# Representation theory of the Lorentz group

The Lorentz group, a Lie group on which special relativity is based, has a variety of representations. Many of these representations are important in theoretical physics in the description of particles in relativistic quantum mechanics, as well as fields in quantum field theory. This representation theory provides a theoretical ground for the concept of spin (which can be either integer or half-integer in the unit of the reduced Planck constant ℏ); representations with half-integer weights are normally constructed out of spinors.

The group may also be represented[how?] in terms of a set of functions defined on the Riemann sphere. These are the Riemann P-functions, which are expressible as hypergeometric functions. The identity component SO(3;1)+ of the Lorentz group is isomorphic to the Möbius group, and hence any representation of the Lorentz group is necessarily a representation of the Möbius group.

The subgroup SO(3) with its representation theory form a simpler theory, but the two are related (see details below) and both are prominent in theoretical physics as descriptions of spin, angular momentum, and other things related to rotation.

## Finite-dimensional representations

Representation theory of groups in general, and Lie groups is particular, is a very rich subject. The full Lorentz group makes no exception. The Lorentz group has some properties that makes it "agreeable" and other that makes it "not very agreeable" within the context of representation theory. The group is semisimple, but it is not simple. It is not connected, and none of its components are simply connected. Perhaps most importantly, the Lorentz group is not compact.

The presence of semisimplicity means the Lorentz group can be much be dealt with without the presence of simplicity - all representations are known from the fact that the Lorentz group is semisimple. But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, can on the contrary, not be dealt with in all the aspects of the same simple framework that applies to simply connected, compact groups.

### 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 Ji of rotations and the three generators Ki of boosts. These two sets of generators, for the standard representation, are both found in explicit form in the section Conventions below. Now first complexify the Lie algebra, and then change basis to the components of:

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

In this new basis, one checks that the components of A = (A1, A2, A3) and B = (B1, B2, B3) satisfy separately the commutation relations of the Lie algebra su(2) and moreover that they commute with each other[1]

$\left[A_i ,A_j\right] = i\varepsilon_{ijk}A_k\,,\quad \left[B_i ,B_j\right] = i\varepsilon_{ijk}B_k\,,\quad \left[A_i ,B_j\right] = 0\,,$

where i, j, k are indices which each take values 1, 2, 3, and εijk is the three-dimensional Levi-Civita symbol. In other words, one has the isomorphism

$\mathfrak{so}(3,1)_C \cong \mathfrak{sl}(2,C) \oplus \mathfrak{sl}(2,C)\,,$[2]

(A1)

where sl(2, C) is the complexification of su(2). The utility of this isomorphism comes from the fact that all irreducible representations of su(2) are, up to isomorphism, 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 its complexification sl(2, C).[3] Via the displayed isomorphism, all irreducible representations of so(3;1)C, and thus those of so(3;1) are known. Since so(3;1) is semisimple,[2] all its representations, not necessarily irreducible, 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 and n, conventionally written as

$(m,n) \equiv D^{(m,n)} \equiv D^{(m)} \otimes D^{(n)}$

where denotes the tensor product of the representations D(m) and D(n), which fix representations of the su(2) subalgebras spanned by the components of A and B respectively. The m and n are the highest weights, respectively, of the complexified su(2) representations, i.e. sl(2,C) representations, of AC and BC. 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 this classification. These are, up to a similarity transformation, uniquely given by

\begin{align} \pi_{m,n}(J_i) & = 1_{(2m+1)}\otimes J^{(n)}_i + J^{(m)}_i\otimes 1_{(2n+1)}\\ \pi_{m,n}(K_i) & = i(1_{(2m+1)}\otimes J^{(n)}_i - J^{(m)}_i \otimes 1_{(2n+1)}), \end{align}[4]

(A2)

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 1n is the n-dimensional unit matrix. Explicit formulas on component form are given at the end of the article.

