Representation theory of the Lorentz group

The development of the representation theory has historically followed the development of the more general theory of representation theory of semisimple groups, largely due to Élie Cartan and Hermann Weyl, but the Lorentz group has also received special attention due to its importance in physics. Notable contributors are physicist E. P. Wigner and mathematician Valentine Bargmann with their Bargmann–Wigner program, one conclusion of which is, roughly, a classification of all unitary representations of the inhomogeneous Lorentz group amounts to a classification of all possible relativistic wave equations. The classification of the irreducible infinite-dimensional representations of the Lorentz group was established by Paul Dirac's doctoral student in theoretical physics, Harish-Chandra, later turned mathematician, in 1947. Closely related work was published independently by Bargmann and Israel Gelfand together with Mark Naimark in the same year.

The full theory of the finite-dimensional representations of the Lie algebra of the Lorentz group is deduced using the general framework of the representation theory of semisimple Lie algebras. The finite-dimensional representations of the connected component ${\displaystyle \text{SO}(3;1{)}^{+}}$ of the full Lorentz group O(3; 1) are obtained by employing the Lie correspondence and the matrix exponential. The full finite-dimensional representation theory of the universal covering group (and also the spin group, a double cover) ${\displaystyle \text{SL}(2,\mathbb{C})}$ of ${\displaystyle \text{SO}(3;1{)}^{+}}$ is obtained, and explicitly given in terms of action on a function space in representations of ${\displaystyle \text{SL}(2,\mathbb{C})}$ and ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$. The representatives of time reversal and space inversion are given in space inversion and time reversal, completing the finite-dimensional theory for the full Lorentz group. The general properties of the (m, n) representations are outlined. Action on function spaces is considered, with the action on spherical harmonics and the Riemann P-functions appearing as examples. The infinite-dimensional case of irreducible unitary representations is realized for the ${\displaystyle \text{SL}(2,\mathbb{C})}$ principal series and the complementary series. Finally, the Plancherel formula for ${\displaystyle \text{SL}(2,\mathbb{C})}$ is given, and representations of SO(3, 1) are classified and realized for Lie algebras.

Finite-dimensional representations

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

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 ${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$, and the irreducible representations of ${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$ are labeled by nonnegative half-integers. Consequently, the irreducible representations of the Lorentz group's Lie algebra are labeled by ordered pairs ${\displaystyle (m,n)}$ of nonnegative half-integers.

This section addresses the irreducible complex linear representations of the complexification ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1{)}_{\mathbb{C}}}$ of the Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ of the Lorentz group. A convenient basis for ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ is given by the three generators J_i of rotations and the three generators K_i of boosts. They are explicitly given in conventions and Lie algebra bases.

The Lie algebra is complexified, and the basis is changed to the components of its two ideals $${\displaystyle \mathbf{A}=\frac{\mathbf{J}+i\mathbf{K}}{2},\mathbf{B}=\frac{\mathbf{J}-i\mathbf{K}}{2}.}$$

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

$${\displaystyle \left[A_i,A_j\right]=i{\epsilon }_{ijk}{A}_k,\left[B_i,B_j\right]=i{\epsilon }_{ijk}{B}_k,\left[A_i,B_j\right]=0,}$$

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

One has the isomorphisms

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

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

Unitarian trick

The Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})\oplus \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ is the Lie algebra of ${\displaystyle \text{SL}(2,\mathbb{C})\times \text{SL}(2,\mathbb{C}).}$ It contains the compact subgroup SU(2) × SU(2) with Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{u}(2)\oplus \mathfrak{s}\mathfrak{u}(2).}$ The latter is a compact real form of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})\oplus \mathfrak{s}\mathfrak{l}(2,\mathbb{C}).}$ Thus from the first statement of the unitarian trick, representations of SU(2) × SU(2) are in one-to-one correspondence with holomorphic representations of ${\displaystyle \text{SL}(2,\mathbb{C})\times \text{SL}(2,\mathbb{C}).}$

By compactness, the Peter–Weyl theorem applies to SU(2) × SU(2), and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of SU(2) × SU(2) are precisely the tensor products of irreducible unitary representations of SU(2).

