Representation theory of the Lorentz group

Representation theory of groups in general, and Lie groups in particular, is a very rich subject. The Lorentz group has some properties that makes it "agreeable" and others that make it "not very agreeable" within the context of representation theory; the group is simple and thus semisimple, but is not connected, and none of its components are simply connected. Furthermore, the Lorentz group is not compact. For finite-dimensional representations, the presence of semisimplicity means that the Lorentz group can be dealt with the same way as other semisimple groups using a well-developed theory. In addition, all representations are built from the irreducible ones, since the Lie algebra possesses the complete reducibility property. But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, cannot be dealt with in all the aspects as in the simple framework that applies to simply connected, compact groups. Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional unitary representations exist. Lack of simple connectedness gives rise to spin representations of the group. The non-connectedness means that, for representations of the full Lorentz group, time reversal and reversal of spatial orientation have to be dealt with separately. History The development of the finite-dimensional representation theory of the Lorentz group mostly follows that of representation theory in general. Lie theory originated with Sophus Lie in 1873. By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing. In 1913 the theorem of highest weight for representations of simple Lie algebras, the path that will be followed here, was completed by Élie Cartan. Richard Brauer was during the period of 1935–38 largely responsible for the development of the Weyl-Brauer matrices describing how spin representations of the Lorentz Lie algebra can be embedded in Clifford algebras. The Lorentz group has also historically received special attention in representation theory due to its exceptional importance in physics (see History of infinite-dimensional unitary representations below). Mathematicians Hermann Weyl and Harish-Chandra and physicists Eugene Wigner and Valentine Bargmann made substantial contributions both to general representation theory and in particular to the Lorentz group. Physicist Paul Dirac was perhaps the first to manifestly knit everything together in a practical application of major lasting importance with the Dirac equation in 1928. Lie algebra , Independent discoverer of Lie algebras. The simple Lie algebras were first classified by him in 1888. The irreducible representations of the Lie algebra of the Lorentz group can be derived by factoring that Lie algebra into a direct product of two subalgebras. Each subalgebra is isomorphic to \mathfrak{su}(2), and the irreducible representations of \mathfrak{su}(2) are labeled by nonnegative half-integers. Consequently, the irreducible representations of the Lorentz group's Lie algebra are labeled by ordered pairs (m,n) of nonnegative half-integers. This section addresses the irreducible complex linear representations of the complexification \mathfrak{so}(3; 1)_\Complex of the Lie algebra \mathfrak{so}(3; 1) of the Lorentz group. A convenient basis for \mathfrak{so}(3; 1) is given by the three generators of rotations and the three generators of boosts. They are explicitly given in conventions and Lie algebra bases. The Lie algebra is complexified, and the basis is changed to the components of its two ideals \mathbf{A} = \frac{\mathbf{J} + i \mathbf{K}}{2},\quad \mathbf{B} = \frac{\mathbf{J} - i \mathbf{K}}{2}. The components of and separately satisfy the commutation relations of the Lie algebra \mathfrak{su}(2) and, moreover, they commute with each other, \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 are indices which each take values , and is the three-dimensional Levi-Civita symbol. Let \mathbf{A}_\Complex and \mathbf{B}_\Complex denote the complex linear span of and respectively. One has the isomorphisms {{NumBlk|| \begin{align} \mathfrak{so}(3; 1) \hookrightarrow \mathfrak{so}(3; 1)_\Complex &\cong \mathbf{A}_\Complex \oplus \mathbf{B}_\Complex \cong \mathfrak{su}(2)_\Complex \oplus \mathfrak{su}(2)_\Complex \\[5pt] &\cong \mathfrak{sl}(2, \Complex) \oplus \mathfrak{sl}(2, \Complex) \\[5pt] &\cong \mathfrak{sl}(2, \Complex) \oplus i\mathfrak{sl}(2, \Complex) = \mathfrak{sl}(2, \Complex)_\Complex \hookleftarrow \mathfrak{sl}(2, \Complex), \end{align} }} where \mathfrak{sl}(2, \Complex) is the complexification of \mathfrak{su}(2) \cong \mathbf{A} \cong \mathbf{B}. The utility of these isomorphisms comes from the fact that all irreducible representations of \mathfrak{su}(2), and hence all irreducible complex linear representations of \mathfrak{sl}(2, \Complex), are known. The irreducible complex linear representation of \mathfrak{sl}(2, \Complex) is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of \mathfrak{sl}(2, \Complex). and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of are precisely the tensor products of irreducible unitary representations of . By appeal to simple connectedness, the second statement of the unitarian trick is applied. The objects in the following list are in one-to-one correspondence: • Holomorphic representations of \text{SL}(2, \Complex) \times \text{SL}(2, \Complex) • Smooth representations of • Real linear representations of \mathfrak{su}(2) \oplus \mathfrak{su}(2) • Complex linear representations of \mathfrak{sl}(2, \Complex) \oplus \mathfrak{sl}(2, \Complex) Tensor products of representations appear at the Lie algebra level as either of {{NumBlk|| \begin{align} \pi_1\otimes\pi_2(X) &= \pi_1(X) \otimes \mathrm{Id}_V + \mathrm{Id}_U \otimes \pi_2(X) && X \in \mathfrak{g} \\ \pi_1\otimes\pi_2(X, Y) &= \pi_1(X) \otimes \mathrm{Id}_V + \mathrm{Id}_U \otimes \pi_2(Y) && (X, Y) \in \mathfrak{g} \oplus \mathfrak{g} \end{align} | }} where is the identity operator. Here, the latter interpretation, which follows from , is intended. The highest weight representations of \mathfrak{sl}(2, \Complex) are indexed by for . (The highest weights are actually , but the notation here is adapted to that of \mathfrak{so}(3; 1).) The tensor products of two such complex linear factors then form the irreducible complex linear representations of \mathfrak{sl}(2, \Complex) \oplus \mathfrak{sl}(2, \Complex). Finally, the \R-linear representations of the real forms of the far left, \mathfrak{so}(3; 1), and the far right, \mathfrak{sl}(2, \Complex), in are obtained from the \Complex-linear representations of \mathfrak{sl}(2, \Complex) \oplus \mathfrak{sl}(2, \Complex) characterized in the previous paragraph. (μ, ν)-representations of sl(2, C) The complex linear representations of the complexification of \mathfrak{sl}(2, \Complex), \mathfrak{sl}(2, \Complex)_\Complex, obtained via isomorphisms in , stand in one-to-one correspondence with the real linear representations of \mathfrak{sl}(2, \Complex). The set of all real linear irreducible representations of \mathfrak{sl}(2, \Complex) are thus indexed by a pair . The complex linear