The highest weight for a sl(2, C) representation coincides with the highest eigenvalue of the representative of the third Pauli matrix σ3sl(2, C) and is in quantum mechanics thus related to the maximum value the spin z-component can take on for a particle whose state vector transform under the representation in question.

### The group

Here V is a finite-dimensional vector space, GL(V) is the set of all invertible linear transformations on V and gl(V) is its Lie algebra. The maps π and Π are Lie algebra and group representations respectively, and exp is the exponential mapping. The diagram commutes only up to a sign if Π is projective.

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. If X is an element of so(3;1) in the standard representation, then

$\Lambda = e^{iX} \equiv \sum_{n=0}^{\infty} \frac{(iX)^n}{n!}$

(G1)

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

$\Pi_U(e^{iX}) = e^{i\pi(X)}, \quad X \in \mathfrak{so}(3;1).$

(G2)

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. But a theorem[5] 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). The qualitative form of the Baker-Campbell-Hausdorff formula then guarantees that it is a group homomorphism, still for X small enough (B).[5] 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. The second problem is that for a given g ∈ SO(3;1)+ there may not be exactly one Xso(3;1) such that g = eiX. Technically, formula (G2) is used to define Π near the identity. For other elements gU 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

$g = g_n = (g_n g_{n-1}^{-1})(g_{n-1}g_{n-2}^{-1})\cdots(g_{2}g_{1}^{-1})(g_{1}g_{0}), \qquad \Pi(g) \equiv \Pi_U(g_{n}g_{n-1}^{-1})\Pi_U(g_{n-1}g_{n-2}^{-1})\cdots\Pi_U(g_{2}g_{1}^{-1})\Pi_U(g_{1}g_{0}), \quad g_0 = 1$[5]

(G3)

where the gi are on the path and the factors on the far right are uniquely defined by (G2) provided that all gigi + 1−1U. For each i take, by the inverse function theorem, the unique Xi such that exp(Xi) = gigi-1-1 and obtain

$\Pi(g) = \Pi_U(e^{iX_n})\Pi_U(e^{iX_{n - 1}})\cdots\Pi_U(e^{iX_2})\Pi_U(e^{iX_1}) = \Pi_U(e^{iX_n}e^{iX_{n - 1}}\cdots e^{iX_2}e^{iX_1}),$

(G4)

where the final step is by the Baker-Campbell-Hausdorff formula. It shows that Π is a group homomorphism for elements close to the identity. 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. Likewise, the formula yields a homomorphism for all elements, whether small or not, as seen using expressions similar to (G2). One simply drops the subscript U in formula (G2).

It turns out that the result is always independent of the partitioning of the path.[5] For simply connected groups, the construction will be independent of the path as well, yielding a well defined representation.[5] In that case formula (G2) can unambiguously be used directly. 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.[6] The result, when using (G2), will then depend on which X in the Lie algebra is used to obtain the representative matrix for g.

The Lorentz group is doubly connected so that its fundamental group π1(SO(3,1)+), whose elements are the path homotopy classes, has two members. Thus not all representations of the Lie algebra will yield representations of the group, but some will instead yield projective representations.[5] Once these conclusions has been reached, and once one knows whether a representation is projective, there is no need not be concerned about paths and partitions. Formula (G2) applies to all group elements and all representations, including the projective ones.

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

$[\Pi(\Lambda_1)\Pi(\Lambda_2)\Pi^{-1}(\Lambda_1\Lambda_2)]^2 = 1\Rightarrow \Pi(\Lambda_1\Lambda_2) = \pm \Pi(\Lambda_1)\Pi(\Lambda_2), \qquad \Lambda_1,\Lambda_2 \in SO(3;1),$[7]

(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. One cannot consistently chose sign to obtain a continuous representation of all of SO(3;1)+, but this is possible to do locally around any point.[8]