By appeal to simple connectedness, the second statement of the unitarian trick is applied. The objects in the following list are in one-to-one correspondence:

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

Tensor products of representations appear at the Lie algebra level as either of

where Id is the identity operator. Here, the latter interpretation, which follows from (G6), is intended. The highest weight representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ are indexed by μ for μ = 0, 1/2, 1, .... (The highest weights are actually 2μ = 0, 1, 2, ..., but the notation here is adapted to that of ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1).}$) The tensor products of two such complex linear factors then form the irreducible complex linear representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})\oplus \mathfrak{s}\mathfrak{l}(2,\mathbb{C}).}$

Finally, the ${\displaystyle \mathbb{R}}$-linear representations of the real forms of the far left, ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$, and the far right, ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C}),}$ in (A1) are obtained from the ${\displaystyle \mathbb{C}}$-linear representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})\oplus \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ characterized in the previous paragraph.

(μ, ν)-representations of sl(2, C)

The complex linear representations of the complexification of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C}),\mathfrak{s}\mathfrak{l}(2,\mathbb{C}{)}_{\mathbb{C}},}$ obtained via isomorphisms in (A1), stand in one-to-one correspondence with the real linear representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C}).}$ The set of all real linear irreducible representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ are thus indexed by a pair (μ, ν). The complex linear ones, corresponding precisely to the complexification of the real linear ${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$ representations, are of the form (μ, 0), while the conjugate linear ones are the (0, ν). All others are real linear only. The linearity properties follow from the canonical injection, the far right in (A1), of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ into its complexification. Representations on the form (ν, ν) or (μ, ν) ⊕ (ν, μ) are given by real matrices (the latter are not irreducible). Explicitly, the real linear (μ, ν)-representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ are $${\displaystyle {\phi }_{\mu ,\nu }(X)=\left({\phi }_{\mu }\otimes \overline{{\phi }_{\nu }}\right)(X)={\phi }_{\mu }(X)\otimes {\mathrm{Id}}_{\nu +1}+{\mathrm{Id}}_{\mu +1}\otimes \overline{{\phi }_{\nu }(X)},X\in \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$$ where ${\phi }_{\mu },\mu =0,\frac{1}{2},1,\frac{3}{2},\dots$ are the complex linear irreducible representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ and ${\displaystyle \overline{{\phi }_{\nu }},\nu =0,\frac{1}{2},1,\frac{3}{2},\dots }$ their complex conjugate representations. (The labeling is usually in the mathematics literature 0, 1, 2, ..., but half-integers are chosen here to conform with the labeling for the ${\displaystyle \mathfrak{s}\mathfrak{o}(3,1)}$ Lie algebra.) Here the tensor product is interpreted in the former sense of (A0). These representations are concretely realized below.

(m, n)-representations of so(3; 1)

Via the displayed isomorphisms in (A1) and knowledge of the complex linear irreducible representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})\oplus \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ upon solving for J and K, all irreducible representations of ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1{)}_{\mathbb{C}},}$ and, by restriction, those of ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ are obtained. The representations of ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible. Since ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ is semisimple, all its representations can be built up as direct sums of the irreducible ones.

Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers m = μ and n = ν, conventionally written as one of $${\displaystyle (m,n)\equiv {\pi }_{(m,n)}:\mathfrak{s}\mathfrak{o}(3;1)\to \mathfrak{g}\mathfrak{l}(V),}$$ where V is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given by

where 1_n is the n-dimensional unit matrix and $${\displaystyle {\mathbf{J}}^{(n)}=\left(J_1^{(n)},J_2^{(n)},J_3^{(n)}\right)}$$ are the (2n + 1)-dimensional irreducible representations of ${\displaystyle \mathfrak{s}\mathfrak{o}(3)\cong \mathfrak{s}\mathfrak{u}(2)}$ also termed spin matrices or angular momentum matrices. These are explicitly given as $${\displaystyle \begin{array}{cc}{\left(J_1^{(j)}\right)}_{a^{\prime }a}& =\frac{1}{2}\left(\sqrt{(j-a)(j+a+1)}{\delta }_{a^{\prime },a+1}+\sqrt{(j+a)(j-a+1)}{\delta }_{a^{\prime },a-1}\right)\\ {\left(J_2^{(j)}\right)}_{a^{\prime }a}& =\frac{1}{2i}\left(\sqrt{(j-a)(j+a+1)}{\delta }_{a^{\prime },a+1}-\sqrt{(j+a)(j-a+1)}{\delta }_{a^{\prime },a-1}\right)\\ {\left(J_3^{(j)}\right)}_{a^{\prime }a}& =a{\delta }_{a^{\prime },a}\end{array}}$$ where δ denotes the Kronecker delta. In components, with −m ≤ a, a′ ≤ m, −n ≤ b, b′ ≤ n, the representations are given by $${\displaystyle \begin{array}{cc}{\left({\pi }_{(m,n)}\left(J_i\right)\right)}_{a^{\prime }{b}^{\prime },ab}& ={\delta }_{b^{\prime }b}{\left(J_i^{(m)}\right)}_{a^{\prime }a}+{\delta }_{a^{\prime }a}{\left(J_i^{(n)}\right)}_{b^{\prime }b}\\ {\left({\pi }_{(m,n)}\left(K_i\right)\right)}_{a^{\prime }{b}^{\prime },ab}& =-i\left({\delta }_{b^{\prime }b}{\left(J_i^{(m)}\right)}_{a^{\prime }a}-{\delta }_{a^{\prime }a}{\left(J_i^{(n)}\right)}_{b^{\prime }b}\right)\end{array}}$$

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

Since complex conjugation exchanges (m, n) and (n, m), the direct sum of representations (m, n) and (n, m) admits real matrix representatives and therefore has particular relevance to physics.

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

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 ${\displaystyle \exp :\mathfrak{g}\to G.}$

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

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 (G2) can be used for all elements of the group. It holds for all ${\displaystyle X\in \mathfrak{g}}$, however, in the general case, e.g. for ${\displaystyle \text{SL}(2,\mathbb{C})}$, not all g ∈ G are in the image of exp.

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

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

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

Fundamental group

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

Proof

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

$${\displaystyle \lambda =u{e}^h,}$$

where u is unitary with determinant one, hence in SU(2), and h is Hermitian with trace zero. The trace and determinant conditions imply: $${\displaystyle \begin{array}{cccc}h& =\left(\begin{array}{cc}c& a-ib\\ a+ib& -c\end{array}\right)& & (a,b,c)\in {\mathbb{R}}^3\\ u& =\left(\begin{array}{cc}d+ie& f+ig\\ -f+ig& d-ie\end{array}\right)& & (d,e,f,g)\in {\mathbb{R}}^4\text{ subject to }{d}^2+e^2+f^2+g^2=1.\end{array}}$$

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

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

explicitly exhibiting that ${\displaystyle \text{SL}(2,\mathbb{C})}$ is simply connected. But ${\displaystyle \text{SO}(3;1)\cong \text{SL}(2,\mathbb{C})/\{\pm I\},}$ where ${\displaystyle \{\pm I\}}$ is the center of ${\displaystyle \text{SL}(2,\mathbb{C})}$. Identifying λ and −λ amounts to identifying u with −u, which in turn amounts to identifying antipodal points on ${\displaystyle {\mathbb{S}}^3.}$ Thus topologically, $${\displaystyle \text{SO}(3;1)\cong {\mathbb{R}}^3\times ({\mathbb{S}}^3/{\mathbb{Z}}_2),}$$