ones, corresponding precisely to the complexification of the real linear \mathfrak{su}(2) representations, are of the form , while the conjugate linear ones are the . {{NumBlk|| \pi_{(m,n)}(J_i) = J^{(m)}_i \otimes 1_{(2n+1)}+1_{(2m+1)} \otimes J^{(n)}_i \pi_{(m,n)}(K_i) = -i \left(J^{(m)}_i \otimes 1_{(2n+1)} - 1_{(2m+1)} \otimes J^{(n)}_i\right), where is the -dimensional unit matrix and \mathbf{J}^{(n)} = \left(J^{(n)}_1, J^{(n)}_2, J^{(n)}_3\right) are the -dimensional irreducible representations of \mathfrak{so}(3) \cong \mathfrak{su}(2) also termed spin matrices or angular momentum matrices. These are explicitly given as \begin{align} \left(J_1^{(j)}\right)_{a'a} &= \frac{1}{2} \left(\sqrt{(j - a)(j + a + 1)}\delta_{a',a + 1} + \sqrt{(j + a)(j - a + 1)}\delta_{a',a - 1}\right) \\ \left(J_2^{(j)}\right)_{a'a} &= \frac{1}{2i}\left(\sqrt{(j - a)(j + a + 1)}\delta_{a',a + 1} - \sqrt{(j + a)(j - a + 1)}\delta_{a',a - 1}\right) \\ \left(J_3^{(j)}\right)_{a'a} &= a\delta_{a',a} \end{align} where denotes the Kronecker delta. In components, with , , the representations are given by \begin{align} \left(\pi_{(m,n)}\left(J_i\right)\right)_{a'b', ab} &= \delta_{b'b} \left(J_i^{(m)}\right)_{a'a} + \delta_{a'a} \left(J_i^{(n)}\right)_{b'b}\\ \left(\pi_{(m,n)}\left(K_i\right)\right)_{a'b', ab} &= -i \left( \delta_{b'b} \left(J_i^{(m)}\right)_{a'a} - \delta_{a'a} \left(J_i^{(n)}\right)_{b'b}\right) \end{align} Common representations Representations of this Lie algebra are used in physics, since various physical quantities of interest are functions, or collections of functions, defined on spacetime that transform together under a representation of an appropriate dimension. • The representation is the one-dimensional trivial representation and is carried by relativistic scalar field theories. • Fermionic supersymmetry generators transform under one of the or representations (Weyl spinors). • The four-momentum of a particle (either massless or massive) transforms under the representation, a four-vector. • A physical example of a (1,1) traceless symmetric tensor field is the traceless part of the energy–momentum tensor . Since complex conjugation exchanges and , the direct sum of representations and admits real matrix representatives and therefore has particular relevance to physics. • is the Dirac spinor representation. See also Weyl spinors below. • is the Rarita–Schwinger field representation. • would be the symmetry of the hypothesized gravitino. It can be obtained from the representation. • is the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation. 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 : \mathfrak{g} \to G. If \pi : \mathfrak{g} \to \mathfrak{gl}(V) for some vector space is a representation, a representation of the connected component of is defined by {{NumBlk||\begin{align} \Pi(g = e^{iX}) &\equiv e^{i\pi(X)}, && X \in \mathfrak g, \quad g = e^{iX} \in \mathrm{im}(\exp),\\ \Pi(g = g_1g_2\cdots g_n) &\equiv \Pi(g_1)\Pi(g_2)\cdots \Pi(g_n), && g \notin \mathrm{im}(\exp), \quad g_1 , g_2, \ldots, g_n \in \mathrm{im}(\exp). \end{align}|}} This definition applies whether the resulting representation is projective or not. Surjectiveness of exponential map for SO(3, 1) From a practical point of view, it is important whether the first formula in can be used for all elements of the group. It holds for all X \in \mathfrak{g}, however, in the general case, e.g. for \text{SL}(2,\Complex), not all are in the image of . But \exp : \mathfrak{so}(3;1) \to \text{SO}(3;1)^+ is surjective. One way to show this is to make use of the isomorphism \text{SO}(3; 1)^+ \cong \text{PGL}(2,\Complex), the latter being the Möbius group. It is a quotient of \text{GL}(n,\Complex) (see the linked article). The quotient map is denoted with p : \text{GL}(n,\Complex) \to \text{PGL}(2,\Complex). The map \exp : \mathfrak{gl}(n, \Complex) \to \text{GL}(n, \Complex) is onto. Apply with being the differential of at the identity. Then \forall X \in \mathfrak{gl}(n, \Complex): \quad p ( \exp (iX)) =\exp ( i \pi (X)). Since the left hand side is surjective (both and are), the right hand side is surjective and hence \exp : \mathfrak{pgl}(2, \Complex) \to \text{PGL}(2, \Complex) is surjective. Finally, recycle the argument once more, but now with the known isomorphism between and \text{PGL}(2, \Complex) to find that is onto for the connected component of the Lorentz group. Fundamental group The Lorentz group is doubly connected, i. e. is a group with two equivalence classes of loops as its elements. {{math proof | proof = To exhibit the fundamental group of , the topology of its covering group \text{SL}(2,\Complex) is considered. By the polar decomposition theorem, any matrix \lambda \in \text{SL}(2,\Complex) may be uniquely expressed as \lambda = ue^h, where is unitary with determinant one, hence in , and is Hermitian with trace zero. The trace and determinant conditions imply: \begin{align} h &= \begin{pmatrix}c&a-ib\\a+ib&-c\end{pmatrix} && (a,b,c) \in \R^3 \\[4pt] u &= \begin{pmatrix}d+ie&f+ig\\-f+ig&d-ie\end{pmatrix} && (d,e,f,g) \in \R^4 \text{ subject to } d^2 + e^2 + f^2 + g^2 = 1. \end{align} The manifestly continuous one-to-one map is a homeomorphism with continuous inverse given by (the locus of is identified with \mathbb{S}^3 \subset \R^4) \begin{cases} \R^3 \times \mathbb{S}^3\to \text{SL}(2, \Complex) \\ (r,s) \mapsto u(s) e^{h(r)} \end{cases} explicitly exhibiting that \text{SL}(2,\Complex) is simply connected. But \text{SO}(3; 1)\cong \text{SL}(2,\Complex)/\{\pm I\}, where \{\pm I\} is the center of \text{SL}(2,\Complex). Identifying and amounts to identifying with , which in turn amounts to identifying antipodal points on \mathbb{S}^3. Thus topologically, Once it is known whether a representation is projective, formula applies to all group elements and all representations, including the projective ones — with the understanding that the representative of a group element will depend on which element in the Lie algebra (the in ) is used to represent the group element in the standard representation. For the Lorentz group, the -representation is projective when is a half-integer. See . For a projective representation of , it holds that the universal covering group of . Since each has two elements, by the above construction, there is a 2:1 covering map . According to covering group theory, the Lie algebras \mathfrak{so}(3; 1), \mathfrak{sl}(2,\Complex) and \mathfrak{g} of are all isomorphic. The covering map is simply given by . An algebraic view For an algebraic view of the universal covering group, let \text{SL}(2,\Complex) act on the set of all Hermitian matrices \mathfrak{h} by the operation is a multiple of the identity, which must be since . The space \mathfrak{h} is mapped to Minkowski space , via {{NumBlk||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 on \mathfrak{h} preserves determinants. The induced representation of \text{SL}(2,\Complex) on \R^4, via the above isomorphism, given by {{NumBlk||\mathbf{p}(A)\overrightarrow{X} = \overrightarrow{AXA^\dagger}|}} preserves the Lorentz inner product since - \det X = \xi_1^2 + \xi_2^2 +\xi_3^2 -\xi_4^2 = x^2 + y^2 +z^2 - t^2. This means that belongs to the full Lorentz group . By the main theorem of connectedness, since \text{SL}(2,\Complex) is connected, its image under in is connected, and hence is contained in . It can be shown that the Lie map of \mathbf{p} : \text{SL}(2,\Complex) \to \text{SO}(3; 1)^+, is a Lie algebra isomorphism: \pi : \mathfrak{sl}(2,\Complex) \to \mathfrak{so}(3; 1). The map is also onto. Thus \text{SL}(2,\Complex), since it is simply connected, is the universal covering group of , isomorphic to the group of above. Non-surjectiveness of exponential mapping for SL(2, C) The exponential mapping \exp : \mathfrak{sl}(2,\Complex) \to \text{SL}(2,\Complex) is not onto. The matrix {{NumBlk||q = \begin{pmatrix} -1&1\\ 0&-1\\ \end{pmatrix}|}} is in \text{SL}(2,\Complex), but there is no Q\in \mathfrak{sl}(2,\Complex) such that . In general, if is an element of a connected Lie group with Lie algebra \mathfrak{g}, then, by , {{NumBlk||g = \exp(X_1) \cdots \exp(X_n), \qquad X_1, \ldots X_n \in \mathfrak{g}.