorentz inner product since

−det X = ξ₁² + ξ₂² + ξ₃² − ξ₄² = x² + y² + z² − t².

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

It can be shown that the Lie map of p : SL(2, C) → SO(3; 1)⁺, π : sl(2, C) → so(3; 1) is a Lie algebra isomorphism (its kernel is {∅} and must therefore be an isomorphism for dimensional reasons). The map P is also onto.

Thus SL(2, C), since it is simply connected, is the universal covering group of SO(3; 1)⁺, isomorphic to the group G of above.

Representations of SL(2, C) and sl(2, C)

The complex linear representations of sl(2, C) and SL(2, C) are more straightforward to obtain than the SO(3; 1)⁺ representations. If π_μ is a representation of su(2) with highest weight μ, then the complexification of π_μ is a complex linear representation of sl(2, C). All complex linear representation of sl(2, C) are of this form. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are obtained by exponentiation. By simple connectedness of SL(2, C), this always yields a representation of the group as opposed to in the SO(3; 1)⁺ case. The real linear representations of sl(2, C) are exactly the (μ, ν)-representations presented earlier. 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.

It is also possible to obtain representations of SL(2, C) directly. This will be done below. Then, using the unitarian trick, going the other way, one finds sl(2, C)-,SU(2)-,su(2)-,SL(2, R)-, and sl(2, R)-representations as well as so(3; 1)-representations (via (A1)) and, possibly projective, SO(3; 1)⁺-representations (via projection from SL(2, C), see below, or exponentiation).

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 sl(2, C) be

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

Concrete realization

The irreducible holomorphic (n + 1)-dimensional representations of SL(2, C), n ≥ 0, can be realized on a set of functions ℙ²_n = {P:C² → C} where each P ∈ ℙ²_n is a homogeneous polynomial of degree n in 2 variables. The elements of ℙ²_n appears as P(z₁, z₂) = c_nz₁ⁿ + c_n−1z₁ⁿ⁻¹z₂ + ... + c_nz₂ⁿ. The action of SL(2, C) is given by

The associated sl(2, C)-action is, using (G6) and the definition above, given by

Defining z(t) = e^−tXz = (z₁(t), z₂(t))^T and using the chain rule one finds

The basis elements of sl(2, C) are then represented by

on the space P ∈ ℙ²_n (all n). By employing the unitarian trick one obtains representations for SU(2), su(2), SL(2, R, and sl(2, R), all are obtained by restriction of either (S2) or (S4). They are formally identical to (S2) or (S4). With a choice of basis for P ∈ ℙ²_n, all these representations become matrix groups or matrix Lie algebras.

The (μ, ν)-representations are realized on a space of polynomials ℙ²_μν in z₁, z₁, z₂, z₂, homogeneous of degree μ in z₁, z₂ and homogeneous of degree ν in z₁, z₂. The representations are given by

By carrying out the same steps as above, one finds

from which the expressions

for the basis elements follow.

Non-surjectiveness of exponential mapping

Unlike in the case exp: so(3; 1) → SO(3; 1)⁺, the exponential mapping exp: sl(2, C) → SL(2, C) is not onto. The conjugacy classes of SL(2, C) are represented by the matrices

but there is no element Q in sl(2, C) such that q = exp(Q).

In general, if g is an element of a connected Lie group G with Lie algebra g, then

This follows from the compactness of a path from the identity to g and the one-to-one nature of exp near the identity. In the case of the matrix q, one may write

The kernel of the covering map p:SL(2, C) → SO(3; 1)⁺ of above is N = {I, −I}, a normal subgroup of SL(2, C)⁺. The composition p ∘ exp: sl(2, C) → SO(3; 1) is onto. If a matrix a is not in the image of exp, then there is a matrix b equivalent to it with respect to p, meaning p(b) = p(a), that is in the image of exp. The condition for equivalence is a⁻¹b ∈ N. In the case of the matrix q, one may solve for p in the equation p⁻¹q = -I ∈ N. One finds

As a corollary, since the covering map p is a homomorphism,the mapping version of the Lie correspondence (G6) can be used to provide a proof of the surjectiveness of exp for so(3; 1). Let σ denote the isomorphism between sl(2, C) and so(3; 1). Refer to the commutative diagram. One has p ∘ exp: sl(2, C) → SO(3; 1) = exp ∘ σ for all X ∈ sl(2, C). Since p ∘ exp is onto, exp ∘ σ is onto, and hence exp: so(3; 1) → SO(3; 1)⁺ is onto as well.