where last factor is not simply connected: Geometrically, it is seen (for visualization purposes, ${\displaystyle {\mathbb{S}}^3}$ may be replaced by ${\displaystyle {\mathbb{S}}^2}$) that a path from u to −u in ${\displaystyle SU(2)\cong {\mathbb{S}}^3}$ is a loop in ${\displaystyle {\mathbb{S}}^3/{\mathbb{Z}}_2}$ since u and −u are antipodal points, and that it is not contractible to a point. But a path from u to −u, thence to u again, a loop in ${\displaystyle {\mathbb{S}}^3}$ and a double loop (considering p(ue^h) = p(−ue^h), where ${\displaystyle p:\text{SL}(2,\mathbb{C})\to \text{SO}(3;1)}$ is the covering map) in ${\displaystyle {\mathbb{S}}^3/{\mathbb{Z}}_2}$ that is contractible to a point (continuously move away from −u "upstairs" in ${\displaystyle {\mathbb{S}}^3}$ and shrink the path there to the point u). Thus π₁(SO(3; 1)) is a group with two equivalence classes of loops as its elements, or put more simply, SO(3; 1) is doubly connected.

Projective representations

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

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

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

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

Covering group SL(2, C)

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

$${\displaystyle \left(\frac{1}{2}{\sigma }_1,\frac{1}{2}{\sigma }_2,\frac{1}{2}{\sigma }_3,\frac{i}{2}{\sigma }_1,\frac{i}{2}{\sigma }_2,\frac{i}{2}{\sigma }_3\right)\equiv (j_1,j_2,j_3,k_1,k_2,k_3),}$$

where the sigmas are the Pauli matrices. From the relations

is obtained

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

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

The exponential mapping ${\displaystyle \exp :\mathfrak{s}\mathfrak{l}(2,\mathbb{C})\to \text{SL}(2,\mathbb{C})}$ is not onto. The matrix

is in ${\displaystyle \text{SL}(2,\mathbb{C}),}$ but there is no ${\displaystyle Q\in \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ such that q = exp(Q).

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

The matrix q can be written

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

The complex linear representations of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ and ${\displaystyle \text{SL}(2,\mathbb{C})}$ are more straightforward to obtain than the ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1{)}^{+}}$ 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 ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ are exactly the (μ, ν)-representations. They can be exponentiated too. The (μ, 0)-representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here).

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

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

Complex linear representations

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

$${\displaystyle P\left(\begin{array}{c}{z}_1\\ z_2\end{array}\right)=c_n{z}_1^n+c_{n-1}{z}_1^{n-1}{z}_2+\cdots +c_0{z}_2^n,c_0,c_1,\dots ,c_n\in \mathbb{Z}.}$$

The action of ${\displaystyle \text{SL}(2,\mathbb{C})}$ is given by

The associated ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$-action is, using (G6) and the definition above, for the basis elements of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C}),}$

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

Real linear representations

The (μ, ν)-representations are realized on a space of polynomials ${\displaystyle {\mathbf{P}}_{\mu ,\nu }^2}$ in ${\displaystyle z_1,\overline{z_1},z_2,\overline{z_2},}$ homogeneous of degree μ in ${\displaystyle z_1,z_2}$ and homogeneous of degree ν in ${\displaystyle \overline{z_1},\overline{z_2}.}$ The representations are given by

By employing (G6) again it is found that

In particular for the basis elements,

Properties of the (m, n) representations

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

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

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 $${\displaystyle \mathfrak{s}\mathfrak{o}(3,1{)}_{\mathbb{C}}\cong {\mathbf{A}}_{\mathbb{C}}\oplus {\mathbf{B}}_{\mathbb{C}},}$$ where each of ${\displaystyle {\mathbf{A}}_{\mathbb{C}}}$ and ${\displaystyle {\mathbf{B}}_{\mathbb{C}}}$ is a copy of ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ (treated here as ${\displaystyle \mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}}}$, as above, with the generators ${\displaystyle A_i}$ and ${\displaystyle B_i}$, respectively).