### The covering group

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

$G = \{(g,[p_g]): g\in SO^+(3,1),[p_g]\in \pi_g\}$

and endow it with the multiplication operation

$(g_1,[p_1])(g_2,[p_2]) = (g_1g_2,[p_{12}]),\quad g_1,g_2\in SO(3;1)^+,\quad [p_1]\in\pi_{g_1}, [p_2]\in \pi_{g_2}, [p_{12}]\in \pi_{g_{12}},\quad p_{12}(t) = p_1(t)\cdot p_2(t).$

The dot on the far right denotes path multiplication. With this multiplication, G is a group and G SL(2,C),[9] the universal covering group of SO+(3;1). By the above construction, there is, since each πg has two elements, a 2:1 covering map p:SL(2.C) → SO(3,1)+. According to covering group theory, the Lie algebras so(3;1) and sl(2,C) are isomorphic and the Π(m,n) representations (whether projective or not) correspond to non-projective representations of the simply connected group SL(2,C).

For an algebraic view of the universal covering group, let SL(2,C) act on the set of all Hermitean 2 × 2 matrices h by the operation[10]

$\mathbf{P}(A): \mathbf{h} \rightarrow \mathbf{h}; \quad X \rightarrow A^\dagger XA, \quad X \in \mathbf{h}, A \in SL(2,C).$

Since X ∈ h is Hermitean, AXA is again Hemitean because (AXA) = AXA†† = AXA, and also A(αX + βY)A = αAXA + βAYA, so the action is linear as well. An element of h may generally be written in the form

$X = \bigl(\begin{smallmatrix} \xi_4 + \xi_3&\xi_1 + i\xi_2\\ \xi_1 - i\xi_2&\xi_4 - \xi_3\\ \end{smallmatrix}\bigr)$

for ξi real, showing that h is a 4-dimensional real vector space. Moreover, (AB)X(AB) = BAXAB meaning that P is a group homomorphism into End(h). Thus P:SL(2,C) → End(h) is a 4-dimensional representation of SL(2,C). Now map h to spacetime R4, endowed with the Lorentz metric, via

$X = (\xi_1,\xi_2,\xi_3,\xi_4) \leftrightarrow \overrightarrow{(\xi_1,\xi_2,\xi_3,\xi_4)} = (x,y,z,t) = \overrightarrow{X}.$

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

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

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 O(3;1). By the main theorem of connectedness, since SL(2,C) is connected, its image under p in O(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 (it's kernel is {∅} and must therefore be an isomorphism for dimensional reasons) and 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 πm is a representation of su(2) with highest weight m, then the complexification of π is a complex linear representation of sl(2,C). All complex linear representation of sl(2,C) are of this form. The group representations 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 kernel of the mapping p of above is {I, -I}. By the first isomorphism theorem, a representation Π of SL(2,C) descends to a representation of SO(3;1)+ if and only if ker p ⊂ ker Π. In particular, if Π is faithful, then there is no corresponding proper representation of SO(3;1)+, but there is a projective one as was shown in a previous section.

### Common representations

m=0 1/2 1 scalar Weyl spinor bispinor self-dual 2-form 2-form field Weyl spinor (right-handed) 4-vector Rarita–Schwinger field anti-self-dual 2-form traceless symmetric tensor Purple: (m, n) complex irreps   Black: (m, n) ⊕ (n, m) Bold: (m, m)

Since for any irrep where mn 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.

### Properties of the (m, n) representations

The irreducible (m, n) representations are (2m + 1)(2n + 1)-dimensional, and they are the only irreducible representations.

Since the angular momentum operator is given by J = A + B, the highest weight (or spin modulo ℏ in quantum mechanics) of the rotation subrepresentation will be m + n.

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. It 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.[4]

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, …, | mn |,[4] 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.

The (m, n) representation is the dual of the (n, m) representation.