SO(3; 1)⁺-representations from SL(2, C)-representations

By the first isomorphism theorem, a representation (Φ, V) of SL(2, C) descends to a representation (Π, V) of SO(3; 1)⁺ if and only if ker p ⊂ ker Φ. Refer to the commutative diagram. If this condition holds, then both elements in the fiber p⁻¹(g), g ∈ SO(3; 1)⁺ will be mapped by Φ to the same representative, and the expression Φ(p⁻¹(g)) makes sense. One may thus define Π: SO(3; 1)⁺ → GL(V), Π(g) = Φ(p⁻¹(g)). In particular, if Π is faithful, i.e. having kernel = I, then there is no corresponding proper representation of SO(3; 1)⁺, but there is a projective one as was shown in a previous section, corresponding to the two possible choices of representative in each fiber p⁻¹(g).

Lie algebra representations of so(3; 1) are obtained from sl(2, C)-representations simply by composition with σ⁻¹.

SL(2, C)-representations from SO(3; 1)⁺-representations

SL(2, C)-representations can be obtained from non-projective SO(3; 1)⁺-representations by composition with the projection map p. These are always representations since they are compositions of group homomorphisms. Such a representation is never faithful because Ker p = {I, −I}. If the SO(3; 1)⁺-representation is projective, then the resulting SL(2, C)-representation would be projective as well. Instead, the isomorphism σ:so(3; 1) → sl(3, C) can be employed, composed with exp:sl(2, C) → SL(2, C). This is always a non-projective representation.

Properties of the (m, n) representations

The (m, n) representations are irreducible, and they are the only irreducible representations.

Irreducibility follows from the unitarian trick and that a representation Π of SU(2) × SU(2) is irreducible if and only if Π = Π_μ ⊗ Π_ν, 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.

Dimension

The (m, n) representations are (2m + 1)(2n + 1)-dimensional. It follows from the Weyl dimension formula. For a Lie algebra g it reads

dim ⁡ π μ = Π α ∈ R + ⟨ α , μ + δ ⟩ Π α ∈ R + ⟨ α , δ ⟩ ,

where R⁺ is the set of positive roots and δ is half the sum of the positive roots. The inner product <⋅,⋅> is that of the Lie algebra g, invariant under the action of the Weyl group on h ⊂ g, the Cartan subalgebra. The roots (really elements of h*) are via this inner product identified with elements of h. For sl(2, C), the formula reduces to dim π_μ = μ + 1 = 2m + 1. By taking tensor products, the result follows.

A quicker approach is, of course, to simply count the dimensions in any concrete realization, such as the one given in representations of SL(2, C) and sl(2,  C).

Faithfulness

If a representation Π of a Lie group G is not faithful, then N = ker Π is a nontrivial normal subgroup because Π(n) = I ⇒ Π(gng⁻¹) = Π(g)Π(n)Π(g)⁻¹ = Π(g)Π(g)⁻¹ = I. 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.

In the case of SO(3; 1)⁺, the first case is excluded since SO(3; 1)⁺ is semi-simple. The second case (and the first case) is excluded because SO(3; 1)⁺ is simple. For the third case, SO(3; 1)⁺ is isomorphic to the quotient SL(2, C)/{I, −I}. But {I, −I} is the center of SL(2, 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, and the center of SO(3; 1)⁺ replaced by the center of sl(3; 1)⁺. The center of any semisimple Lie algebra is trivial and 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 SL(2, ℂ) is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding SL(2, ℂ) representation is not faithful, but is 2:1.

Non-unitarity

The (m, n) Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary. 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. There is a topological proof of this. 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, ker U, 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, U(G) ≈ G., and U(G) is 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 If U(G) were closed, it would be compact, and then G would be compact, contrary to assumption.

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. 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.

Restriction to SO(3)

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  |, 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.

Spinors

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-integral. 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 23/2 + 1 = 4 and (2 + 1)(21/2 + 1) = 6 respectively. Note that, 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.

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.

Dual representations

To see if the dual representation of an irreducible representation is isomorphic to the original representation one can consider the following theorems:

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.
2. Two irreducible representations are isomorphic if and only if they have the same highest weight.
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.
4. If π_μ0 is an irreducible representation with highest weight μ₀, then π*_μ0 has highest weight w₀ ⋅ (−μ).

Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. One sees that if −I is an element of the Weyl group of a semisimple Lie algebra, then w₀ = −I. In the case of sl(2, C), the Weyl group is W = {I, −I}. It follows that each π_μ, μ = 0, 1, … is isomorphic to its dual π_μ*. The root system of sl(2, C) ⊕ sl(2, C) is shown in the figure to the right. The Weyl group is generated by {w_γ} where w_γ is reflection in the plane orthogonal to γ as γ ranges over all roots. One sees that w_α ⋅w_β = −I so −I ∈ W. Then using the fact that if π, σ are Lie algebra representations and π ≈ σ, then Π ≈ Σ. The conclusion for SO(3; 1)⁺ is

π m , n ∗ ≅ π m , n , Π m , n ∗ ≅ Π m , n , 2 m , 2 n ∈ N .

Complex conjugate representations

If π is a representation of a Lie algebra, then π 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. In general, every irreducible representation π of sl(n, C) can be written uniquely as π = π⁺ + π⁻, where

π ± ( X ) = 1 2 ( π ( X ) ± i π ( i − 1 X ) ) ,

with π⁺ holomorphic (complex linear) and π⁻ anti-holomorphic (conjugate linear). For sl(2, C), since π_μ is holomorphic, π_μ is anti-holomorphic. Direct examination of the explicit expressions for π_{μ, 0} and π_{0, ν} in equation (S8) below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression (S8) also allows for identification of π⁺ and π⁻ for π_{μ, ν} as π⁺_{μ, ν} = π_μ^{⊕_{ν + 1}} and π⁻_{μ, ν} = π_ν^{⊕_{μ + 1}}.

Using the above identities (interpreted as pointwise addition of functions), for SO(3; 1)⁺ yields

π m , n ¯ = π m , n + + π m , n − ¯ = π m ⊕ 2 n + 1 ¯ + π n ¯ ⊕ 2 m + 1 ¯ = π n ⊕ 2 m + 1 + π m ¯ ⊕ 2 n + 1 = π n , m + + π n , m − = π n , m , Π m , n ¯ = Π n , m , 2 m , 2 n ∈ N ,

where the statement for the group representations follow from exp(X) = exp(X). It follows that the irreducible representations (m, n) have real matrix representatives if and only if m = n. Reducible representations on the form (m, n) ⊕ (n, m) have real matrices too.

Induced representations on the Clifford algebra and the Dirac spinor representation

In general representation theory, if (π, V) is a representation of a Lie algebra g, then there is an associated representation of g on End V, also denoted π, given by

Likewise, a representation (Π, V) of a group G yields a representation Π on End  V of G, still denoted Π, given by

Applying this to the Lorentz group, if (Π, V) is a projective representation, then direct calculation using (G4) shows that the induced representation on End V is, in fact, a proper representation, i.e. a representation without phase factors.

In quantum mechanics this means that if (π, H) or (Π, H) is a representation acting on some Hilbert space H, then the corresponding induced representation acts on the set of linear operators on H. As an example, the induced representation of the projective spin (1/2, 0) ⊕ (0, 1/2) representation on End(H) is the non-projective 4-vector (1/2, 1/2) representation.

For simplicity, consider now only the "discrete part" of End H, that is, given a basis for H, the set of constant matrices of various dimension, including possibly infinite dimensions. A general element of the full End H is the sum of tensor products of a matrix from the simplified End H and an operator from the left out part. The left out part consists of functions of spacetime, differential and integral operators and the like. See Dirac operator for an illustrative example. Also left out are operators corresponding to other degrees of freedom not related to spacetime, such as gauge degrees of freedom in gauge theories.

The induced 4-vector representation of above on this simplified End H has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. (Note the different metric convention in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, Cℓ_3,1(R), whose complexification is M₄(C), generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the (0, 0), a pseudoscalar irrep, also the (0, 0), but with parity inversion eigenvalue −1, see the next section below, the already mentioned vector irrep, (1/2, ,1/2), a pseudovector irrep, (1/2, 1/2) with parity inversion eigenvalue +1 (not −1), and a tensor irrep, (1, 0) ⊕ (0, 1). The dimensions add up to 1 + 1 + 4 + 4 + 6 = 16. In other words,

where, as is customary, a representation is confused with its representation space. This is, in fact, a reasonably convenient way to show that the algebra spanned by the gammas is 16-dimensional.