With this normalization, quadratic Casimirs for the two factors may be taken as $${\displaystyle {\mathrm{\Omega }}_A=A_1^2+A_2^2+A_3^2,{\mathrm{\Omega }}_B=B_1^2+B_2^2+B_3^2.}$$ Equivalently, if ${\displaystyle A_{\pm }=A_1\pm i{A}_2}$ and ${\displaystyle B_{\pm }=B_1\pm i{B}_2}$, then $${\displaystyle {\mathrm{\Omega }}_A=A_3^2+\frac{1}{2}(A_{+}{A}_{-}+A_{-}{A}_{+}),{\mathrm{\Omega }}_B=B_3^2+\frac{1}{2}(B_{+}{B}_{-}+B_{-}{B}_{+}).}$$ Since the two factors commute, ${\displaystyle {\mathrm{\Omega }}_A}$ and ${\displaystyle {\mathrm{\Omega }}_B}$ are central in the universal enveloping algebra of ${\displaystyle \mathfrak{s}\mathfrak{o}(3,1{)}_{\mathbb{C}}}$. A convenient normalized choice of algebraically independent quadratic Casimir operators for the Lorentz algebra is therefore $${\displaystyle C_1=2({\mathrm{\Omega }}_A+{\mathrm{\Omega }}_B),C_2={\mathrm{\Omega }}_A-{\mathrm{\Omega }}_B.}$$

In terms of the rotation and boost generators, this becomes $${\displaystyle C_1={\mathbf{J}}^2-{\mathbf{K}}^2,C_2=i\mathbf{J}\cdot \mathbf{K}.}$$ If (m,n) denotes the irreducible representation obtained from the spin-m representation of ${\displaystyle \mathfrak{a}}$ and the spin-n representation of ${\displaystyle \mathfrak{b}}$, then $${\displaystyle {\mathrm{\Omega }}_A\mapsto m(m+1),{\mathrm{\Omega }}_B\mapsto n(n+1),}$$ and hence $${\displaystyle C_1\mapsto 2(m(m+1)+n(n+1)),C_2\mapsto m(m+1)-n(n+1).}$$ In particular, the conjugate pair (m,n) and (n,m) have the same value of ${\displaystyle C_1}$ but opposite values of ${\displaystyle C_2}$; when m=n, the second Casimir vanishes.

Dimension

The (m, n) representations are (2m + 1)(2n + 1)-dimensional. This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of ${\displaystyle \text{SL}(2,\mathbb{C})}$ and ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$. For a Lie general algebra ${\displaystyle \mathfrak{g}}$ the Weyl dimension formula, $${\displaystyle \dim {\pi }_{\rho }=\frac{{\mathrm{\Pi }}_{\alpha \in R^{+}}⟨\alpha ,\rho +\delta ⟩}{{\mathrm{\Pi }}_{\alpha \in R^{+}}⟨\alpha ,\delta ⟩},}$$ applies, where R⁺ is the set of positive roots, ρ is the highest weight, and δ is half the sum of the positive roots. The inner product ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is that of the Lie algebra ${\displaystyle \mathfrak{g},}$ invariant under the action of the Weyl group on ${\displaystyle \mathfrak{h}\subset \mathfrak{g},}$ the Cartan subalgebra. The roots (really elements of ${\displaystyle {\mathfrak{h}}^{\ast }}$) are via this inner product identified with elements of ${\displaystyle \mathfrak{h}.}$ For ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C}),}$ the formula reduces to dim π_μ = 2μ + 1 = 2m + 1, where the present notation must be taken into account. The highest weight is 2μ. By taking tensor products, the result follows.

Faithfulness

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

1. N is non-discrete and abelian.
2. N is non-discrete and non-abelian.
3. N is discrete. In this case N ⊂ Z, where Z is the center of G.

