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.^[31]

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.^{[nb 9]}^[32] 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.^[33] Lack of simple connectedness gives rise to spin representations of the group.^[34] 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.^[35]^[36]

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.^[37]^[38] By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing.^[39]^[40] In 1913 the theorem of highest weight for representations of simple Lie algebras, the path that will be followed here, was completed by Élie Cartan.^[41]^[42] Richard Brauer was during the period of 1935–38 largely responsible for the development of the Weyl-Brauer matrices describing how spin representations of the Lorentz Lie algebra can be embedded in Clifford algebras.^[43]^[44] The Lorentz group has also historically received special attention in representation theory, see History of infinite-dimensional unitary representations below, due to its exceptional importance in physics. Mathematicians Hermann Weyl^[41]^[45]^[37]^[46]^[47] and Harish-Chandra^[48]^[49] and physicists Eugene Wigner^[50]^[51] and Valentine Bargmann^[52]^[53]^[54] made substantial contributions both to general representation theory and in particular to the Lorentz group.^[55] Physicist Paul Dirac was perhaps the first to manifestly knit everything together in a practical application of major lasting importance with the Dirac equation in 1928.^[56]^[57]^{[nb 10]}

The Lie algebra

This section addresses the irreducible complex linear representations of the complexification ${\displaystyle {\mathfrak {so))(3;1)_{\mathbb {C} ))$ 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 $J i$ of rotations and the three generators $K i$ of boosts. They are explicitly given in conventions and Lie algebra bases.

The Lie algebra is complexified, and the basis is changed to the components of its two ideals^[58]

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

The components of $A = (A 1, A 2, A 3)$ and $B = (B 1, B 2, B 3)$ separately satisfy the commutation relations of the Lie algebra ${\mathfrak {su))(2)$ and, moreover, they commute with each other,^[59]

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

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

One has the isomorphisms^[60]^{[nb 11]}

{\begin{aligned}{\mathfrak {so))(3;1)\hookrightarrow {\mathfrak {so))(3;1)_{\mathbb {C} }&\cong \mathbf {A} _{\mathbb {C} }\oplus \mathbf {B} _{\mathbb {C} }\cong {\mathfrak {su))(2)_{\mathbb {C} }\oplus {\mathfrak {su))(2)_{\mathbb {C} }\\[5pt]&\cong {\mathfrak {sl))(2,\mathbb {C} )\oplus {\mathfrak {sl))(2,\mathbb {C} )\\[5pt]&\cong {\mathfrak {sl))(2,\mathbb {C} )\oplus i{\mathfrak {sl))(2,\mathbb {C} )={\mathfrak {sl))(2,\mathbb {C} )_{\mathbb {C} }\hookleftarrow {\mathfrak {sl))(2,\mathbb {C} ),\end{aligned))

(A1)

where ${\mathfrak {sl))(2,\mathbb {C} )$ 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 ${\mathfrak {sl))(2,\mathbb {C} ),$ are known. The irreducible complex linear representation of ${\mathfrak {sl))(2,\mathbb {C} )$ is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of ${\mathfrak {sl))(2,\mathbb {C} ).$

The unitarian trick

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

By compactness, the Peter–Weyl theorem applies to $SU(2) \times SU(2)$ ,^[61] and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of $SU(2) \times SU(2)$ are precisely the tensor products of irreducible unitary representations of $SU(2)$ .^[62]

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

Holomorphic representations of ${\text{SL))(2,\mathbb {C} )\times {\text{SL))(2,\mathbb {C} )$
Smooth representations of $SU(2) \times SU(2)$
Real linear representations of ${\mathfrak {su))(2)\oplus {\mathfrak {su))(2)$
Complex linear representations of ${\mathfrak {sl))(2,\mathbb {C} )\oplus {\mathfrak {sl))(2,\mathbb {C} )$

Tensor products of representations appear at the Lie algebra level as either of^{[nb 12]}

{\begin{aligned}\pi _{1}\otimes \pi _{2}(X)&=\pi _{1}(X)\otimes \mathrm {Id} _{V}+\mathrm {Id} _{U}\otimes \pi _{2}(X)&&X\in {\mathfrak {g))\\\pi _{1}\otimes \pi _{2}(X,Y)&=\pi _{1}(X)\otimes \mathrm {Id} _{V}+\mathrm {Id} _{U}\otimes \pi _{2}(Y)&&(X,Y)\in {\mathfrak {g))\oplus {\mathfrak {g))\end{aligned))

(A0)

where $Id$ is the identity operator. Here, the latter interpretation, which follows from (G6), is intended. The highest weight representations of ${\mathfrak {sl))(2,\mathbb {C} )$ are indexed by $μ$ for $μ = 0, 1/2, 1, ...$ . (The highest weights are actually $2 μ = 0, 1, 2, ...$ , 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 ${\mathfrak {sl))(2,\mathbb {C} )\oplus {\mathfrak {sl))(2,\mathbb {C} ).$

Finally, the $\mathbb {R}$ -linear representations of the real forms of the far left, ${\mathfrak {so))(3;1)$ , and the far right, ${\mathfrak {sl))(2,\mathbb {C} ),$ ^{[nb 13]} in (A1) are obtained from the $\mathbb {C}$ -linear representations of ${\mathfrak {sl))(2,\mathbb {C} )\oplus {\mathfrak {sl))(2,\mathbb {C} )$ characterized in the previous paragraph.

The (μ, ν)-representations of sl(2, C)

The complex linear representations of the complexification of ${\mathfrak {sl))(2,\mathbb {C} ),{\mathfrak {sl))(2,\mathbb {C} )_{\mathbb {C} },$ obtained via isomorphisms in (A1), stand in one-to-one correspondence with the real linear representations of ${\mathfrak {sl))(2,\mathbb {C} ).$ ^[63] The set of all real linear irreducible representations of ${\mathfrak {sl))(2,\mathbb {C} )$ are thus indexed by a pair $(μ, ν)$ . The complex linear ones, corresponding precisely to the complexification of the real linear ${\mathfrak {su))(2)$ representations, are of the form $(μ, 0)$ , while the conjugate linear ones are the $(0, ν)$ .^[63] All others are real linear only. The linearity properties follow from the canonical injection, the far right in (A1), of ${\mathfrak {sl))(2,\mathbb {C} )$ into its complexification. Representations on the form $(ν, ν)$ or $(μ, ν) \oplus (ν, μ)$ are given by real matrices (the latter are not irreducible). Explicitly, the real linear $(μ, ν)$ -representations of ${\mathfrak {sl))(2,\mathbb {C} )$ are

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

where

{\textstyle \varphi _{\mu },\mu =0,{\tfrac {1}{2)),1,{\tfrac {3}{2)),\ldots }

are the complex linear irreducible representations of

{\mathfrak {sl))(2,\mathbb {C} )

and

{\overline {\varphi _{\nu ))},\nu =0,{\tfrac {1}{2)),1,{\tfrac {3}{2)),\ldots

their complex conjugate representations. (The labeling is usually in the mathematics literature

0, 1, 2, ...