The (1/2, 0) ⊕ (0, 1/2) spin representation

The six-dimensional representation space of the tensor (1, 0) ⊕ (0, 1)-representation inside Cℓ_3,1(R) has two roles. In particular, letting

where {γ^μ ∈ Cℓ_3,1(R): μ = 0,1,2,3} are the gamma matrices, the {σ^μν ∈ Cℓ_3,1(R)} , only 6 of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,

and hence constitute a representation (in addition to being a representation space) sitting inside Cℓ_3,1(R), the (1/2, 0) ⊕ (0, 1/2) spin representation. For details, see bispinor and Dirac algebra.

The conclusion is that every element of the complexified Cℓ_3,1(R) in End H (i.e. every complex 4×4 matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on C⁴, making it a space of bispinors.

Reducible representations

There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained in a standard manner by taking direct sums, tensor products, and quotients of the irreducible representations. Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e.g. GL(n, ℝ). These representations are in general not irreducible, and are not discussed here. It is to be noted though that the Lorenz group and its Lie algebra have the complete reducibility property. This means that every representation reduces to a direct sum of irreducible representations.

Space inversion and time reversal

The (possibly projective) (m, n) representation is irreducible as a representation SO(3; 1)⁺, the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If m = n it can be extended to a representation of all of O(3; 1), the full Lorentz group, including space parity inversion and time reversal. The representations (m, n) ⊕ (n, m) can be extended likewise.

Space parity inversion

For space parity inversion, one considers the adjoint action Ad_P of P ∈ SO(3; 1) on so(3; 1), where P is the standard representative of space parity inversion, P = diag(1, −1, −1, −1), given by

It is these properties of K and J under P that motivate the terms vector for K and pseudovector or axial vector for J. In a similar way, if π is any representation of so(3; 1) and Π is its associated group representation, then Π(SO(3; 1)⁺) acts on the representation of π by the adjoint action, π(X) ↦ Π(g) π(X) Π(g)⁻¹ for X ∈ so(3; 1), g ∈ SO(3; 1)⁺. If P is to be included in Π, then consistency with (F1) requires that

holds, where A and B are defined as in the first section. This can hold only if A_i and B_i have the same dimensions, i.e. only if m = n. When m ≠ n then (m, n) ⊕ (n, m) can be extended to an irreducible representation of SO(3; 1)⁺, the orthocronous Lorentz group. The parity reversal representative Π(P) does not come automatically with the general construction of the (m, n) representations. It must be specified separately. The matrix β = i γ⁰ (or a multiple of modulus −1 times it) may be used in the (1/2, 0) ⊕ (0, 1/2) representation.

If parity is included with a minus sign (the 1×1 matrix [−1]) in the (0,0) representation, it is called a pseudoscalar representation.

Time reversal

Time reversal T = diag(−1, 1, 1, 1), acts similarly on so(3; 1) by

By explicitly including a representative for T, as well as one for P, one obtains a representation of the full Lorentz group SO(3; 1). A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the P^μ, in addition to the Jⁱ and Kⁱ generate the group. These are interpreted as generators of translations. The time-component P⁰ is the Hamiltonian H. The operator T satisfies the relation

in analogy to the relations above with so(3; 1) replaced by the full Poincaré algebra. By just cancelling the i's, the result THT⁻¹ = −H would imply that for every state Ψ with positive energy E in a Hilbert space of quantum states with time-reversal invariance, there would be a state Π(T⁻¹)Ψ with negative energy −E. Such states do not exist. The operator Π(T) is therefore chosen antilinear and antiunitary, so that it anticommutes with i, resulting in THT⁻¹ = +H, and its action on Hilbert space likewise becomes antilinear and antiunitary. It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix. This is mathematically sound, see Wigner's theorem, but if one is very strict with terminology, Π is not a representation.

When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, (1/2, 0) ⊕ (0, 1/2), is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.

The third discrete symmetry entering in the CPT theorem along with P and T, charge conjugation symmetry C, has nothing directly to do with Lorentz invariance.

Action on function spaces

In the classification of the irreducible finite-dimensional representations of above it was never specified precisely how a representative of a group or Lie algebra element acts on vectors in the representation space. The action can be anything as long as it is linear. The point silently adopted was that after a choice of basis in the representation space, everything becomes matrices anyway.