|}} The matrix can be written {{NumBlk||\begin{align} &\exp(-X)\exp(i\pi H) \\ {}={} &\exp \left(\begin{pmatrix} 0&-1\\ 0&0\\ \end{pmatrix}\right) \exp \left(i\pi \begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix} \right) \\[6pt] {}={} &\begin{pmatrix} 1&-1\\ 0&1 \\ \end{pmatrix} \begin{pmatrix} -1&0\\ 0&-1\\ \end{pmatrix}\\[6pt] {}={} &\begin{pmatrix} -1&1\\ 0&-1\\ \end{pmatrix} \\ {}={} &q. \end{align} | }} Realization of representations of and and their Lie algebras The complex linear representations of \mathfrak{sl}(2,\Complex) and \text{SL}(2,\Complex) are more straightforward to obtain than the \mathfrak{so}(3; 1)^+ representations. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are related to the complex linear Lie algebra representations by exponentiation. The real linear representations of \mathfrak{sl}(2,\Complex) are exactly the -representations. They can be exponentiated too. The -representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here). The mathematical convention is used in this section for convenience. Lie algebra elements differ by a factor of , and there is no factor of in the exponential mapping compared to the physics convention used elsewhere. Let the basis of \mathfrak{sl}(2,\Complex) be {{NumBlk||H = \begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix}, \quad X = \begin{pmatrix} 0&1\\ 0&0\\ \end{pmatrix}, \quad Y = \begin{pmatrix} 0&0\\ 1&0\\ \end{pmatrix}. |}} This choice of basis and the notation is standard in the mathematical literature. Complex linear representations The irreducible holomorphic -dimensional representations \text{SL}(2,\Complex), n \geqslant 2, can be realized on the space of homogeneous polynomial of degree in 2 variables \mathbf{P}^2_n, the elements of which are P\begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} = c_n z_1^n + c_{n-1} z_1^{n-1}z_2 + \cdots + c_0 z_2^n, \quad c_0, c_1, \ldots, c_n \in \mathbb Z. The action of \text{SL}(2,\Complex) is given by {{NumBlk||(\phi_n(g)P)\begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} = \left [\phi_n \begin{pmatrix} a&b\\ c&d\\ \end{pmatrix} P\right ] \begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} = P\left( \begin{pmatrix} a&b\\ c&d\\ \end{pmatrix}^{-1} \begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} \right ), \qquad P \in \mathbf{P}^2_n.|}} The associated \mathfrak{sl}(2,\Complex)-action is, using and the definition above, for the basis elements of \mathfrak{sl}(2,\Complex), {{NumBlk||\phi_n(H) = -z_1\frac{\partial}{\partial z_1} + z_2\frac{\partial}{\partial z_2}, \quad \phi_n(X) = -z_2\frac{\partial}{\partial z_1}, \quad \phi_n(Y) = -z_1\frac{\partial}{\partial z_2}.|}} With a choice of basis for P \in \mathbf{P}^2_{n}, these representations become matrix Lie algebras. Real linear representations The -representations are realized on a space of polynomials \mathbf{P}^2_{\mu,\nu} in z_1, \overline{z_1}, z_2, \overline{z_2}, homogeneous of degree in z_1, z_2 and homogeneous of degree in \overline{z_1}, \overline{z_2}. {{NumBlk||(\phi_{\mu,\nu}(g)P)\begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} =\left [\phi_{\mu,\nu} \begin{pmatrix} a&b\\ c&d\\ \end{pmatrix} P\right ] \begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} = P \left( \begin{pmatrix} a&b\\ c&d\\ \end{pmatrix}^{-1} \begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} \right ), \quad P \in \mathbf{P}^2_{\mu,\nu}.|}} By employing again it is found that {{NumBlk||\begin{align} \phi_{\mu,\nu}(E)P = &- \frac{\partial P}{\partial z_1} \left (E_{11}z_1 + E_{12}z_2 \right ) - \frac{\partial P}{\partial z_2} \left (E_{21}z_1 + E_{22}z_2 \right) \\ &- \frac{\partial P}{\partial \overline{z_1}}\left (\overline{E_{11}}\overline{z_1} + \overline{E_{12}}\overline{z_2} \right ) -\frac{\partial P}{\partial \overline{z_2}} \left (\overline{E_{21}}\overline{z_1} + \overline{E_{22}}\overline{z_2} \right ) \end{align}, \quad E \in \mathfrak{sl}(2, \mathbf{C}).|}} In particular for the basis elements, {{NumBlk||\begin{align} \phi_{\mu,\nu}(H) &= -z_1\frac{\partial}{\partial z_1} + z_2\frac{\partial}{\partial z_2}-\overline{z_1}\frac{\partial}{\partial \overline{z_1}} + \overline{z_2}\frac{\partial}{\partial \overline{z_2}} \\ \phi_{\mu,\nu}(X) &= -z_2\frac{\partial}{\partial z_1} - \overline{z_2}\frac{\partial}{\partial \overline{z_1}} \\ \phi_{\mu,\nu}(Y) &= -z_1\frac{\partial}{\partial z_2} - \overline{z_1}\frac{\partial}{\partial \overline{z_2}} \end{align}|}} Properties of the (m, n) representations The representations, defined above via (as restrictions to the real form \mathfrak{sl}(3, 1)) of tensor products of irreducible complex linear representations and of \mathfrak{sl}(2,\Complex), are irreducible, and they are the only irreducible representations. and that a representation of is irreducible if and only if , where are irreducible representations of . • Uniqueness follows from that the are the only irreducible representations of , which is one of the conclusions of the theorem of the highest weight. Casimir operators For the Lorentz Lie algebra, the Casimir operators are central elements of the universal enveloping algebra, hence by Schur's lemma they act by scalars on each irreducible representation. For the finite-dimensional theory it is convenient to pass to the complexification and use the decomposition \mathfrak{so}(3,1)_\mathbb C \cong \mathbf A_{\mathbb C} \oplus \mathbf B_{\mathbb C}, where each of and \mathbf B_{\mathbb C} is a copy of \mathfrak{sl}(2,\mathbb C) (treated here as \mathfrak{su}(2)_{\mathbb C}, as above, with the generators A_i and B_i, respectively). With this normalization, quadratic Casimirs for the two factors may be taken as \Omega_A=A_1^2+A_2^2+A_3^2,\qquad \Omega_B=B_1^2+B_2^2+B_3^2. Equivalently, if A_\pm=A_1\pm iA_2 and B_\pm=B_1\pm iB_2, then \Omega_A=A_3^2+\frac12(A_+A_-+A_-A_+),\qquad \Omega_B=B_3^2+\frac12(B_+B_-+B_-B_+). Since the two factors commute, \Omega_A and \Omega_B are central in the universal enveloping algebra of \mathfrak{so}(3,1)_\mathbb C. A convenient normalized choice of algebraically