, but half-integers are chosen here to conform with the labeling for the

{\mathfrak {so))(3,1)

Lie algebra.) Here the tensor product is interpreted in the former sense of (A0). These representations are concretely realized below.

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

Via the displayed isomorphisms in (A1) and knowledge of the complex linear irreducible representations of ${\mathfrak {sl))(2,\mathbb {C} )\oplus {\mathfrak {sl))(2,\mathbb {C} )$ upon solving for $J$ and $K$ , all irreducible representations of ${\mathfrak {so))(3;1)_{\mathbb {C} },$ and, by restriction, those of ${\mathfrak {so))(3;1)$ are obtained. The representations of ${\mathfrak {so))(3;1)$ obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible.^[60] Since ${\mathfrak {so))(3;1)$ is semisimple,^[60] all its representations can be built up as direct sums of the irreducible ones.

Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers $m = μ$ and $n = ν$ , conventionally written as one of

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

where

V

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

{\displaystyle \pi _{(m,n)}(J_{i})=J_{i}^{(m)}\otimes 1_{(2n+1)}+1_{(2m+1)}\otimes J_{i}^{(n)))

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

(A2)

where $1 n$ is the $n$ -dimensional unit matrix and

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

are the

(2 n + 1)

-dimensional irreducible representations of

{\mathfrak {so))(3)\cong {\mathfrak {su))(2)

also termed spin matrices or angular momentum matrices. These are explicitly given as^[64]

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

where

δ

denotes the Kronecker delta. In components, with

- m \leq a, a' \leq m

- n \leq b, b' \leq n

, the representations are given by^[65]

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

Common representations

Irreducible representations for small $(m, n)$ . Dimension in parentheses.
	$m = 0$	$1 / 2$	$1$	$3 / 2$
$n = 0$	Scalar (1)	Left-handed Weyl spinor (2)	Self-dual 2-form (3)	(4)
$1 / 2$	Right-handed Weyl spinor (2)	4-vector (4)	(6)	(8)
$1$	Anti-self-dual 2-form (3)	(6)	Traceless symmetric tensor (9)	(12)
$3 / 2$	(4)	(8)	(12)	(16)

The $(0, 0)$ representation is the one-dimensional trivial representation and is carried by relativistic scalar field theories.
Fermionic supersymmetry generators transform under one of the $(0, 1 / 2)$ or $(1 / 2, 0)$ representations (Weyl spinors).^[29]
The four-momentum of a particle (either massless or massive) transforms under the $(1 / 2, 1 / 2)$ representation, a four-vector.
A physical example of a (1,1) traceless symmetric tensor field is the traceless^{[nb 15]} part of the energy–momentum tensor $T μν$ .^[66]^{[nb 16]}

Off-diagonal direct sums

Since for any irreducible representation for which $m \neq n$ it is essential to operate over the field of complex numbers, the direct sum of representations $(m, n)$ and $(n, m)$ have particular relevance to physics, since it permits to use linear operators over real numbers.

$(1 / 2, 0) \oplus (0, 1 / 2)$ is the bispinor representation. See also Dirac spinor and Weyl spinors and bispinors below.
$(1, 1 / 2) \oplus (1 / 2, 1)$ is the Rarita–Schwinger field representation.
$(3 / 2, 0) \oplus (0, 3 / 2)$ would be the symmetry of the hypothesized gravitino.^{[nb 17]} It can be obtained from the $(1, 1 / 2) \oplus (1 / 2, 1)$ representation.
$(1, 0) \oplus (0, 1)$ is the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation.

The group

The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence.^[67] The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras.^[68] The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted $\exp :{\mathfrak {g))\to G.$

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

{\begin{aligned}\Pi (g=e^{iX})&\equiv e^{i\pi (X)},&&X\in {\mathfrak {g)),\quad g=e^{iX}\in \mathrm {im} (\exp ),\\\Pi (g=g_{1}g_{2}\cdots g_{n})&\equiv \Pi (g_{1})\Pi (g_{2})\cdots \Pi (g_{n}),&&g\notin \mathrm {im} (\exp ),\quad g_{1},g_{2},\ldots ,g_{n}\in \mathrm {im} (\exp ).\end{aligned))

(G2)

This definition applies whether the resulting representation is projective or not.

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 $X\in {\mathfrak {g))$ , however, in the general case, e.g. for ${\text{SL))(2,\mathbb {C} )$ , not all $g \in G$ are in the image of $exp$ .

But ${\displaystyle \exp :{\mathfrak {so))(3;1)\to {\text{SO))(3;1)^{+))$ is surjective. One way to show this is to make use of the isomorphism ${\text{SO))(3;1)^{+}\cong {\text{PGL))(2,\mathbb {C} ),$ the latter being the Möbius group. It is a quotient of ${\text{GL))(n,\mathbb {C} )$ (see the linked article). The quotient map is denoted with $p:{\text{GL))(n,\mathbb {C} )\to {\text{PGL))(2,\mathbb {C} ).$ The map $\exp :{\mathfrak {gl))(n,\mathbb {C} )\to {\text{GL))(n,\mathbb {C} )$ is onto.^[69] Apply (Lie) with $π$ being the differential of $p$ at the identity. Then

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

Since the left hand side is surjective (both $exp$ and $p$ are), the right hand side is surjective and hence $\exp :{\mathfrak {pgl))(2,\mathbb {C} )\to {\text{PGL))(2,\mathbb {C} )$ is surjective.^[70] Finally, recycle the argument once more, but now with the known isomorphism between $SO(3; 1) +$ and ${\text{PGL))(2,\mathbb {C} )$ to find that $exp$ is onto for the connected component of the Lorentz group.

Fundamental group

The Lorentz group is doubly connected, i. e. $π 1 (SO(3; 1))$ 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 ${\text{SL))(2,\mathbb {C} )$ is considered. By the polar decomposition theorem, any matrix $\lambda \in {\text{SL))(2,\mathbb {C} )$ may be uniquely expressed as^[71]

\lambda =ue^{h},

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

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

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

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

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

{\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 $p:{\text{SL))(2,\mathbb {C} )\to {\text{SO))(3;1)$ is the covering map) in ${\displaystyle \mathbb {S} ^{3}/\mathbb {Z} _{2))$ that is contractible to a point (continuously move away from $- u$ "upstairs" in ${\displaystyle \mathbb {S} ^{3))$ and shrink the path there to the point $u$ ).^[72] Thus $π 1 (SO(3; 1))$ 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 $π 1 (SO(3; 1) +)$ has two elements, some representations of the Lie algebra will yield projective representations.^[73]^{[nb 18]} Once it is known whether a representation is projective, formula (G2) applies to all group elements and all representations, including the projective ones — with the understanding that the representative of a group element will depend on which element in the Lie algebra (the $X$ in (G2)) is used to represent the group element in the standard representation.