If V is a vector space of functions of a finite number of variables n, then the action on a scalar function f ∈ V given by

produces another function Πf ∈ V. Here Π_x is an n-dimensional representation, and Π is a possibly infinite-dimensional representation. A special case of this construction is when V is a space of functions defined on the group G itself, viewed as a n-dimensional manifold embedded in Rⁿ. This is the setting in which the Peter–Weyl theorem and the Borel–Weil theorem are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations. The completeness of the characters in this sense can thus be used to prove the existence of the highest weight representations. The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. SL(2, C); in the present case, there is a one-to-one correspondence between representations of SU(2) and holomorphic representations of SL(2, C). (A group representation is called holomorphic if its corresponding Lie algebra representation is complex linear.) This theorem too can be used to demonstrate the existence of the highest weight representations.

Euclidean rotations

The subgroup SO(3) of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space L²(S²) = span{Y^ℓ_m, ℓ ∈ N⁺, −ℓ ≤ m ≤ ℓ }, where the Y^ℓ_m are spherical harmonics. Its elements are square integrable complex-valued functions on the sphere. The inner product on this space is given by

If f is an arbitrary square integrable function defined on the unit sphere S², then it can be expressed as

where the expansion coefficients are given by

The Lorentz group action restricts to that of SO(3) and is expressed as

This action is unitary, meaning that

The D^(ℓ) can be obtained from the D^(m, n) of above using Clebsch–Gordan decomposition, but they are more easily directly expressed as an exponential of an odd-dimensional su(2)-representation (the 3-dimensional one is exactly so(3)). In this case the space L²(S²) decomposes neatly into an infinite direct sum of irreducible odd finite-dimensional representations V_{2i + 1}, i = 0, 1, … according to

This is characteristic of infinite-dimensional unitary representations of SO(3). If Π is an infinite-dimensional unitary representation on a separable Hilbert space, then it decomposes as a direct sum of finite-dimensional unitary representations. Such a representation is thus never irreducible. All irreducible finite-dimensional representations (Π, V) can be made unitary by an appropriate choice of inner product,

⟨ f , g ⟩ U ≡ ∫ S O ( 3 ) ⟨ Π ( R ) f , Π ( R ) g ⟩ d g = 1 8 π 2 ∫ 0 2 π ∫ 0 π ∫ 0 2 π ⟨ Π ( R ) f , Π ( R ) g ⟩ sin ⁡ θ d φ d θ d ψ , f , g ∈ V ,

where the integral is the unique invariant integral over SO(3) normalized to 1, here expressed using the Euler angles parametrization. The inner product inside the integral is any inner product on V.

The Möbius group

The identity component of the Lorentz group is isomorphic to the Möbius group M. This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection, the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere.

In the plane, a Möbius transformation characterized by the complex numbers a, b, c, d acts on the plane according to

and can be represented by complex matrices

since multiplication by a nonzero complex scalar does not change f. These are elements of SL(2,  ℂ) and are unique up to a sign (since ±Π_f give the same f), hence M ≈ SL(2,  ℂ)/{I,  −I} ≈ SO(3; 1)⁺.

The Riemann P-functions

The Riemann P-functions, solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as

where the a,  b,  c,  α,  β,  γ,  α′,  β′,  γ′ are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is

The set of constants 0, ∞, 1 in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation. Its exponents, i. e. solutions of the indicial equation, for expansion around the singular point 0 are 0 and 1 − c ,corresponding to the two linearly independent solutions, and for expansion around the singular point 1 they are 0 and c − a − b. Similarly, the exponents for ∞ are a and b for the two solutions.

One has thus

where the condition (sometimes called Riemann's identity)

α + α ′ + β + β ′ + γ + γ ′ = 1

on the exponents of the solutions of Riemann's differential equation has been used to define γ′.

The first set of constants on the left hand side in (T1), a, b, c denotes the regular singular points of Riemann's differential equation. The second set, α, β, γ, are the corresponding exponents at a, b, c for one of the two linearly independent solutions, and, accordingly, α′, β′, γ′ are exponents at a, b, c for the second solution.

Define an action of the Lorentz group on the set of all Riemann P-functions by first setting

where A,  B,  C,  D are the entries in

for Λ = p(λ) ∈ SO(3; 1)⁺ a Lorentz transformation.