independent quadratic Casimir operators for the Lorentz algebra is therefore C_1=2(\Omega_A+\Omega_B),\qquad C_2=\Omega_A-\Omega_B. In terms of the rotation and boost generators, this becomes C_1=\mathbf J^2-\mathbf K^2,\qquad C_2=i\,\mathbf J\cdot\mathbf K. If denotes the irreducible representation obtained from the spin- representation of \mathfrak a and the spin- representation of \mathfrak b, then \Omega_A\mapsto m(m+1),\qquad \Omega_B\mapsto n(n+1), and hence C_1\mapsto 2\bigl(m(m+1)+n(n+1)\bigr),\qquad C_2\mapsto m(m+1)-n(n+1). In particular, the conjugate pair and have the same value of C_1 but opposite values of C_2; when , the second Casimir vanishes. Dimension The representations are -dimensional. This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of \text{SL}(2,\Complex) and \mathfrak{sl}(2, \Complex). For a Lie general algebra \mathfrak{g} the Weyl dimension formula, \dim\pi_\rho = \frac{\Pi_{\alpha \in R^+} \langle\alpha, \rho + \delta \rangle}{\Pi_{\alpha \in R^+} \langle\alpha, \delta \rangle}, applies, where is the set of positive roots, is the highest weight, and is half the sum of the positive roots. The inner product \langle \cdot, \cdot \rangle is that of the Lie algebra \mathfrak{g}, invariant under the action of the Weyl group on \mathfrak{h} \subset \mathfrak{g}, the Cartan subalgebra. The roots (really elements of \mathfrak{h}^*) are via this inner product identified with elements of \mathfrak{h}. For \mathfrak{sl}(2,\Complex), the formula reduces to , where the present notation must be taken into account. The highest weight is . By taking tensor products, the result follows. Faithfulness If a (nontrivial) representation of a Lie group is not faithful, then is a nontrivial normal subgroup. There are three relevant cases. • is non-discrete and abelian. • is non-discrete and non-abelian. • is discrete. In this case , where is the center of . In the case of , the first case is excluded since is semi-simple. The second case (and the first case) is excluded because is simple. For the third case, is isomorphic to the quotient \text{SL}(2,\Complex)/\{\pm I\}. But \{\pm I\} is the center of \text{SL}(2,\Complex). It follows that the center of is trivial, and this excludes the third case. The conclusion is that every representation and every projective representation for 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 replaced by the center of \mathfrak{sl}(3; 1)^+The center of any semisimple Lie algebra is trivial and \mathfrak{so}(3; 1) is semi-simple and simple, and hence has no non-trivial ideals. A related fact is that if the corresponding representation of \text{SL}(2,\Complex) is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding \text{SL}(2,\Complex) representation is not faithful, but is . Non-unitarity The 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. Let , where is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group . Then where is the compact subgroup of consisting of unitary transformations of . The kernel of is a normal subgroup of . Since is simple, is either all of , in which case is trivial, or is trivial, in which case is faithful. In the latter case is a diffeomorphism onto its image, and is a Lie group. This would mean that is an embedded non-compact Lie subgroup of the compact group . This is impossible with the subspace topology on since all embedded Lie subgroups of a Lie group are closed If were closed, it would be compact, and then 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 and used in the construction are Hermitian. This means that is Hermitian, but 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 representation is, however, unitary when restricted to the rotation subgroup , but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an representation have -invariant subspaces of highest weight (spin) , where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) is -dimensional. So for example, the (, ) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively. Since the angular momentum operator is given by , the highest spin in quantum mechanics of the rotation sub-representation will be 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 -invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the representation has spin if is half-integer. The simplest are and , the Weyl-spinors of dimension . Then, for example, and are a spin representations of dimensions and respectively. According to the above paragraph, there are subspaces with spin both and in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under . It cannot be ruled out in general, however, that representations with multiple 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 representations for any (under ) 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 of \mathfrak{sl}(2,\Complex) \oplus \mathfrak{sl}(2,\Complex). The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation: • 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. • Two irreducible representations are isomorphic if and only if they have the same highest weight. • For each semisimple Lie algebra there exists a unique element of the Weyl group such that if is a dominant integral weight, then is again a dominant integral weight. • If \pi_{\mu_0} is an irreducible representation with highest weight , then \pi^*_{\mu_0} has highest weight . It follows that each is isomorphic to its dual \pi^*_{\mu}. The root system of \mathfrak{sl}(2,\Complex) \oplus \mathfrak{sl}(2,\Complex) is shown in the figure to the right. The Weyl group is generated by \{w_{\gamma}\} where w_\gamma is reflection in the plane orthogonal to as ranges over all roots. Inspection shows that so . Using the fact that if are Lie algebra representations and , then , the conclusion for is \pi_{m, n}^{*} \cong \pi_{m, n}, \quad \Pi_{m, n}^{*} \cong \Pi_{m, n}, \quad 2m, 2n \in \mathbf{N}. Complex conjugate representations If is a representation of a Lie algebra, then \overline{\pi} is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication. In general, every irreducible representation of \mathfrak{sl}(n,\Complex) can be written uniquely as , where \pi^\pm(X) = \frac{1}{2}\left(\pi(X) \pm i\pi\left(i^{-1}X\right)\right), with \pi^+ holomorphic (complex linear) and \pi^- anti-holomorphic (conjugate linear). For \mathfrak{sl}(2,\Complex), since \pi_\mu is holomorphic, \overline{\pi_\mu} is anti-holomorphic. Direct examination of the explicit expressions for \pi_{\mu, 0} and \pi_{0, \nu} in equation below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression also allows for identification of \pi^+ and \pi^- for \pi_{\mu, \nu} as \pi^+_{\mu, \nu} = \pi_\mu^{\oplus_{\nu+1}},\qquad \pi^-_{\mu, \nu} = \overline{\pi_\nu^{\oplus_{\mu+1}}}. Using the above identities (interpreted as pointwise addition of functions), for yields \begin{align} \overline{\pi_{m, n}} &= \overline{\pi_{m, n}^+ + \pi_{m, n}^-}=\overline{\pi_m^{\oplus_{2n + 1}}} + \overline{\overline{\pi_n}^{\oplus_{2m + 1}}} \\ &=\pi_n^{\oplus_{2m + 1}} + \overline{\pi_m}^{\oplus_{2n + 1}} = \pi_{n, m}^+ + \pi_{n, m}^- = \pi_{n, m} \\ & &&2m, 2n \in \mathbb{N} \\ \overline{\Pi_{m, n}} &= \Pi_{n, m} \end{align} where the statement for the group representations follow from . It follows that the irreducible representations have real matrix representatives if and only if . Reducible representations on the form have real matrices too. Adjoint representation, Clifford algebra, and Dirac spinor representation and wife Ilse 1970. Brauer generalized the spin representations of Lie algebras sitting inside Clifford algebras to spin higher than . Photo courtesy of MFO. In general representation theory, if is a representation of a Lie algebra \mathfrak{g}, then there is an associated representation of \mathfrak{g}, on , also denoted , given by {{NumBlk||\pi(X)(A) = [\pi(X), A],\qquad A \in \operatorname{End}(V),\ X \in \mathfrak{g}.