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

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

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

Non-unitarity

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

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

Restriction to SO(3)

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

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

Spinors

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

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

Dual representations

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

1. The set of weights of the dual representation of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation.
2. Two irreducible representations are isomorphic if and only if they have the same highest weight.
3. For each semisimple Lie algebra there exists a unique element w₀ of the Weyl group such that if μ is a dominant integral weight, then w₀ ⋅ (−μ) is again a dominant integral weight.
4. If ${\displaystyle {\pi }_{{\mu }_0}}$ is an irreducible representation with highest weight μ₀, then ${\displaystyle {\pi }_{{\mu }_0}^{\ast }}$ has highest weight w₀ ⋅ (−μ).

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

Complex conjugate representations

If π is a representation of a Lie algebra, then ${\displaystyle \overline{\pi }}$ is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication. In general, every irreducible representation π of ${\displaystyle \mathfrak{s}\mathfrak{l}(n,\mathbb{C})}$ can be written uniquely as π = π⁺ + π⁻, where $${\displaystyle {\pi }^{\pm }(X)=\frac{1}{2}\left(\pi (X)\pm i\pi \left(i^{-1}X\right)\right),}$$ with ${\displaystyle {\pi }^{+}}$ holomorphic (complex linear) and ${\displaystyle {\pi }^{-}}$ anti-holomorphic (conjugate linear). For ${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C}),}$ since ${\displaystyle {\pi }_{\mu }}$ is holomorphic, ${\displaystyle \overline{{\pi }_{\mu }}}$ is anti-holomorphic. Direct examination of the explicit expressions for ${\displaystyle {\pi }_{\mu ,0}}$ and ${\displaystyle {\pi }_{0,\nu }}$ in equation (S8) below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression (S8) also allows for identification of ${\displaystyle {\pi }^{+}}$ and ${\displaystyle {\pi }^{-}}$ for ${\displaystyle {\pi }_{\mu ,\nu }}$ as $${\displaystyle {\pi }_{\mu ,\nu }^{+}={\pi }_{\mu }^{{\oplus }_{\nu +1}},{\pi }_{\mu ,\nu }^{-}=\overline{{\pi }_{\nu }^{{\oplus }_{\mu +1}}}.}$$

Using the above identities (interpreted as pointwise addition of functions), for SO(3; 1)⁺ yields $${\displaystyle \begin{array}{cc}\overline{{\pi }_{m,n}}& =\overline{{\pi }_{m,n}^{+}+{\pi }_{m,n}^{-}}=\overline{{\pi }_m^{{\oplus }_{2n+1}}}+\overline{{\overline{{\pi }_n}}^{{\oplus }_{2m+1}}}\\ & ={\pi }_n^{{\oplus }_{2m+1}}+{\overline{{\pi }_m}}^{{\oplus }_{2n+1}}={\pi }_{n,m}^{+}+{\pi }_{n,m}^{-}={\pi }_{n,m}\\ & & & 2m,2n\in \mathbb{N}\\ \overline{{\mathrm{\Pi }}_{m,n}}& ={\mathrm{\Pi }}_{n,m}\end{array}}$$ where the statement for the group representations follow from exp(X) = exp(X). It follows that the irreducible representations (m, n) have real matrix representatives if and only if m = n. Reducible representations on the form (m, n) ⊕ (n, m) have real matrices too.

Adjoint representation, Clifford algebra, and Dirac spinor representation

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

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

If π and Π are the standard representations on ${\displaystyle {\mathbb{R}}^4}$ and if the action is restricted to ${\displaystyle \mathfrak{s}\mathfrak{o}(3,1)\subset \text{End}({\mathbb{R}}^4),}$ 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 ${\displaystyle {\mathbb{R}}^n}$ or ${\displaystyle {\mathbb{C}}^n}$) 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 (Π, V) is a projective representation, then direct calculation using (G5) shows that the induced representation on End(V) is a proper representation, i.e. a representation without phase factors.