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

For a projective representation $Π$ of $SO(3; 1) +$ , it holds that^[72]

\left[\Pi (\Lambda _{1})\Pi (\Lambda _{2})\Pi ^{-1}(\Lambda _{1}\Lambda _{2})\right]^{2}=1\Rightarrow \Pi (\Lambda _{1}\Lambda _{2})=\pm \Pi (\Lambda _{1})\Pi (\Lambda _{2}),\quad \Lambda _{1},\Lambda _{2}\in \mathrm {SO} (3;1),

(G5)

since any loop in $SO(3; 1) +$ traversed twice, due to the double connectedness, is contractible to a point, so that its homotopy class is that of a constant map. It follows that $Π$ is a double-valued function. It is not possible to consistently choose a sign to obtain a continuous representation of all of $SO(3; 1) +$ , but this is possible locally around any point.^[33]

The covering group SL(2, C)

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

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

where the sigmas are the Pauli matrices. From the relations

{\displaystyle [\sigma _{i},\sigma _{j}]=2i\epsilon _{ijk}\sigma _{k))

(J1)

is obtained

[j_{i},j_{j}]=i\epsilon _{ijk}j_{k},\quad [j_{i},k_{j}]=i\epsilon _{ijk}k_{k},\quad [k_{i},k_{j}]=-i\epsilon _{ijk}j_{k},

(J2)

which are exactly on the form of the $3$ -dimensional version of the commutation relations for ${\mathfrak {so))(3;1)$ (see conventions and Lie algebra bases below). Thus, the map $J i \leftrightarrow j i$ , $K i \leftrightarrow k i$ , extended by linearity is an isomorphism. Since ${\text{SL))(2,\mathbb {C} )$ is simply connected, it is the universal covering group of $SO(3; 1) +$ .

More on covering groups in general and the

{\text{SL))(2,\mathbb {C} )

covering of the Lorentz group in particular

A geometric view[edit]

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

Let $p g (t), 0 \leq t \leq 1$ be a path from $1 \in SO(3; 1) +$ to $g \in SO(3; 1) +$ , denote its homotopy class by $[p g]$ and let $π g$ be the set of all such homotopy classes. Define the set

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

(C1)

and endow it with the multiplication operation

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

(C2)

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

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

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

An algebraic view[edit]

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

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

(C3)

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

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

(C4)

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

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

(C5)

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

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

(C6)

preserves the Lorentz inner product since

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

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

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

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

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

The exponential mapping $\exp :{\mathfrak {sl))(2,\mathbb {C} )\to {\text{SL))(2,\mathbb {C} )$ is not onto.^[76] The matrix

q={\begin{pmatrix}-1&1\\0&-1\\\end{pmatrix))

(S6)

is in ${\text{SL))(2,\mathbb {C} ),$ but there is no $Q\in {\mathfrak {sl))(2,\mathbb {C} )$ such that $q = exp(Q)$ .^{[nb 22]}

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

g=\exp(X_{1})\cdots \exp(X_{n}),\qquad X_{1},\ldots X_{n}\in {\mathfrak {g)).

(S7)

The matrix $q$ can be written

{\begin{aligned}&\exp(-X)\exp(i\pi H)\\{}={}&\exp \left({\begin{pmatrix}0&-1\\0&0\\\end{pmatrix))\right)\exp \left(i\pi {\begin{pmatrix}1&0\\0&-1\\\end{pmatrix))\right)\\[6pt]{}={}&{\begin{pmatrix}1&-1\\0&1\\\end{pmatrix)){\begin{pmatrix}-1&0\\0&-1\\\end{pmatrix))\\[6pt]{}={}&{\begin{pmatrix}-1&1\\0&-1\\\end{pmatrix))\\{}={}&q.\end{aligned))

(S8)

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

The complex linear representations of ${\mathfrak {sl))(2,\mathbb {C} )$ and ${\text{SL))(2,\mathbb {C} )$ are more straightforward to obtain than the ${\displaystyle {\mathfrak {so))(3;1)^{+))$ representations. They can be (and usually are) written down from scratch. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are related to the complex linear Lie algebra representations by exponentiation. The real linear representations of ${\mathfrak {sl))(2,\mathbb {C} )$ are exactly the $(μ, ν)$ -representations. They can be exponentiated too. The $(μ, 0)$ -representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here).

The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of $i$ and there is no factor of $i$ in the exponential mapping compared to the physics convention used elsewhere. Let the basis of ${\mathfrak {sl))(2,\mathbb {C} )$ be^[77]

H={\begin{pmatrix}1&0\\0&-1\\\end{pmatrix)),\quad X={\begin{pmatrix}0&1\\0&0\\\end{pmatrix)),\quad Y={\begin{pmatrix}0&0\\1&0\\\end{pmatrix)).

(S1)

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

Complex linear representations

The irreducible holomorphic $(n + 1)$ -dimensional representations ${\text{SL))(2,\mathbb {C} ),n\geqslant 2,$ can be realized on the space of homogeneous polynomial of degree $n$ in 2 variables $\mathbf {P} _{n}^{2},$ ^[78]^[79] 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 ${\text{SL))(2,\mathbb {C} )$ is given by^[80]^[81]

(\phi _{n}(g)P){\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix))=\left[\phi _{n}{\begin{pmatrix}a&b\\c&d\\\end{pmatrix))P\right]{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix))=P\left({\begin{pmatrix}a&b\\c&d\\\end{pmatrix))^{-1}{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix))\right),\qquad P\in \mathbf {P} _{n}^{2}.

(S2)

The associated ${\mathfrak {sl))(2,\mathbb {C} )$ -action is, using (G6) and the definition above, for the basis elements of ${\mathfrak {sl))(2,\mathbb {C} ),$ ^[82]

\phi _{n}(H)=-z_{1}{\frac {\partial }{\partial z_{1))}+z_{2}{\frac {\partial }{\partial z_{2))},\quad \phi _{n}(X)=-z_{2}{\frac {\partial }{\partial z_{1))},\quad \phi _{n}(Y)=-z_{1}{\frac {\partial }{\partial z_{2))}.

(S5)

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

Real linear representations

The $(μ, ν)$ -representations are realized on a space of polynomials ${\displaystyle \mathbf {P} _{\mu ,\nu }^{2))$ in $z_{1},{\overline {z_{1))},z_{2},{\overline {z_{2))},$ homogeneous of degree $μ$ in ${\displaystyle z_{1},z_{2))$ and homogeneous of degree $ν$ in ${\overline {z_{1))},{\overline {z_{2))}.$ ^[79] The representations are given by^[83]

(\phi _{\mu ,\nu }(g)P){\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix))=\left[\phi _{\mu ,\nu }{\begin{pmatrix}a&b\\c&d\\\end{pmatrix))P\right]{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix))=P\left({\begin{pmatrix}a&b\\c&d\\\end{pmatrix))^{-1}{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix))\right),\quad P\in \mathbf {P} _{\mu ,\nu }^{2}.

(S6)

By employing (G6) again it is found that

{\begin{aligned}\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{aligned)),\quad E\in {\mathfrak {sl))(2,\mathbf {C} ).