|}} Likewise, a representation of a group yields a representation on of , still denoted , given by {{NumBlk||\Pi(g)(A) = \Pi(g)A\Pi(g)^{-1},\qquad A \in \operatorname{End}(V),\ g \in G.|}} If and are the standard representations on \R^4 and if the action is restricted to \mathfrak{so}(3, 1) \subset \text{End}(\R^4), then the two above representations are the adjoint representation of the Lie algebra and the adjoint representation of the group respectively. The corresponding representations (some \R^n or \Complex^n) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular. Applying this to the Lorentz group, if is a projective representation, then direct calculation using shows that the induced representation on is a proper representation, i.e. a representation without phase factors. In quantum mechanics this means that if or is a representation acting on some Hilbert space , then the corresponding induced representation acts on the set of linear operators on . As an example, the induced representation of the projective spin representation on is the non-projective 4-vector (, ) representation. For simplicity, consider only the "discrete part" of , that is, given a basis for , the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4-vector representation of above on this simplified has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, \mathcal{Cl}_{3,1}(\R), whose complexification is \text{M}(4, \Complex), generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the , a pseudoscalar irrep, also the , but with parity inversion eigenvalue , see the next section below, the already mentioned vector irrep, , a pseudovector irrep, with parity inversion eigenvalue +1 (not −1), and a tensor irrep, . The dimensions add up to . In other words, {{NumBlk||\mathcal{Cl}_{3,1}(\R) = (0,0) \oplus \left(\frac{1}{2}, \frac{1}{2}\right) \oplus [(1, 0) \oplus (0, 1)] \oplus \left(\frac{1}{2}, \frac{1}{2}\right)_p \oplus (0, 0)_p,|}} where, as is customary, a representation is confused with its representation space. spin representation The six-dimensional representation space of the tensor -representation inside \mathcal{Cl}_{3,1}(\R) has two roles. The {{NumBlk||\sigma^{\mu\nu} = -\frac{i}{4} \left[\gamma^\mu, \gamma^\nu\right],|}} where \gamma^0, \ldots, \gamma^3 \in \mathcal{Cl}_{3,1}(\R) are the gamma matrices, the sigmas, only 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, {{NumBlk||\left[\sigma^{\mu\nu}, \sigma^{\rho\tau}\right] = i \left(\eta^{\tau\mu}\sigma^{\rho\nu} + \eta^{\nu\tau}\sigma^{\mu\rho} - \eta^{\rho\mu}\sigma^{\tau\nu} - \eta^{\nu\rho}\sigma^{\mu\tau}\right),|}} and hence constitute a representation (in addition to spanning a representation space) sitting inside \mathcal{Cl}_{3,1}(\R), the spin representation. For details, see Dirac spinor and Dirac algebra. The conclusion is that every element of the complexified \mathcal{Cl}_{3,1}(\R) in (i.e. every complex 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 \Complex^4, making it a space of Dirac spinors. Reducible representations Other representations can be deduced from the irreducible ones, such as those obtained 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. \text{GL}(n,\R) and the Poincaré group. These representations are in general not irreducible. The Lorentz 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) representation is irreducible as a representation , the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If it can be extended to a representation of all of , the full Lorentz group, including space parity inversion and time reversal. The representations can be extended likewise. Space parity inversion For space parity inversion, the adjoint action of on \mathfrak{so}(3; 1) is considered, where is the standard representative of space parity inversion, , given by {{NumBlk||\mathrm{Ad}_P(J_i) = PJ_iP^{-1} = J_i, \qquad \mathrm{Ad}_P(K_i) = PK_iP^{-1} = -K_i.|}} It is these properties of and under that motivate the terms vector for and pseudovector or axial vector for . In a similar way, if is any representation of \mathfrak{so}(3; 1) and is its associated group representation, then acts on the representation of by the adjoint action, for X \in \mathfrak{so}(3; 1), . If is to be included in , then consistency with requires that {{NumBlk||\Pi(P)\pi(B_i)\Pi(P)^{-1} = \pi(A_i)|}} holds, where and are defined as in the first section. This can hold only if and have the same dimensions, i.e. only if . When then can be extended to an irreducible representation of , the orthochronous Lorentz group. The parity reversal representative does not come automatically with the general construction of the representations. It must be specified separately. The matrix (or a multiple of modulus −1 times it) may be used in the representation. If parity is included with a minus sign (the matrix ) in the representation, it is called a pseudoscalar representation. Time reversal Time reversal , acts similarly on \mathfrak{so}(3; 1) by {{NumBlk||\mathrm{Ad}_T(J_i) = TJ_iT^{-1} = -J_i, \qquad \mathrm{Ad}_T(K_i) = TK_iT^{-1} = K_i.|}} By explicitly including a representative for , as well as one for , a representation of the full Lorentz group is obtained. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the , in addition to the and generate the group. These are interpreted as generators of translations. The time-component is the Hamiltonian . The operator satisfies the relation {{NumBlk||\mathrm{Ad}_{T}(iH) = TiHT^{-1} = -iH|}} in analogy to the relations above with \mathfrak{so}(3; 1) replaced by the full Poincaré algebra. By just cancelling the 's, the result would imply that for every state with positive energy in a Hilbert space of quantum states with time-reversal invariance, there would be a state with negative energy . Such states do not exist. The operator is therefore chosen antilinear and antiunitary, so that it anticommutes with , resulting in , 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 is then antiunitary rather than a complex-linear representation operator. 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, , is usually taken to include both space parity and time inversions. Without space parity inversion, it is a reducible rather than irreducible representation. The third discrete symmetry entering in the CPT theorem along with and , charge conjugation symmetry , has nothing directly to do with Lorentz invariance. == Action on function spaces ==