### Induced representations

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

$\pi(X)(A) = [\pi(X),A], \quad A\in \mathrm{End}\,(V),\ X\in\mathfrak{g}.$

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

$\Pi(g)(A) = \Pi(g)A\Pi(g)^{-1}, \quad A\in \mathrm{End}\,(V),\ g\in G.$

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.[4]

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.[4] (Note the different metric convention in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, Cℓ3,1(R)M4(C)End(H) spanned by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), a pseudoscalar irrep, a vector irrep, a pseudovector irrep, and a tensor irrep.[4] The dimensions add up to 1 + 1 + 4 + 4 + 6 = 16. This is, in fact, a reasonably convenient way to show that the algebra spanned by the gammas is 16-dimensional.[4] An algebraic proof of this fact is fairly lengthy.[12] For details, see Dirac algebra.

The conclusion is that every linear operator (complex 4×4 matrix) in the simplified End(H) has well defined Lorentz transformation properties. The "left out" part transforms under the infinite-dimensional representations or, if not related to spacetime, not at all under Lorentz transformations, but instead they transform under the corresponding gauge group. The gauge transformations in turn leave the spacetime parts unaffected.

There is also a multitude of other representations that can be said being "induced" by the irreducible ones, such as those obtained in a standard manner by taking direct sums, tensor products, duals, quotients, etc of the irreducible representations. These are not discussed here.

### The full Lorentz group

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.[4] This follows from considering the adjoint action AdP of P ∈ O(3;1) on so(3;1), where P is the standard representative of space parity inversion, P = diag(1, −1, −1, −1), given by

$\mathrm{Ad}_P(J_i) = PJ_iP^{-1} = J_i, \qquad \mathrm{Ad}_P(K_i) = PK_iP^{-1} = -K_i.$

(F1)

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 Xso(3;1), g ∈ SO(3;1)+. If P is to be included in Π, then consistency with (F1) requires that

$\Pi(P)\pi(B_i)\Pi(P)^{-1} = \pi(A_i)$

(F2)

holds, where A and B are defined as in the first section. This can hold only if Ai and Bi have the same dimensions, i.e. only if m = n. When mn then (m, n) ⊕ (n, m) can be extended to an irreducible representation of O+(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 = iγ0 may be used in the (1/2, 0) ⊕ (0, 1/2)[4] representation. If parity is included in the (0,0) representation, it is called a pseudoscalar representation.

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

$\mathrm{Ad}_T(J_i) = TJ_iT^{-1} = -J_i, \qquad \mathrm{Ad}_P(K_i) = TK_iT^{-1} = K_i.$

(F3)

By explicitly including a representative for T, as well as one for P, one obtains a representation of the full Lorentz group O(3;1). The matrix Π(T) is antiunitary and its action on Hilbert space is antilinear.[13]

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.[14]

## Infinite-dimensional unitary representations

### History

The Lorentz group SO(3,1)[clarification needed] and its double cover SL(2, C) also have infinite dimensional unitary representations, first studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) (at the instigation of Paul Dirac). 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 hyperboloid 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, and thus have the form

$\chi_{\nu,k}(re^{i\theta})=r^{i\nu} e^{ik\theta},$

for k an integer and ν real. The representations are irreducible; the only repetitions occur when k is replaced by k. By definition the representations are realised on L2 sections of line bundles on G / B = S2, 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 realised as an induced representation of K using the identification G / B = K / T, where T = BK is the maximal torus in K consisting of diagonal matrices with | z | = 1. It is the representation induced from the 1-dimensional representation zk 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

$\pi_{\nu,k}\begin{pmatrix}a& b\\ c& d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-i\nu} \left({cz+d\over |cz+d|}\right)^{-k}f\left({az+b\over cz+d}\right).$

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 ireducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition G = BBsB where s is the Weyl group element $\begin{pmatrix}0& -1\\ 1& 0\end{pmatrix}$.[15]
• The action of the Lie algebra $\mathfrak{g}$ 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 $\mathfrak{g}$-module.[16][17]

### Complementary series

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

$(f,g)=\int \int {f(z) \overline{g(w)}\, dz\, dw\over |z-w|^{2-t}}.$

with the action given by

$\pi_{t}\begin{pmatrix}a& b\\ c& d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-t} f\left({az+b\over cz+d}\right).$