(S7)

In particular for the basis elements,

{\begin{aligned}\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{aligned))

(S8)

Properties of the (m, n) representations

The $(m, n)$ representations, defined above via (A1) (as restrictions to the real form ${\mathfrak {sl))(3,1)$ ) of tensor products of irreducible complex linear representations $π m = μ$ and $π n = ν$ of ${\mathfrak {sl))(2,\mathbb {C} ),$ are irreducible, and they are the only irreducible representations.^[61]

Irreducibility follows from the unitarian trick^[84] and that a representation $Π$ of $SU(2) \times SU(2)$ is irreducible if and only if $Π = Π μ \otimes Π ν$ ,^{[nb 23]} 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.^[85]

Dimension

The $(m, n)$ representations are $(2 m + 1)(2 n + 1)$ -dimensional.^[86] This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of ${\text{SL))(2,\mathbb {C} )$ and ${\mathfrak {sl))(2,\mathbb {C} )$ . For a Lie general algebra ${\mathfrak {g))$ the Weyl dimension formula,^[87]

\dim \pi _{\rho }={\frac {\Pi _{\alpha \in R^{+))\langle \alpha ,\rho +\delta \rangle }{\Pi _{\alpha \in R^{+))\langle \alpha ,\delta \rangle )),

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

\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

{\displaystyle {\mathfrak {h))^{*))

are via this inner product identified with elements of

{\mathfrak {h)).

For

{\mathfrak {sl))(2,\mathbb {C} ),

the formula reduces to

dim π μ = 2 μ + 1 = 2 m + 1

, where the present notation must be taken into account. The highest weight is

2 μ

.^[88] By taking tensor products, the result follows.

Faithfulness

If a representation $Π$ of a Lie group $G$ is not faithful, then $N = ker Π$ is a nontrivial normal subgroup.^[89] There are three relevant cases.

$N$ is non-discrete and abelian.
$N$ is non-discrete and non-abelian.
$N$ is discrete. In this case $N \subset Z$ , where $Z$ is the center of $G$ .^{[nb 24]}

In the case of $SO(3; 1) +$ , the first case is excluded since $SO(3; 1) +$ is semi-simple.^{[nb 25]} The second case (and the first case) is excluded because $SO(3; 1) +$ is simple.^{[nb 26]} For the third case, $SO(3; 1) +$ is isomorphic to the quotient ${\text{SL))(2,\mathbb {C} )/\{\pm I\}.$ But ${\displaystyle \{\pm I\))$ is the center of ${\text{SL))(2,\mathbb {C} ).$ It follows that the center of $SO(3; 1) +$ is trivial, and this excludes the third case. The conclusion is that every representation $Π : SO(3; 1) + \to GL(V)$ and every projective representation $Π : SO(3; 1) + \to PGL(W)$ 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,^[90] and the center of $SO(3; 1) +$ replaced by the center of ${\displaystyle {\mathfrak {sl))(3;1)^{+))$ The center of any semisimple Lie algebra is trivial^[91] 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 ${\text{SL))(2,\mathbb {C} )$ is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding ${\text{SL))(2,\mathbb {C} )$ 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.^{[nb 27]} 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.^[33] There is a topological proof of this.^[92] Let $u : G \to GL(V)$ , where $V$ is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group $G$ . Then $u (G) \subset U(V) \subset GL(V)$ 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,^[93] $u (G) ≅ G$ 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 $u (G) \subset U(V)$ since all embedded Lie subgroups of a Lie group are closed^[94] If $u (G)$ were closed, it would be compact,^{[nb 28]} and then $G$ would be compact,^{[nb 29]} contrary to assumption.^{[nb 30]}

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.^[95] 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.^[96]

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) $m + n, m + n - 1, ..., | m - n |$ ,^[97] where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) $j$ is $(2 j + 1)$ -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.^[98]

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 $2\cdot 3 / 2 + 1 = 4$ and $(2 + 1)(2\cdot 1 / 2 + 1) = 6$ 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.^[99]

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.^[100]

Dual representations

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.^[101]
Two irreducible representations are isomorphic if and only if they have the same highest weight.^{[nb 31]}
For each semisimple Lie algebra there exists a unique element $w 0$ of the Weyl group such that if $μ$ is a dominant integral weight, then $w 0 \cdot (- μ)$ is again a dominant integral weight.^[102]
If $\pi _{\mu _{0))$ is an irreducible representation with highest weight $μ 0$ , then ${\displaystyle \pi _{\mu _{0))^{*))$ has highest weight $w 0 \cdot (- μ)$ .^[102]

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 $w 0 = - I$ . In the case of ${\mathfrak {sl))(2,\mathbb {C} ),$ the Weyl group is $W = {I, - I}$ .^[103] It follows that each $π μ, μ = 0, 1, ...$ is isomorphic to its dual $\pi _{\mu }^{*}.$ The root system of ${\mathfrak {sl))(2,\mathbb {C} )\oplus {\mathfrak {sl))(2,\mathbb {C} )$ is shown in the figure to the right.^{[nb 32]} 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.^{[nb 33]} Inspection shows that $w α \cdot w β = - I$ so $- I \in W$ . Using the fact that if $π, σ$ are Lie algebra representations and $π ≅ σ$ , then $Π ≅ Σ$ ,^[104] the conclusion for $SO(3; 1) +$ 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 $π$ 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.^[105] In general, every irreducible representation $π$ of ${\mathfrak {sl))(n,\mathbb {C} )$ can be written uniquely as $π = π + + π -$ , where^[106]

\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

{\mathfrak {sl))(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 ))

\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 $SO(3; 1) +$ yields

{\begin{aligned}{\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{aligned))

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) \oplus (n, m)

have real matrices too.

The adjoint representation, the Clifford algebra, and the Dirac spinor representation

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

\pi (X)(A)=[\pi (X),A],\qquad A\in \operatorname {End} (V),\ X\in {\mathfrak {g)).

(I1)

Likewise, a representation $(Π, V)$ of a group $G$ yields a representation $Π$ on $End(V)$ of $G$ , still denoted $Π$ , given by^[107]

\Pi (g)(A)=\Pi (g)A\Pi (g)^{-1},\qquad A\in \operatorname {End} (V),\ g\in G.

(I2)

If $π$ and $Π$ are the standard representations on ${\displaystyle \mathbb {R} ^{4))$ and if the action is restricted to ${\mathfrak {so))(3,1)\subset {\text{End))(\mathbb {R} ^{4}),$ 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 $(1 / 2, 0) \oplus (0, 1 / 2)$ representation on $End(H)$ is the non-projective 4-vector (1/2, 1/2) representation.^[108]

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.^[109] (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, ${\mathcal {Cl))_{3,1}(\mathbb {R} ),$ whose complexification is ${\text{M))(4,\mathbb {C} ),$ 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, $(1, 0) \oplus (0, 1)$ .^[110] The dimensions add up to $1 + 1 + 4 + 4 + 6 = 16$ . In other words,

{\mathcal {Cl))_{3,1}(\mathbb {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},

(I3)

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

The $(1 / 2, 0) \oplus (0, 1 / 2)$ spin representation

The six-dimensional representation space of the tensor $(1, 0) \oplus (0, 1)$ -representation inside ${\mathcal {Cl))_{3,1}(\mathbb {R} )$ has two roles. The^[111]

\sigma ^{\mu \nu }=-{\frac {i}{4))\left[\gamma ^{\mu },\gamma ^{\nu }\right],

(I4)

where $\gamma ^{0},\ldots ,\gamma ^{3}\in {\mathcal {Cl))_{3,1}(\mathbb {R} )$ 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,^[112]

\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),

(I5)

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

The conclusion is that every element of the complexified ${\mathcal {Cl))_{3,1}(\mathbb {R} )$ 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 $\mathbb {C} ^{4},$ making it a space of bispinors.