If is a vector space of functions of a finite number of variables , then the action on a scalar function f \in V given by {{NumBlk||(\Pi(g)f)(x) = f\left(\Pi_x(g)^{-1} x\right),\qquad x \in \R^n, f \in V|}} produces another function . Here is an -dimensional representation, and is a possibly infinite-dimensional representation. A special case of this construction is when is a space of functions defined on the a linear group itself, viewed as a -dimensional manifold embedded in \R^{m^2} (with the dimension of the matrices). 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. {{NumBlk||f(\theta, \varphi) = \sum_{l = 1}^\infty\sum_{m = -l}^l f_{lm} Y^l_m(\theta, \varphi),|}} where the are generalized Fourier coefficients. The Lorentz group action restricts to that of and is expressed as {{NumBlk||\begin{align} (\Pi(R)f)(\theta(x), \varphi(x)) &= \sum_{l = 1}^\infty\sum_{m, = -l}^l\sum_{m' = -l}^lD^{(l)}_{mm'}(R) f_{lm'} Y^l_m \left(\theta\left(R^{-1} x\right), \varphi\left(R^{-1}x\right) \right), \\[5pt] & R \in \mathrm{SO}(3), x \in \mathbb{S}^2, \end{align} |}} where the are obtained from the representatives of odd dimension of the generators of rotation. Möbius group The identity component of the Lorentz group is isomorphic to the Möbius group . 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 acts on the plane according to {{NumBlk||f(z) = \frac{a z + b}{c z + d}, \qquad ad - bc \neq 0.|}} and can be represented by complex matrices {{NumBlk||\Pi_f = \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \lambda \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad \lambda \in \Complex - \{0\}, \operatorname{det} \Pi_f = 1, |}} since multiplication by a nonzero complex scalar does not change . These are elements of \text{SL}(2,\Complex) and are unique up to a sign (since give the same ), hence \text{SL}(2, \Complex) / \{\pm I\} \cong \text{SO}(3; 1)^+. 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 {{NumBlk||\begin{align} w(z) &= P \left\{ \begin{matrix} a & b & c & \\ \alpha & \beta & \gamma & \; z \\ \alpha' & \beta' & \gamma' & \end{matrix} \right\} \\ &= \left(\frac{z - a}{z - b}\right)^\alpha \left(\frac{z - c}{z - b}\right)^\gamma P \left\{ \begin{matrix} 0 & \infty & 1 & \\ 0 & \alpha + \beta + \gamma & 0 & \;\frac{(z - a)(c - b)}{(z - b)(c - a)} \\ \alpha' - \alpha & \alpha + \beta' + \gamma & \gamma' - \gamma & \end{matrix} \right\}\end{align}, |}} where the are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is {{NumBlk||P \left\{ \begin{matrix} 0 & \infty & 1 & \\ 0 & a & 0 & \;z \\ 1 - c & b & c - a - b & \end{matrix} \right\} = {}_2 F_1(a,\, b;\, c;\, z). |}} The set of constants 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 are and ,corresponding to the two linearly independent solutions, and for expansion around the singular point they are and . Similarly, the exponents for are and for the two solutions. One has thus {{NumBlk||w(z) = \left(\frac{z - a}{z - b}\right)^\alpha \left(\frac{z - c}{z - b}\right)^\gamma {}_2F_1 \left(\alpha + \beta + \gamma,\, \alpha + \beta' + \gamma;\, 1 + \alpha - \alpha';\, \frac{(z - a)(c - b)}{(z - b)(c - a)}\right),|}} where the condition (sometimes called Riemann's identity) \alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1 on the exponents of the solutions of Riemann's differential equation has been used to define . The first set of constants on the left hand side in , denotes the regular singular points of Riemann's differential equation. The second set, , are the corresponding exponents at for one of the two linearly independent solutions, and, accordingly, are exponents at for the second solution. Define an action of the Lorentz group on the set of all Riemann P-functions by first setting {{NumBlk||u(\Lambda)(z) = \frac{Az + B}{Cz + D},|}} where are the entries in {{NumBlk||\lambda = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in \text{SL}(2, \Complex),|}} for a Lorentz transformation. Define where is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Möbius 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 {{NumBlk||[\Pi(\Lambda) P](u) = P \left\{ \begin{matrix} \eta & \zeta & \theta & \\ \alpha & \beta & \gamma & \;u \\ \alpha' & \beta' & \gamma' & \end{matrix} \right\}, |}} where {{NumBlk||\eta = \frac{Aa + B}{Ca + D} \quad \text{ and } \quad \zeta = \frac{Ab + B}{Cb + D} \quad \text{ and } \quad \theta = \frac{Ac + B}{Cc + D}.|}} ==Infinite-dimensional unitary representations== History The Lorentz group and its double cover \text{SL}(2,\Complex) also have infinite dimensional unitary representations, studied independently by , and at the instigation of Paul Dirac. This trail of development begun with where he devised matrices and necessary for description of higher spin (compare Dirac matrices), elaborated upon by , see also , and proposed precursors of the Bargmann-Wigner equations. In 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 and , based on an analogue for \text{SL}(2,\Complex) of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in and . The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3-dimensional hyperbolic space sitting in Minkowski 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 , , and the posthumous text of . Principal series for SL(2, C) The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup  of G = \text{SL}(2,\Complex). Since the one-dimensional representations of correspond to the representations of the diagonal matrices, with non-zero complex entries and , they thus have the form \chi_{\nu,k}\begin{pmatrix}z& 0\\ c& z^{-1}\end{pmatrix}=r^{i\nu} e^{ik\theta}, for an integer, real and with . The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when is replaced by . By definition the representations are realized on sections of line bundles on G/B = \mathbb{S}^2, which is isomorphic to the Riemann sphere. When , these representations constitute the so-called spherical principal series. The restriction of a principal series to the maximal compact subgroup of  can also be realized as an induced representation of  using the identification , where is the maximal torus in  consisting of diagonal matrices with . It is the representation induced from the 1-dimensional representation , and is independent of . By Frobenius reciprocity, on  they decompose as a direct sum of the irreducible representations of  with dimensions with a non-negative integer. Using the identification between the Riemann sphere minus a point and \Complex, the principal series can be defined directly on L^2(\Complex) 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 . 