Define

where P is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Mobius transformation of the argument is that of shifting the poles to new locations, hence changing the critical points, but there is no change in the exponents of the differential equation the new function satisfies. The new function is expressed as

where

History

The Lorentz group SO(3; 1)⁺ and its double cover SL(2, C) also have infinite dimensional unitary representations, studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) at the instigation of Paul Dirac. This trail of development begun with Dirac (1936) where he devised matrices U and B necessary for description of higher spin (compare Dirac matrices), elaborated upon by Fierz (1939), see also Fierz & Pauli (1939), and proposed precursors of the Bargmann-Wigner equations. In Dirac (1945) he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors. These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinite-dimensional generalization of spinors in his 1947 paper.

The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by Harish-Chandra (1951) and Gelfand & Graev (1953), based on an analogue for SL(2, C) of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in Rühl (1970) and Knapp (2001).

The theory of spherical functions for the Lorentz group, required for harmonic analysis on the 3-dimensional unit quasi-sphere in Minkowski space, or equivalently 3-dimensional hyperbolic space, is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on R. This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008).

Principal series

The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup B of G = SL(2, C). Since the one-dimensional representations of B correspond to the representations of the diagonal matrices, with non-zero complex entries z and z⁻¹, they thus have the form

χ ν , k ( z 0 c z − 1 ) = r i ν e i k θ ,

for k an integer, ν real and with z = re^iθ. The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when k is replaced by −k. By definition the representations are realized on L² sections of line bundles on G/B = S², which is isomorphic to the Riemann sphere. When k = 0, these representations constitute the so-called spherical principal series.

The restriction of a principal series to the maximal compact subgroup K = SU(2) of G can also be realized as an induced representation of K using the identification G / B = K / T, where T = B ∩ K is the maximal torus in K consisting of diagonal matrices with | z | = 1. It is the representation induced from the 1-dimensional representation z^k T, and is independent of ν. By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions | k | + 2m + 1 with m a non-negative integer.

Using the identification between the Riemann sphere minus a point and C, the principal series can be defined directly on L²(C) by the formula

Irreducibility can be checked in a variety of ways:

The representation is already irreducible on B. This can be seen directly, but is also a special case of general results on irreducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition G = B ∪ B s B where s is the Weyl group element

( 0 − 1 1 0 ) .

The action of the Lie algebra g of G can be computed on the algebraic direct sum of the irreducible subspaces of K can be computed explicitly and the it can be verified directly that the lowest-dimensional subspace generates this direct sum as a g -module.

Complementary series

The for 0 < t < 2, the complementary series is defined on L² functions f on C for the inner product

( f , g ) = ∫ ∫ f ( z ) g ( w ) ¯ d z d w | z − w | 2 − t .

with the action given by

The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of K, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of K = SU(2). Irreducibility can be proved by analyzing the action of g on the algebraic sum of these subspaces or directly without using the Lie algebra.

Plancherel theorem

The only irreducible unitary representations of SL(2, C) are the principal series, the complementary series and the trivial representation. Since −I acts as (−1)^k on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even.

To decompose the left regular representation of G on L²(G), only the principal series are required. This immediately yields the decomposition on the subrepresentations L²(G/±I), the left regular representation of the Lorentz group, and L²(G/K), the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with k = 0.)

The left and right regular representation λ and ρ are defined on L²(G) by

λ ( g ) f ( x ) = f ( g − 1 x ) , ρ ( g ) f ( x ) = f ( x g ) .

Now if f is an element of C_c(G), the operator π_ν,k(f) defined by

π ν , k ( f ) = ∫ G f ( g ) π ( g ) d g

is Hilbert–Schmidt. Define a Hilbert space H by

H = ⨁ k ≥ 0 H S ( L 2 ( C ) ) ⊗ L 2 ( R , c k ( ν 2 + k 2 ) 1 / 2 d ν ) ,

where

c 0 = 1 / 4 π 3 / 2 , c k = 1 / ( 2 π ) 3 / 2 ( k ≠ 0 )

and HS(L²(C)) denotes the Hilbert space of Hilbert–Schmidt operators on L²(C). Then the map U defined on C_c(G) by

U ( f ) ( ν , k ) = π ν , k ( f )

extends to a unitary of L²(G) onto H.