Reducible representations

There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained 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. ${\text{GL))(n,\mathbb {R} )$ 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. The reducible representations will therefore not be discussed.

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 $(m, n) \oplus (n, m)$ can be extended likewise.^[113]

Space parity inversion

For space parity inversion, the adjoint action $Ad P$ of $P \in SO(3; 1)$ on ${\mathfrak {so))(3;1)$ is considered, where $P$ is the standard representative of space parity inversion, $P = diag(1, -1, -1, -1)$ , given by

\mathrm {Ad} _{P}(J_{i})=PJ_{i}P^{-1}=J_{i},\qquad \mathrm {Ad} _{P}(K_{i})=PK_{i}P^{-1}=-K_{i}.

(F1)

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

\Pi (P)\pi (B_{i})\Pi (P)^{-1}=\pi (A_{i})

(F2)

holds, where $A$ and $B$ are defined as in the first section. This can hold only if $A i$ and $B i$ have the same dimensions, i.e. only if $m = n$ . When $m \neq n$ then $(m, n) \oplus (n, m)$ 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 $β = i γ 0$ (or a multiple of modulus −1 times it) may be used in the $(1 / 2, 0) \oplus (0, 1 / 2)$ ^[114] representation.

If parity is included with a minus sign (the $1\times1$ 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 ${\mathfrak {so))(3;1)$ by^[115]

\mathrm {Ad} _{T}(J_{i})=TJ_{i}T^{-1}=-J_{i},\qquad \mathrm {Ad} _{T}(K_{i})=TK_{i}T^{-1}=K_{i}.

(F3)

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 $J i$ and $K i$ generate the group. These are interpreted as generators of translations. The time-component $P 0$ is the Hamiltonian $H$ . The operator $T$ satisfies the relation^[116]

\mathrm {Ad} _{T}(iH)=TiHT^{-1}=-iH

(F4)

in analogy to the relations above with ${\mathfrak {so))(3;1)$ replaced by the full Poincaré algebra. By just cancelling the $i$ 's, the result $THT -1 = - H$ would imply that for every state $Ψ$ with positive energy $E$ in a Hilbert space of quantum states with time-reversal invariance, there would be a state $Π(T -1)Ψ$ with negative energy $- E$ . Such states do not exist. The operator $Π(T)$ is therefore chosen antilinear and antiunitary, so that it anticommutes with $i$ , resulting in $THT -1 = H$ , and its action on Hilbert space likewise becomes antilinear and antiunitary.^[117] It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix.^[118] This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, $Π$ is not a representation.

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

The third discrete symmetry entering in the CPT theorem along with $P$ and $T$ , charge conjugation symmetry $C$ , has nothing directly to do with Lorentz invariance.^[119]

Action on function spaces

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

(\Pi (g)f)(x)=f\left(\Pi _{x}(g)^{-1}x\right),\qquad x\in \mathbb {R} ^{n},f\in V

(H1)

produces another function $Π f \in V$ . Here $Π x$ is an $n$ -dimensional representation, and $Π$ is a possibly infinite-dimensional representation. A special case of this construction is when $V$ is a space of functions defined on the a linear group $G$ itself, viewed as a $n$ -dimensional manifold embedded in $\mathbb {R} ^{m^{2))$ (with $m$ the dimension of the matrices).^[120] 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.^[61] The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. ${\text{SL))(2,\mathbb {C} ).$

The following exemplifies action of the Lorentz group and the rotation subgroup on some function spaces.

Euclidean rotations

Main articles: Rotation group SO(3) § Spherical harmonics, and Spherical harmonics

The subgroup $SO(3)$ of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space

L^{2}\left(\mathbb {S} ^{2}\right)=\operatorname {span} \left\{Y_{m}^{l},l\in \mathbb {N} ^{+},-l\leqslant m\leqslant l\right\},

where ${\displaystyle Y_{m}^{l))$ are the spherical harmonics. An arbitrary square integrable function $f$ on the unit sphere can be expressed as^[121]

f(\theta ,\varphi )=\sum _{l=1}^{\infty }\sum _{m=-l}^{l}f_{lm}Y_{m}^{l}(\theta ,\varphi ),

(H2)

where the $f lm$ are generalized Fourier coefficients.

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

{\begin{aligned}(\Pi (R)f)(\theta (x),\varphi (x))&=\sum _{l=1}^{\infty }\sum _{m,=-l}^{l}\sum _{m'=-l}^{l}D_{mm'}^{(l)}(R)f_{lm'}Y_{m}^{l}\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{aligned))

(H4)

where the $D l$ are obtained from the representatives of odd dimension of the generators of rotation.

The Möbius group

Main articles: Möbius transformation and Lorentz group § Relation to the Möbius group

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

In the plane, a Möbius transformation characterized by the complex numbers $a, b, c, d$ acts on the plane according to^[122]

f(z)={\frac {az+b}{cz+d)),\qquad ad-bc\neq 0

(M1)

and can be represented by complex matrices

\Pi _{f}={\begin{pmatrix}A&B\\C&D\end{pmatrix))=\lambda {\begin{pmatrix}a&b\\c&d\end{pmatrix)),\qquad \lambda \in \mathbb {C} -\{0\},\operatorname {det} \Pi _{f}=1,

(M2)

since multiplication by a nonzero complex scalar does not change $f$ . These are elements of ${\text{SL))(2,\mathbb {C} )$ and are unique up to a sign (since $\pmΠ f$ give the same $f$ ), hence ${\text{SL))(2,\mathbb {C} )/\{\pm I\}\cong {\text{SO))(3;1)^{+}.$

The Riemann P-functions

Main article: Riemann's differential equation

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^[123]

{\begin{aligned}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{aligned)),

(T1)

where the $a, b, c, α, β, γ, α', β', γ'$ are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is^[124]

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).