The complementary series are irreducible and inequivalent. 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 analysing the action of $\mathfrak{g}$ on the algebraic sum of these subspaces[16][17] or directly without using the Lie algebra.[18][19]

### 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 (−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

$\lambda(g)f(x)=f(g^{-1}x),\,\,\rho(g)f(x)=f(xg).$

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

$\pi_{\nu,k}(f)=\int_G f(g)\pi(g)\, dg$

is Hilbert–Schmidt. We define a Hilbert space H by

$H=\bigoplus_{k\ge 0} HS(L^2(C)) \otimes L^2(R, c_k(\nu^2 + k^2)^{1/2} d\nu),$

where

$c_0=1/4\pi^{3/2}, \,\, c_k=1/(2\pi)^{3/2}\,\,(k\ne 0)$

and HS(L2(C)) denotes the Hilbert space of Hilbert–Schmidt operators on L2(C).[20] Then the map U defined on Cc(G) by

$U(f)(\nu,k) = \pi_{\nu,k}(f)$

extends to a unitary of L2(G) onto H.

The map U satisfies

$U(\lambda(x)\rho(y)f)(\nu,k) = \pi_{\nu,k}(x)^{-1} \pi_{\nu,k}(f)\pi_{\nu,k}(y).$

If f1, f2 are in Cc(G) then

$(f_1,f_2) = \sum_{k\ge 0} c_k^2 \int_{-\infty}^\infty {\rm Tr}(\pi_{\nu,k}(f_1)\pi_{\nu,k}(f_2)^*) (\nu^2 +k^2) \, d\nu.$

Thus if f = f1f2* denotes the convolution of f1 and f2*, and $f_2^*(g)=\overline{f_2(g^{-1})}$, then

$f(1) = \sum_{k\ge 0} c_k^2 \int_{-\infty}^\infty {\rm Tr}(\pi_{\nu,k}(f)) (\nu^2+k^2)\, d\nu.$

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 fi in L2(G). By a theorem of Jacques Dixmier and Paul Malliavin, every function f in $C^\infty_c(G)$ 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.[21]

## Explicit formulas

### 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 in this article. These choices are arbitrary, but once they are made, fixed. The rationale is to allow the use of a single reference[4] for several related formulas. One possible choice of basis for the Lie algebra (which is not fixed by the reference) is, in the 4-vector representation, given by

\begin{align} J_1 &= J^{23} = -J^{32} = i\biggl(\begin{smallmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&0&1&0\\ \end{smallmatrix}\biggr),\\ J_2 &= J^{31} = -J^{13} = i\biggl(\begin{smallmatrix} 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&-1&0&0\\ \end{smallmatrix}\biggr),\\ J_3 &= J^{12} = -J^{21} = i\biggl(\begin{smallmatrix} 0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ K_1 &= J^{01} = J^{10} = i\biggl(\begin{smallmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ K_2 &= J^{02} = J^{20} = i\biggl(\begin{smallmatrix} 0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ K_3 &= J^{03} = J^{30} = i\biggl(\begin{smallmatrix} 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ \end{smallmatrix}\biggr). \end{align}

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

$[J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\sigma\mu}J^{\rho\nu} + \eta^{\nu\sigma}J^{\mu\rho} - \eta^{\rho\mu}J^{\sigma\nu} -\eta^{\nu\rho}J^{\mu\sigma}).$[4]

In three-dimensional notation, these are

$[J_i,J_j] = i\epsilon_{ijk}J_k, \quad [J_i,K_j] = i\epsilon_{ijk}K_k, \quad [K_i,K_j] = -i\epsilon_{ijk}J_k.$[4]

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.