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

For simplicity, consider 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. 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. (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, ${\displaystyle {\mathcal{C}\mathcal{l}}_{3,1}(\mathbb{R}),}$ whose complexification is ${\displaystyle \text{M}(4,\mathbb{C}),}$ generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the (0, 0), a pseudoscalar irrep, also the (0, 0), but with parity inversion eigenvalue −1, see the next section below, the already mentioned vector irrep, (⁠1/2⁠, ⁠1/2⁠), a pseudovector irrep, (⁠1/2⁠, ⁠1/2⁠) with parity inversion eigenvalue +1 (not −1), and a tensor irrep, (1, 0) ⊕ (0, 1). The dimensions add up to 1 + 1 + 4 + 4 + 6 = 16. In other words,

where, as is customary, a representation is confused with its representation space.

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

The six-dimensional representation space of the tensor (1, 0) ⊕ (0, 1)-representation inside ${\displaystyle {\mathcal{C}\mathcal{l}}_{3,1}(\mathbb{R})}$ has two roles. The

where ${\displaystyle {\gamma }^0,\dots ,{\gamma }^3\in {\mathcal{C}\mathcal{l}}_{3,1}(\mathbb{R})}$ are the gamma matrices, the sigmas, only 6 of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,

and hence constitute a representation (in addition to spanning a representation space) sitting inside ${\displaystyle {\mathcal{C}\mathcal{l}}_{3,1}(\mathbb{R}),}$ the (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠) spin representation. For details, see Dirac spinor and Dirac algebra.

The conclusion is that every element of the complexified ${\displaystyle {\mathcal{C}\mathcal{l}}_{3,1}(\mathbb{R})}$ in End(H) (i.e. every complex 4×4 matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on ${\displaystyle {\mathbb{C}}^4,}$ 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. ${\displaystyle \text{GL}(n,\mathbb{R})}$ 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) (m, n) representation is irreducible as a representation SO(3; 1)⁺, the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If m = n it can be extended to a representation of all of O(3; 1), the full Lorentz group, including space parity inversion and time reversal. The representations (m, n) ⊕ (n, m) can be extended likewise.

Space parity inversion

For space parity inversion, the adjoint action Ad_P of P ∈ SO(3; 1) on ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ is considered, where P is the standard representative of space parity inversion, P = diag(1, −1, −1, −1), given by

It is these properties of K and J under P that motivate the terms vector for K and pseudovector or axial vector for J. In a similar way, if π is any representation of ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ 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 ${\displaystyle X\in \mathfrak{s}\mathfrak{o}(3;1),}$ g ∈ SO(3; 1)⁺. If P is to be included in Π, then consistency with (F1) requires that

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

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

Time reversal

Time reversal T = diag(−1, 1, 1, 1), acts similarly on ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ by

By explicitly including a representative for T, as well as one for P, a representation of the full Lorentz group O(3; 1) 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 P^μ, in addition to the Jⁱ and Kⁱ generate the group. These are interpreted as generators of translations. The time-component P⁰ is the Hamiltonian H. The operator T satisfies the relation

in analogy to the relations above with ${\displaystyle \mathfrak{s}\mathfrak{o}(3;1)}$ replaced by the full Poincaré algebra. By just cancelling the i's, the result THT⁻¹ = −H would imply that for every state Ψ with positive energy E in a Hilbert space of quantum states with time-reversal invariance, there would be a state Π(T⁻¹)Ψ with negative energy −E. Such states do not exist. The operator Π(T) is therefore chosen antilinear and antiunitary, so that it anticommutes with i, resulting in THT⁻¹ = H, and its action on Hilbert space likewise becomes antilinear and antiunitary. It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix. This is mathematically sound, see Wigner's theorem, but Π(T) 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, (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠), 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 P and T, charge conjugation symmetry C, has nothing directly to do with Lorentz invariance.