(T2)

The set of constants $0, \infty, 1$ in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation.^[125] Its exponents, i. e. solutions of the indicial equation, for expansion around the singular point $0$ are $0$ and $1 - c$ ,corresponding to the two linearly independent solutions,^{[nb 34]} and for expansion around the singular point $1$ they are $0$ and $c - a - b$ .^[126] Similarly, the exponents for $\infty$ are $a$ and $b$ for the two solutions.^[127]

One has thus

w(z)=\left({\frac {z-a}{z-b))\right)^{\alpha }\left({\frac {z-c}{z-b))\right)^{\gamma }{}_{2}F_{1}\left(\alpha +\beta +\gamma ,\,\alpha +\beta '+\gamma ;\,1+\alpha -\alpha ';\,{\frac {(z-a)(c-b)}{(z-b)(c-a)))\right),

(T3)

where the condition (sometimes called Riemann's identity)^[128]

\alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1

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

γ'

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

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

u(\Lambda )(z)={\frac {Az+B}{Cz+D)),

(T4)

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

\lambda ={\begin{pmatrix}A&B\\C&D\end{pmatrix))\in {\text{SL))(2,\mathbb {C} ),

(T5)

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

Define

[\Pi (\Lambda )P](z)=P[u(\Lambda )(z)],

(T6)

where $P$ 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

[\Pi (\Lambda )P](u)=P\left\((\begin{matrix}\eta &\zeta &\theta &\\\alpha &\beta &\gamma &\;u\\\alpha '&\beta '&\gamma '&\end{matrix))\right\},

(T6)

where

\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)).

(T7)

Infinite-dimensional unitary representations

History

The Lorentz group $SO(3; 1) +$ and its double cover ${\text{SL))(2,\mathbb {C} )$ also have infinite dimensional unitary representations, studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) at the instigation of Paul Dirac.^[129]^[130] This trail of development begun with Dirac (1936) where he devised matrices $U$ and $B$ necessary for description of higher spin (compare Dirac matrices), elaborated upon by Fierz (1939), see also Fierz & Pauli (1939), and proposed precursors of the Bargmann-Wigner equations.^[131] In Dirac (1945) he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors.^{[nb 35]} These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinite-dimensional generalization of spinors in his 1947 paper.

The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by Harish-Chandra (1951) and Gelfand & Graev (1953), based on an analogue for ${\text{SL))(2,\mathbb {C} )$ of the integration formula of Hermann Weyl for compact Lie groups.^[132] Elementary accounts of this approach can be found in Rühl (1970) and Knapp (2001).

The theory of spherical functions for the Lorentz group, required for harmonic analysis on the 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 $\mathbb {R} .$ This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008).

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 $B$ of $G={\text{SL))(2,\mathbb {C} ).$ Since the one-dimensional representations of $B$ correspond to the representations of the diagonal matrices, with non-zero complex entries $z$ and $z -1$ , they thus have the form

\chi _{\nu ,k}{\begin{pmatrix}z&0\\c&z^{-1}\end{pmatrix))=r^{i\nu }e^{ik\theta },

for

k

an integer,

ν

real and with

z = re iθ

. The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when

k

is replaced by

- k

. By definition the representations are realized on

L 2

sections of line bundles on

G/B=\mathbb {S} ^{2},

which is isomorphic to the Riemann sphere. When

k = 0

, these representations constitute the so-called spherical principal series.

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

Using the identification between the Riemann sphere minus a point and $\mathbb {C} ,$ the principal series can be defined directly on $L^{2}(\mathbb {C} )$ by the formula^[133]

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

Irreducibility can be checked in a variety of ways:

The representation is already irreducible on $B$ . This can be seen directly, but is also a special case of general results on irreducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition $G = B \cup BsB$ where $s$ is the Weyl group element^[134] ${\begin{pmatrix}0&-1\\1&0\end{pmatrix))$ .
The action of the Lie algebra ${\mathfrak {g))$ of $G$ can be computed on the algebraic direct sum of the irreducible subspaces of $K$ can be computed explicitly and the it can be verified directly that the lowest-dimensional subspace generates this direct sum as a ${\mathfrak {g))$ -module.^[8]^[135]

Complementary series for $SL(2, C)$

The for $0 < t < 2$ , the complementary series is defined on $L^{2}(\mathbb {C} )$ for the inner product^[136]

(f,g)_{t}=\iint {\frac {f(z){\overline {g(w)))}{|z-w|^{2-t))}\,dz\,dw,

with the action given by^[137]^[138]

\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 $K$ , each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of $K = SU(2)$ . Irreducibility can be proved by analyzing the action of ${\mathfrak {g))$ on the algebraic sum of these subspaces^[8]^[135] or directly without using the Lie algebra.^[139]^[140]

Plancherel theorem for SL(2, C)

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

To decompose the left regular representation of $G$ on $L^{2}(G)$ 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 $k = 0$ .)

The left and right regular representation $λ$ and $ρ$ are defined on $L^{2}(G)$ by

{\begin{aligned}(\lambda (g)f)(x)&=f\left(g^{-1}x\right)\\(\rho (g)f)(x)&=f(xg)\end{aligned))

Now if $f$ is an element of $C c (G)$ , 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

H

H=\bigoplus _{k\geqslant 0}{\text{HS))\left(L^{2}(\mathbb {C} )\right)\otimes L^{2}\left(\mathbb {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}(\mathbb {C} )\right)

denotes the Hilbert space of Hilbert–Schmidt operators on

L^{2}(\mathbb {C} ).

^{[nb 36]} Then the map

U

defined on

C c (G)

U(f)(\nu ,k)=\pi _{\nu ,k}(f)

extends to a unitary of

L^{2}(G)

onto

H

The map $U$ 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 $f 1, f 2$ are in $C c (G)$ 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 ${\displaystyle f=f_{1}*f_{2}^{*))$ denotes the convolution of ${\displaystyle f_{1))$ and $f_{2}^{*},$ and $f_{2}^{*}(g)={\overline {f_{2}(g^{-1}))),$ then^[141]

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 $f$ . It can be extended to much wider classes of functions satisfying mild differentiability conditions.^[61]

Classification of representations of $SO(3, 1)$

The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to assume they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinite-dimensional representation $Π H$ on a Hilbert space $H$ of $SO(3; 1) +$ is at hand.^[142] Since $SO(3)$ is a subgroup, $Π H$ is a representation of it as well. Each irreducible subrepresentation of $SO(3)$ is finite-dimensional, and the $SO(3)$ representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of $SO(3)$ if $Π H$ is unitary.^[143]

The steps are the following:^[144]

Choose a suitable basis of common eigenvectors of $J 2$ and $J 3$ .
Compute matrix elements of $J 1, J 2, J 3$ and $K 1, K 2, K 3$ .
Enforce Lie algebra commutation relations.
Require unitarity together with orthonormality of the basis.^{[nb 37]}

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 $j 0$ would correspond the lowest occurring eigenvalue $j (j + 1)$ of $J 2$ in the representation, equal to $| m - n |$ , and $j 1$ would correspond to the highest occurring eigenvalue, equal to $m + n$ . In the infinite-dimensional case, $j 0 \geq 0$ retains this meaning, but $j 1$ does not.^[66] For simplicity, it is assumed that a given $j$ occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown^[145] 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 $J 1, J 2, J 3$ and $K 1, K 2, K 3$ forming the basis of the Lie algebra of ${\mathfrak {so))(3;1).$ The matrix elements of ${\displaystyle J_{\pm }=J_{1}\pm iJ_{2))$ and ${\displaystyle J_{3))$ (the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by^[146]^[147]