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 -ma,a′m, -nb,b′n, the representations are given by

\begin{align} (\pi_{m,n}(J_i))_{a'b' , ab} &= \delta_{b'b}(J_i^{(m)})_{a'a} + \delta_{a'a}(J_i^{(n)})_{b'b},\\ (\pi_{m,n}(K_i))_{a'b' , ab} &= i(\delta_{a'a}(J_i^{(n)})_{b'b} - \delta_{b'b}(J_i^{(m)})_{a'a}), \end{align}[4]

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

\begin{align} (J_3^{(j)})_{a'a} &= a\delta_{a'a},\\ (J_1^{(j)} \pm iJ_2^{(j)})_{a'a} &= \sqrt{(j \mp a)(j \pm a + 1)}\delta_{a',a \pm 1}. \end{align}[4]

### Weyl spinors and bispinors

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

$J_i^{(\frac{1}{2})} = \frac{1}{2}\sigma_i$

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

\begin{align} \pi_{(\frac{1}{2},0)}(J_i) & = \frac{1}{2}(\sigma_i\otimes 1_{(1)} + 1_{(2)}\otimes J^{(0)}_i) = \frac{1}{2}\sigma_i\quad\pi_{(\frac{1}{2},0)}(K_i) = \frac{i}{2}(1_{(2)}\otimes J^{(0)}_i - \sigma_i \otimes 1_{(1)}) = -\frac{i}{2}\sigma_i,\\ \pi_{(0,\frac{1}{2})}(J_i) & = \frac{1}{2}(J^{(0)}_i\otimes 1_{(2)} + 1_{(1)}\otimes \sigma_i) = \frac{1}{2}\sigma_i\quad\pi_{(0,\frac{1}{2})}(K_i) = \frac{i}{2}(1_{(1)}\otimes\sigma_i - J^{(0)}_i \otimes 1_{(2)}) = +\frac{i}{2}\sigma_i. \end{align}

(W1)

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) VL and VR, whose elements ΨL and ΨR are called left- and right-handed Weyl spinors respectively. Given (π(1/2,0),VL) and (π(0,1/2),VR) one may form their direct sum as representations,

\begin{align} \pi_{(\frac{1}{2},0) \oplus (0,\frac{1}{2})}(J_i) &= \frac{1}{2}\biggl(\begin{matrix} \sigma_i&0\\ 0&\sigma_i\\ \end{matrix}\biggr),\\ \pi_{(\frac{1}{2},0) \oplus (0,\frac{1}{2})}(K_i) &= \frac{i}{2} \biggl(\begin{matrix} \sigma_i&0\\ 0&-\sigma_i\\ \end{matrix}\biggr)\\ \end{align}.[22]

(D1)

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 L, ΨR) of (VLVR), 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.

## Notes

1. ^ Weinberg 2003, Chapter 5
2. ^ a b Hall 2003, Chapter 6
3. ^ Hall 2003, Chapter 4
4. Weinberg 2002, Chapter 5
5. Hall 2003, Chapter 3
6. ^ Weinberg 2002, Appendix B, Chapter 2
7. ^ Weinberg 2002, Section 2.7, Chapter 2
8. ^ Wigner 1937
9. ^ Wigner 1937, p.27
10. ^ Weinberg 2003, Section 2.7
11. ^ The "traceless" property can be expressed as Sαβgαβ = 0, or Sαα = 0, or Sαβgαβ = 0 depending on the presentation of the field: covariant, mixed, and contravariant respectively.
12. ^ Greiner, W (2000), Relativistic Quantum Mechanics, ISBN 3-540-67457-8
13. ^ Weinberg 2002 Chapter 2
14. ^ Weinberg 2002}, Chapter 3
15. ^ Knapp 2001, Chapter II
16. ^ a b Harish-Chandra 1947
17. ^ a b Taylor 1986
18. ^ Gelfand & Naimark 1947
19. ^ Takahashi 1963, p. 343
20. ^ Note that for a Hilbert space H, HS(H) may be identified canonically with the Hilbert space tensor product of H and its conjugate space.
21. ^ Knapp 2001
22. ^ Weinberg 2002, Equations (5.4.19) and (5.4.20)

## References

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