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 where is the Weyl group element \begin{pmatrix}0& -1\\ 1& 0\end{pmatrix}. • The action of the Lie algebra \mathfrak{g} of  can be computed on the algebraic direct sum of the irreducible subspaces of  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. Complementary series for The for , the complementary series is defined on L^2(\Complex) for the inner product (f,g)_t =\iint \frac{f(z) \overline{g(w)}}{|z-w|^{2-t}} \, dz\, dw, 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 representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of , each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of . Irreducibility can be proved by analyzing the action of \mathfrak{g} on the algebraic sum of these subspaces Plancherel theorem for SL(2, C) The only irreducible unitary representations of \text{SL}(2,\Complex) are the principal series, the complementary series and the trivial representation. Since acts as on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided is taken to be even. To decompose the left regular representation of  on L^2(G) only the principal series are required. This immediately yields the decomposition on the subrepresentations L^2(G/\{\pm I\}), the left regular representation of the Lorentz group, and L^2(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 .) The left and right regular representation and are defined on L^2(G) by \begin{align} (\lambda(g)f)(x) &= f\left(g^{-1}x\right) \\ (\rho(g)f) (x) &= f(xg) \end{align} Now if is an element of , the operator \pi_{\nu, k}(f) defined by \pi_{\nu, k}(f) = \int_G f(g)\pi(g)\, dg is Hilbert–Schmidt. Define a Hilbert space  by H = \bigoplus_{k\geqslant 0} \text{HS} \left(L^2(\Complex)\right) \otimes L^2 \left(\R, c_k\sqrt{\nu^2 + k^2} d\nu \right), where c_k = \begin{cases} \frac{1}{4\pi^{3/2}} & k = 0 \\ \frac{1}{(2\pi)^{3/2}} & k \neq 0 \end{cases} and \text{HS}\left(L^2(\Complex)\right) denotes the Hilbert space of Hilbert–Schmidt operators on L^2(\Complex). Then the map  defined on by U(f)(\nu, k) = \pi_{\nu,k}(f) extends to a unitary of L^2(G) onto . The map  satisfies the intertwining property U(\lambda(x)\rho(y)f)(\nu,k) = \pi_{\nu,k}(x)^{-1} \pi_{\nu,k}(f)\pi_{\nu,k}(y). If are in then by unitarity (f_1, f_2) = \sum_{k\geqslant 0} c_k^2 \int_{-\infty}^\infty \operatorname{Tr} \left(\pi_{\nu,k}(f_1)\pi_{\nu,k}(f_2)^*\right) \left(\nu^2 + k^2\right) \, d\nu. Thus if f = f_1 * f_2^* denotes the convolution of f_1 and f_2^*, and f_2^*(g)=\overline{f_2(g^{-1})}, then f(1) = \sum_{k\geqslant 0} c_k^2 \int_{-\infty}^\infty \operatorname{Tr} \left(\pi_{\nu,k}(f) \right) \left(\nu^2 + k^2\right)\, 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 f_i \in L^2(G). By a theorem of Jacques Dixmier and Paul Malliavin, every smooth compactly supported function on G is a finite sum of convolutions of similar functions, the inversion formula holds for such . It can be extended to much wider classes of functions satisfying mild differentiability conditions. Since is a subgroup, is a representation of it as well. Each irreducible subrepresentation of is finite-dimensional, and the representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of if is unitary. The steps are the following: • Choose a suitable basis of common eigenvectors of and . • Compute matrix elements of and . • Enforce Lie algebra commutation relations. • Require unitarity together with orthonormality of the basis. Step 1 One suitable choice of basis and labeling is given by \left |j_0\, j_1;j\, m\right\rangle. If this were a finite-dimensional representation, then would correspond the lowest occurring eigenvalue of in the representation, equal to , and would correspond to the highest occurring eigenvalue, equal to . In the infinite-dimensional case, retains this meaning, but does not. 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 and forming the basis of the Lie algebra of \mathfrak{so}(3; 1). The matrix elements of J_\pm = J_1 \pm iJ_2 and J_3 (the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by \begin{align} \left\langle j\, m \right|J_+ \left| j\, m - 1 \right\rangle = \left\langle j\, m - 1 \right|J_- \left| j\, m \right\rangle &= \sqrt{(j + m)(j - m + 1)}, \\ \left\langle j\, m \right|J_3 \left| j\, m \right\rangle &= m, \end{align} where the labels and 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 \epsilon_{ijk} K_k, the triple 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 \begin{align} K^{(1)}_0 &= K_3,\\ K^{(1)}_{\pm 1} &= \mp\frac{1}{\sqrt 2}(K_1 \pm iK_2), \end{align} where the superscript signifies that the defined quantities are the components of a spherical tensor operator of rank (which explains the factor as well) and the subscripts are referred to as in formulas below, are given by \begin{align} \left\langle j' m'\left|K^{(1)}_0 \right|j\,m\right\rangle &= \left \langle j' \, m' \,k = 1 \,q = 0 | j \, m \right \rangle \left \langle j \left \| K^{(1)} \right \| j' \right \rangle,\\ \left\langle j' m'\left|K^{(1)}_{\pm 1}\right |j\,m\right\rangle &= \left \langle j' \, m' \, k= 1 \,q = \pm 1 | j \, m \right \rangle \left \langle j \left \| K^{(1)} \right \| j' \right \rangle. \end{align} Here, the first factors on the right-hand sides are Clebsch–Gordan coefficients for coupling with to get . The second factors are the reduced matrix elements. They do not depend on or , but depend on and, of course, . Step 3 The next step is to demand that the Lie algebra relations hold, i.e. that [K_\pm, K_3] = \pm J_\pm, \quad [K_+, K_-] = -2J_3. This results in a set of equations for which the solutions are \begin{align} \left \langle j \left \| K^{(1)} \right \| j \right \rangle &= i\frac{j_1j_0}{\sqrt{j(j+1)}},\\ \left \langle j \left \| K^{(1)} \right \| j-1 \right \rangle &= -B_j\xi_j\sqrt{j(2j-1)},\\ \left \langle j-1 \left \| K^{(1)} \right \| j \right \rangle &= B_j\xi_j^{-1}\sqrt{j(2j+1)}, \end{align} where B_j = \sqrt{\frac{(j^2 - j_0^2)(j^2 - j_1^2)}{j^2(4j^2 - 1)}}, \quad j_0=0, \tfrac{1}{2}, 1, \ldots \quad \text{and} \quad j_1, \xi_j \in \Complex. 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 and . Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning K_\pm^\dagger = K_\mp,\quad K_3^\dagger = K_3. This translates to \begin{align} \left \langle j \left \| K^{(1)} \right \| j \right \rangle &= \overline{\left \langle j \left \| K^{(1)} \right \| j \right \rangle},\\ \left \langle j \left \| K^{(1)} \right \| j - 1 \right \rangle &= -\overline{\left \langle j - 1 \left \| K^{(1)} \right \| j \right \rangle}, \end{align} leading to \begin{align} j_0 \left(j_1 + \overline{j_1}\right) &= 0, \\ \left|B_j\right| \left(\left|\xi_j\right|^2 - e^{-2i\beta_j}\right) &= 0, \end{align} where is the angle of on polar form. For follows \left|\xi_j\right|^2 = 1 and \xi_j = 1 is chosen by convention. There are two possible cases: • \underline{j_1 + \overline{j_1} = 0.} In this case , real, \left \langle j \left \| K^{(1)} \right \| j \right \rangle = \frac{\nu j_0}{j(j + 1)} \quad \text{and} \quad B_j = \sqrt{\frac{(j^2 - j_0^2)(j^2 + \nu^2)}{4j^2 - 1}} This is the principal series. Its elements are denoted (j_0, \nu), 2j_0 \in \N, \nu \in \R. • \underline{j_0=0.} It follows: \left \langle j \left \| K^{(1)} \right \| j \right \rangle = 0 \quad \text{and} \quad B_j = \sqrt{\frac{j^2 - \nu^2}{4j^2 - 1}} Since , is real and positive for , leading to . This is complementary series. Its elements are denoted This shows that the representations of above are all infinite-dimensional irreducible unitary representations. == Explicit formulas ==