{\begin{aligned}\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{aligned))

where the labels

j 0

and

j 1

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

(K 1, K 2, K 3) \equiv K

is a vector operator^[148] and the Wigner–Eckart theorem^[149] applies for computation of matrix elements between the states represented by the chosen basis.^[150] The matrix elements of

{\begin{aligned}K_{0}^{(1)}&=K_{3},\\K_{\pm 1}^{(1)}&=\mp {\frac {1}{\sqrt {2))}(K_{1}\pm iK_{2}),\end{aligned))

where the superscript $(1)$ signifies that the defined quantities are the components of a spherical tensor operator of rank $k = 1$ (which explains the factor $\sqrt 2$ as well) and the subscripts $0, \pm1$ are referred to as $q$ in formulas below, are given by^[151]

{\begin{aligned}\left\langle j'm'\left|K_{0}^{(1)}\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_{\pm 1}^{(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{aligned))

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

Step 3

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

[K_{\pm },K_{3}]=\pm J_{\pm },\quad [K_{+},K_{-}]=-2J_{3}.

This results in a set of equations^[152] for which the solutions are^[153]

{\begin{aligned}\left\langle j\left\|K^{(1)}\right\|j\right\rangle &=i{\frac {j_{1}j_{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{aligned))

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 \mathbb {C} .

Step 4

The imposition of the requirement of unitarity of the corresponding representation of the group restricts the possible values for the arbitrary complex numbers $j 0$ and $ξ j$ . 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^[154]

{\begin{aligned}\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{aligned))

leading to^[155]

{\begin{aligned}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{aligned))

where

β j

is the angle of

B j

on polar form. For

| B j | \neq 0

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 $j 1 = - iν$ , $ν$ real,^[156] ${\displaystyle \left\langle j\left\|K^{(1)}\right\|j\right\rangle ={\frac {\nu j_{0)){j(j+1)))\quad {\text{and))\quad B_{j}={\sqrt {\frac {(j^{2}-j_{0}^{2})(j^{2}+\nu ^{2})}{4j^{2}-1))))$ This is the principal series. Its elements are denoted $(j_{0},\nu ),2j_{0}\in \mathbb {N} ,\nu \in \mathbb {R} .$
${\underline {j_{0}=0.))$ It follows:^[157] ${\displaystyle \left\langle j\left\|K^{(1)}\right\|j\right\rangle =0\quad {\text{and))\quad B_{j}={\sqrt {\frac {j^{2}-\nu ^{2)){4j^{2}-1))))$ Since $B 0 = B j 0$ , $B 2 j$ is real and positive for $j = 1, 2, ...$ , leading to $-1 \leq ν \leq 1$ . This is complementary series. Its elements are denoted $(0, ν), -1 \leq ν \leq 1$

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

{\begin{aligned}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{aligned))

The commutation relations of the Lie algebra ${\mathfrak {so))(3;1)$ are:^[158]

\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^[159]

\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 $J$ 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 and bispinors

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

{\displaystyle J_{i}^{\left({\frac {1}{2))\right)}={\frac {1}{2))\sigma _{i))

in the general expression (G1), and by using the trivial relations

11 = 1

and

J (0) = 0

, it follows

{\begin{aligned}\pi _{\left({\frac {1}{2)),0\right)}(J_{i})&={\frac {1}{2))\left(\sigma _{i}\otimes 1_{(1)}+1_{(2)}\otimes J_{i}^{(0)}\right)={\frac {1}{2))\sigma _{i}\\\pi _{\left({\frac {1}{2)),0\right)}(K_{i})&={\frac {-i}{2))\left(1_{(2)}\otimes J_{i}^{(0)}-\sigma _{i}\otimes 1_{(1)}\right)={\frac {i}{2))\sigma _{i}\\[6pt]\pi _{\left(0,{\frac {1}{2))\right)}(J_{i})&={\frac {1}{2))\left(J_{i}^{(0)}\otimes 1_{(2)}+1_{(1)}\otimes \sigma _{i}\right)={\frac {1}{2))\sigma _{i}\\\pi _{\left(0,{\frac {1}{2))\right)}(K_{i})&={\frac {-i}{2))\left(1_{(1)}\otimes \sigma _{i}-J_{i}^{(0)}\otimes 1_{(2)}\right)={\frac {-i}{2))\sigma _{i}\end{aligned))

(W1)

These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) $V L$ and $V R$ , whose elements $Ψ L$ and $Ψ R$ are called left- and right-handed Weyl spinors respectively. Given

\left(\pi _{\left({\frac {1}{2)),0\right)},V_{\text{L))\right)\quad {\text{and))\quad \left(\pi _{\left(0,{\frac {1}{2))\right)},V_{\text{R))\right)

their direct sum as representations is formed,^[160]

{\begin{aligned}\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{aligned))

(D1)

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

Open problems

The classification and characterization of the representation theory of the Lorentz group was completed in 1947. But in association with the Bargmann–Wigner programme, there are yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations.

The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the little group of the 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.^[161]^[162] 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.^[163]

One open problem is the completion of the Bargmann–Wigner programme for the isometry group $SO(D - 2, 1)$ of the de Sitter spacetime $dS D -2$ . Ideally, the physical components of wave functions would be realized on the hyperboloid $dS D -2$ of radius $μ > 0$ embedded in ${\displaystyle \mathbb {R} ^{D-2,1))$ and the corresponding $O(D -2, 1)$ covariant wave equations of the infinite-dimensional unitary representation to be known.^[162]

Remarks

^ The way in which one represents the spacetime symmetries may take many shapes depending on the theory at hand. While not being the present topic, some details will be provided in footnotes labeled "nb", and in the section applications.
^ Weinberg 2002, p. 1 "If it turned out that a system could not be described by a quantum field theory, it would be a sensation; if it turned out it did not obey the rules of quantum mechanics and relativity, it would be a cataclysm."
^ In 1945 Harish-Chandra came to see Dirac in Cambridge. Harish-Chandra became convinced that theoretical physics was not the field he should be in. He had found an error in a proof by Dirac in his work on the Lorentz group. Dirac said "I am not interested in proofs but only interested in what nature does." Harish-Chandra later wrote "This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move over to mathematics." Dirac did however suggest the topic of Harish-Chandra's thesis, the classification of the irreducible infinite-dimensional representations of the Lorentz group. See Dalitz & Peierls 1986
^ See formula (1) in S-matrix#From free particle states for how free multi-particle states transform.
^ Weinberg 2002, Equations 5.1.4–5. Weinberg deduces the necessity of creation and annihilation operators from another consideration, the cluster decomposition principle, Weinberg (2002, Chapter 4.)
^ A prescription for how the particle should behave under CPT symmetry may be required as well.
^ For instance, there are versions (free field equations, i.e. without interaction terms) of the Klein–Gordon equation, the Dirac equation, the Maxwell equations, the Proca equation, the Rarita–Schwinger equation, and the Einstein field equations that can systematically be deduced by starting from a given representation of the Lorentz group. In general, these are collectively the quantum field theory versions of the Bargmann–Wigner equations.
See Weinberg (2002, Chapter 5), Tung (1985, Section 10.5.2) and references given in these works.
It should be remarked that high spin theories ( $s > 1$ ) encounter difficulties. See Weinberg (2002, Section 5.8), on general $(m, n)$ fields, where this is discussed in some depth, and references therein. High spin particles do without a doubt exist, e.g. nuclei, the known ones are just not elementary.
^ For part of their representation theory, see Bekaert & Boulanger (2006), which is dedicated to representation theory of the Poincare group. These representations are obtained by the method of induced representations or, in physics parlance, the method of the little group, pioneered by Wigner in 1939 for this type of group and put on firm mathematical footing by George Mackey in the fifties.
^ Hall (2015, Section 4.4.)