The map U satisfies the intertwining property

U ( λ ( x ) ρ ( y ) f ) ( ν , k ) = π ν , k ( x ) − 1 π ν , k ( f ) π ν , k ( y ) .

If f₁, f₂ are in C_c(G) then by unitarity

Thus if f = f₁ ∗ f₂* denotes the convolution of f₁ and f₂*, and f 2 ∗ ( g ) = f 2 ( g − 1 ) ¯ , then

The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively. The Plancherel formula extends to all f_i in L₂(G). By a theorem of Jacques Dixmier and Paul Malliavin, every function f in C c ∞ ( G ) is a finite sum of convolutions of similar functions, the inversion formula holds for such f. It can be extended to much wider classes of functions satisfying mild differentiability conditions.

Classification

The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to assume they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinite-dimensional representation Π_H on a Hilbert space H of SO(3; 1)⁺ is at hand. Since SO(3) is a subgroup, Π_H is a representation of it as well. Each irreducible subrepresentation of SO(3) is finite-dimensional, and the SO(3) representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of SO(3) if Π_H is unitary.

The steps are the following:

1. Chose a suitable basis of common eigenvectors of J² and J₃.
2. Compute matrix elements of J₁, J₂, J₃ and K₁, K₂, K₃.
3. Enforce Lie algebra commutation relations.
4. Require unitarity together with orthonormality of the basis.

Step 1

One may suitably choose a basis and label the basis vectors by

| j 0 j 1 ; j m ⟩ .

If this was a finite-dimensional representation, then j₀ would correspond the lowest occurring eigenvalue j(j + 1) of J² in the representation, equal to |m − n|, and j₁ would correspond to the highest occurring eigenvalue, equal to m + n. In the infinite-dimensional case, j₀ ≥ 0 retains this meaning, but j₁ does not. One assumes for simplicity that a given j occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown that the assumption is possible to avoid (with a slightly more complicated calculation) with the same results.

Step 2

The next step is to compute the matrix elements of the operators J₁, J₂, J₃ and K₁, K₂, K₃ forming the basis of the Lie algebra of so(3; 1). The matrix elements of

J ± = J 1 ± i J 2 , J 3

(here one is operating in the comlpexified Lie algebra) are known from the representation theory of the rotation group, and are given by

⟨ j m | J + | j m − 1 ⟩ = ⟨ j m − 1 | J − | j m ⟩ = ( j + m ) ( j − m + 1 ) , ⟨ j , m | J 3 | j m ⟩ = m ,

where the labels j₀ and j₁ have been dropped since they are the same for all basis vectors in the representation.

Due to the commutation relations

[ J i , K j ] = i ϵ i j k K k ,

the triple (K_i, K_i, K_i) ≡ K is a vector operator and the Wigner–Eckart theorem applies for computation of matrix elements between the states represented by the chosen basis. The matrix elements of

K 0 ( 1 ) = K 3 K ± 1 ( 1 ) = ∓ 1 2 ( K 1 ± i K 2 ) ,

where the superscript (1) signifies that the defined quantities are the components of a spherical tensor operator of rank k = 1 (which explains the factor √2 as well) and the subscripts 0, ±1 are referred to as q in formulas below, are given by

⟨ j ′ m ′ | K 0 ( 1 ) | j m ⟩ = ⟨ j ′ m ′ k = 1 q = 0 | j m ⟩ ⟨ j ∥ K ( 1 ) ∥ j ′ ⟩ ⟨ j ′ m ′ | K ± 1 ( 1 ) | j m ⟩ = ⟨ j ′ m ′ k = 1 q = ± 1 | j m ⟩ ⟨ j ∥ K ( 1 ) ∥ j ′ ⟩ .

Here the first factors on the right hand sides are Clebsch–Gordan coefficients for coupling j′ with k to get j. The second factors are the reduced matrix elements. They do not depend on m, m′ or q, but depend on j, j′ and, of course, K. For a complete list of non-vanishing equations, see Harish-Chandra (1947, p. 375).

Step 3

The next step is to demand that the Lie algebra relations hold, i.e. that

[ K ± , K 3 ] = ± J ± , [ K + , K − ] = − 2 J 3 .