Conventions and Lie algebra bases The metric of choice is given by , 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: \begin{align} J_1 = J^{23} = -J^{32} &= i\begin{pmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&0&1&0 \end{pmatrix},& K_1 = J^{01} = -J^{10} &= i\begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix},\\[8pt] J_2 = J^{31} = -J^{13} &= i\begin{pmatrix} 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&-1&0&0 \end{pmatrix},& K_2 = J^{02} = -J^{20} &= i\begin{pmatrix} 0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0 \end{pmatrix},\\[8pt] J_3 = J^{12} = -J^{21} &= i\begin{pmatrix} 0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0 \end{pmatrix},& K_3 = J^{03} = -J^{30} &= i\begin{pmatrix} 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0 \end{pmatrix}.\\[8pt] \end{align} The commutation relations of the Lie algebra \mathfrak{so}(3; 1) are: \left[J^{\mu\nu}, J^{\rho\sigma}\right] = i\left( \eta^{\sigma\mu}J^{\rho\nu} + \eta^{\nu\sigma}J^{\mu\rho} - \eta^{\rho\mu}J^{\sigma\nu} - \eta^{\nu\rho} J^{\mu\sigma} \right). In three-dimensional notation, these are \left[J_i, J_j\right] = i\epsilon_{ijk}J_k,\quad \left[J_i, K_j\right] = i\epsilon_{ijk}K_k,\quad \left[K_i, K_j\right] = -i\epsilon_{ijk}J_k. The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol above and in the sequel should be observed. For example, a typical boost and a typical rotation exponentiate as, \exp (-i\xi K_1)=\begin{pmatrix} \cosh \xi &\sinh \xi &0&0\\ \sinh \xi &\cosh \xi &0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, \qquad \exp (-i\theta J_1)=\begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&\cos \theta &-\sin\theta\\ 0&0&\sin\theta&\cos\theta \end{pmatrix}, symmetric and orthogonal, respectively. Weyl spinors transform under the -representation. Dirac discovered the gamma matrices in his search for a relativistically invariant equation, then already known to mathematicians. {{NumBlk||\begin{align} \pi_{\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)}\left(J_i\right) &= \frac{1}{2} \begin{pmatrix} \sigma_i&0\\ 0&\sigma_i \end{pmatrix} \\[8pt] \pi_{\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)}\left(K_i\right) &= \frac{i}{2} \begin{pmatrix} \sigma_i&0\\ 0&-\sigma_i \end{pmatrix} \end{align} | }} This is, up to a similarity transformation, the Dirac spinor representation of \mathfrak{so}(3; 1). It acts on the 4-component elements of , by matrix multiplication. The representation may also be obtained in a more general and basis-independent way from the action of the corresponding Clifford algebra and its spin group on a spinor module. These expressions for Dirac spinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of \mathfrak{so}(3; 1). Expressions for the group representations are obtained by exponentiation. == Physics 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 Poincaré 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 and the expinors of Dirac and Harish-Chandra, respectively—consistent with relativity and quantum mechanics, but they have found no proven physical application. 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. The equations that describe the fields 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. 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 and Dirac spinors respectively. The electromagnetic field is also a relativistic wave function according to this definition. The infinite-dimensional representations may be used in the analysis of scattering. Quantum field theory In quantum field theory, 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 (with modest requirements imposed, see the reference) of the system using the canonical formalism, from which a realization of the generators of the Lorentz group may be deduced. 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 an -component field operator: A relativistic field operator is a set of operator valued functions on spacetime which transforms under proper Poincaré transformations according to \Psi^\alpha(x) \to \Psi'^\alpha(x) = U[\Lambda, a]\Psi^\alpha(x) U^{-1} \left[\Lambda, a\right] = D{\left[\Lambda^{-1}\right]^\alpha}_\beta \Psi^\beta (\Lambda x + a) Here is the unitary operator representing on the Hilbert space on which is defined and is an -dimensional representation of the Lorentz group. The transformation rule is the second Wightman axiom of quantum field theory. By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass and spin (or helicity), it is deduced that {{NumBlk||\Psi^\alpha(x) = \sum_\sigma \int dp \left(a(\mathbf{p}, \sigma) u^\alpha(\mathbf{p}, \sigma) e^{ip \cdot x} + a^\dagger(\mathbf{p}, \sigma) v^\alpha(\mathbf{p}, \sigma) e^{-ip \cdot x} \right),|}} where are interpreted as creation and annihilation operators respectively. The creation operator transforms according to a^\dagger(\mathbf{p}, \sigma) \rightarrow a'^\dagger \left(\mathbf{p}, \sigma\right) = U[\Lambda]a^\dagger(\mathbf{p}, \sigma) U \left[\Lambda^{-1}\right] = a^\dagger(\Lambda \mathbf{p}, \rho) D^{(s)}{\left[R(\Lambda, \mathbf{p})^{-1}\right]^\rho}_\sigma, and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin of the particle. The connection between the two are the wave functions, also called coefficient functions u^\alpha(\mathbf{p}, \sigma) e^{ip \cdot x},\quad v^\alpha(\mathbf{p}, \sigma) e^{-ip \cdot x} that carry both the indices operated on by Lorentz transformations and the indices operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection. To exhibit the connection, subject both sides of equation to a Lorentz transformation resulting in for e.g. , {D[\Lambda]^\alpha}_{\alpha'} u^{\alpha'}(\mathbf{p}, \lambda) = {D^{(s)}[R(\Lambda, \mathbf{p})]^{\lambda'}}_\lambda u^\alpha \left(\Lambda \mathbf{p}, \lambda'\right), where is the non-unitary Lorentz group representative of and is a unitary representative of the so-called Wigner rotation associated to and that derives from the representation of the Poincaré group, and is the spin of the particle. All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the 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 dimensions, the generalized Lorentz groups of the appropriate dimension take the place of . 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 -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 only possible dimension of spacetime in such theories is 10. ==Open problems==

The classification and characterization of the representation theory of the Lorentz group was completed in 1947. 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 Poincaré 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 is the completion of the Bargmann–Wigner programme for the isometry group of the de Sitter spacetime . Ideally, the physical components of wave functions would be realized on the hyperboloid of radius embedded in \R^{D-2, 1} and the corresponding covariant wave equations of the infinite-dimensional unitary representation to be known. == See also ==

• Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006). • Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. ==References==