This results in a set of equations for which the solutions are

⟨ j ∥ K ( 1 ) ∥ j ⟩ = i j 1 j 0 j ( j + 1 ) , ⟨ j ∥ K ( 1 ) ∥ j − 1 ⟩ = − B j ξ j j ( 2 j − 1 ) , ⟨ j − 1 ∥ K ( 1 ) ∥ j ⟩ = B j ξ j − 1 j ( 2 j + 1 ) ,

where

B j = ( j 2 − j 0 2 ) ( j 2 − j 1 2 ) j 2 ( 4 j 2 − 1 ) ,

and

j 0 = 0 , 1 2 , 1 , … , j 1 ∈ C , ξ j ∈ C .

Step 4

The imposition of the requirement of unitarity of the corresponding representation of the group restricts the possible values for the arbitrary complex numbers j₀ and ξ_j. Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning

K ± † = K ∓ , K 3 † = K 3 .

This translates to

⟨ j ∥ K ( 1 ) ∥ j ⟩ = ⟨ j ∥ K ( 1 ) ∥ j ⟩ ¯ , ⟨ j ∥ K ( 1 ) ∥ j − 1 ⟩ = − ⟨ j − 1 ∥ K ( 1 ) ∥ j ⟩ ¯ ,

leading to

j 0 ( j 1 + j 1 ¯ ) = 0 , | B j | ( | ξ j | 2 − e − 2 i β j ) = 0 ,

where β_j is the angle of B_j on polar form. For |B_j| ≠ 0 one has |ξ|2
j = 1, and ξ_j = 1 is chosen by convention. There are two possible cases. The first with j₁ + j₁ = 0 gives, with j₁ = − iν, ν real,

⟨ j ∥ K ( 1 ) ∥ j ⟩ = ν j 0 j ( j + 1 ) , B j = ( j 2 − j 0 2 ) ( j 2 + ν 2 ) 4 j 2 − 1 .

This is principal series and the elements may be denoted (j₀, ν), 2j₀ ∈ ℕ, ν ∈ ℝ. For the other possibility, j₀ = 0, one has

⟨ j ∥ K ( 1 ) ∥ j ⟩ = 0 , B j = ( j 2 − ν 2 ) 4 j 2 − 1 .

One needs to require that B2
j is real and positive for j = 1, 2, ... (because B₀ = B_j0), leading to −1 ≤ ν ≤ 1. This is complementary series and its elements may be denoted (0, ν), −1 ≤ ν ≤ 1.

This shows that the representations of above are all infinite-dimensional irreducible unitary representations.

Conventions and Lie algebra bases

The metric of choice is given by η = diag(−1, 1, 1, 1), and the physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of basis for the Lie algebra is, in the 4-vector representation, given by

J 1 = J 23 = − J 32 = i ( 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 1 0 ) , J 2 = J 31 = − J 13 = i ( 0 0 0 0 0 0 0 1 0 0 0 0 0 − 1 0 0 ) , J 3 = J 12 = − J 21 = i ( 0 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 ) , K 1 = J 01 = J 10 = i ( 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ) , K 2 = J 02 = J 20 = i ( 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ) , K 3 = J 03 = J 30 = i ( 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ) .

The commutation relations of the Lie algebra so(3; 1) are

[ J μ ν , J ρ σ ] = i ( η σ μ J ρ ν + η ν σ J μ ρ − η ρ μ J σ ν − η ν ρ J μ σ ) .

In three-dimensional notation, these are

[ J i , J j ] = i ϵ i j k J k , [ J i , K j ] = i ϵ i j k K k , [ K i , K j ] = − i ϵ i j k J k .

The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol J above and in the sequel should be observed.

Weyl spinors and bispinors

By taking, in turn, m = 1/2, n = 0 and m = 0, n = 1/2 and by setting

J i ( 1 2 ) = 1 2 σ i

in the general expression (G1), and by using the trivial relations 1₁ = 1 and J⁽⁰⁾ = 0, one obtains

These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) V_L and V_R, whose elements Ψ_L and Ψ_R are called left- and right-handed Weyl spinors respectively. Given (π(1/2,0), V_L) and (π(0,1/2), V_R) one may form their direct sum as representations,

This is, up to a similarity transformation, the (1/2,0) ⊕ (0,1/2) Dirac spinor representation of so(3; 1). It acts on the 4-component elements (Ψ_L, Ψ_R) of (V_L ⊕ V_R), called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of so(3; 1). Expressions for the group representations are obtained by exponentiation.