One says that a group has the complete reducibility property if every representation decomposes as a direct sum of irreducible representations.
^ Dirac suggested the topic of Wigner (1939) as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in Dirac (1945) (Langlands 1985), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations (Dalitz & Peierls 1986).
^ Knapp 2001 The rather mysterious looking third isomorphism is proved in chapter 2, paragraph 4.
^ Tensor products of representations, $π g \otimes π h$ of ${\mathfrak {g))\oplus {\mathfrak {h))$ can, when both factors come from the same Lie algebra ${\mathfrak {h))={\mathfrak {g)),$ either be thought of as a representation of ${\mathfrak {g))$ or ${\mathfrak {g))\oplus {\mathfrak {g))$ .
^ When complexifying a complex Lie algebra, it should be thought of as a real Lie algebra of real dimension twice its complex dimension. Likewise, a real form may actually also be complex as is the case here.
^ Combine Weinberg (2002, Equations 5.6.7–8, 5.6.14–15) with Hall (2015, Proposition 4.18) about Lie algebra representations of group tensor product representations.
^ The "traceless" property can be expressed as $S αβ g αβ = 0$ , or $S α α = 0$ , or $S αβ g αβ = 0$ depending on the presentation of the field: covariant, mixed, and contravariant respectively.
^ This doesn't necessarily come symmetric directly from the Lagrangian by using Noether's theorem, but it can be symmetrized as the Belinfante–Rosenfeld stress–energy tensor.
^ This is provided parity is a symmetry. Else there would be two flavors, $(3 / 2, 0)$ and $(0, 3 / 2)$ in analogy with neutrinos.
^ The terminology differs between mathematics and physics. In the linked article term projective representation has a slightly different meaning than in physics, where a projective representation is thought of as a local section (a local inverse) of the covering map from the covering group onto the group being covered, composed with a proper representation of the covering group. Since this can be done (locally) continuously in two ways in the case at hand as explained below, the terminology of a double-valued or two-valued representation is natural.
^ In particular, $A$ commutes with the Pauli matrices, hence with all of $SU(2)$ making Schur's lemma applicable.
^ Meaning the kernel is trivial, to see this recall that the kernel of a Lie algebra homomorphism is an ideal and hence a subspace. Since $p$ is $2:1$ and both ${\text{SL))(2,\mathbb {C} )$ and $SO(3; 1) +$ are $6$ -dimensional, the kernel must be $0$ -dimensional, hence ${0}.$
^ The exponential map is one-to-one in a neighborhood of the identity in ${\text{SL))(2,\mathbb {C} ),$ hence the composition $\exp \circ \sigma \circ \log :{\text{SL))(2,\mathbb {C} )\to {\text{SO))(3;1)^{+},$ where $σ$ is the Lie algebra isomorphism, is onto an open neighborhood $U \subset SO(3; 1) +$ containing the identity. Such a neighborhood generates the connected component.
^ Rossmann 2002 From Example 4 in section 2.1 : This can be seen as follows. The matrix $q$ has eigenvalues ${-1, -1}$ , but it is not diagonalizable. If $q = exp(Q)$ , then $Q$ has eigenvalues $λ, - λ$ with $λ = iπ + 2 πik$ for some $k$ because elements of ${\mathfrak {sl))(2,\mathbb {C} )$ are traceless. But then $Q$ is diagonalizable, hence $q$ is diagonalizable, which is a contradiction.
^ Rossmann 2002, Proposition 10, paragraph 6.3. This is easiest proved using character theory.
^ Any discrete normal subgroup of a path connected group $G$ is contained in the center $Z$ of $G$ .

Hall 2015, Exercise 11, chapter 1.
^ A semisimple Lie group does not have any non-discrete normal abelian subgroups. This can be taken as the definition of semisimplicity.
^ A simple group does not have any non-discrete normal subgroups.
^ By contrast, there is a trick, also called Weyl's unitarian trick, but unrelated to the unitarian trick of above showing that all finite-dimensional representations are, or can be made, unitary. If $(Π, V)$ is a finite-dimensional representation of a compact Lie group $G$ and if $(\cdot, \cdot)$ is any inner product on $V$ , define a new inner product $(\cdot, \cdot) Π$ by $(x, y) Π = \int G (Π(g) x, Π(g) y dμ (g)$ , where $μ$ is Haar measure on $G$ . Then $Π$ is unitary with respect to $(\cdot, \cdot) Π$ . See Hall (2015, Theorem 4.28.)

Another consequence is that every compact Lie group has the complete reducibility property, meaning that all its finite-dimensional representations decompose as a direct sum of irreducible representations. Hall (2015, Definition 4.24., Theorem 4.28.)

It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in Greiner & Müller (1994, Section 15.2.).
^ Lee 2003 Lemma A.17 (c). Closed subsets of compact sets are compact.
^ Lee 2003 Lemma A.17 (a). If $f : X \to Y$ is continuous, $X$ is compact, then $f (X)$ is compact.
^ The non-unitarity is a vital ingredient in the proof of the Coleman–Mandula theorem, which has the implication that, contrary to in non-relativistic theories, there can exist no ordinary symmetry relating particles of different spin. See Weinberg (2000)
^ This is one of the conclusions of Cartan's theorem, the theorem of the highest weight.
Hall (2015, Theorems 9.4–5.)
^ Hall 2015, Section 8.2 The root system is the union of two copies of $A 1$ , where each copy resides in its own dimensions in the embedding vector space.
^ Rossmann 2002 This definition is equivalent to the definition in terms of the connected Lie group whose Lie algebra is the Lie algebra of the root system under consideration.
^ See Simmons (1972, Section 30.) for precise conditions under which two Frobenius method yields two linearly independent solutions. If the exponents do not differ by an integer, this is always the case.
^ "This is as close as one comes to the source of the theory of infinite-dimensional representations of semisimple and reductive groups...", Langlands (1985, p. 204.), referring to an introductory passage in Dirac's 1945 paper.
^ Note that for a Hilbert space $H$ , $HS(H)$ may be identified canonically with the Hilbert space tensor product of $H$ and its conjugate space.
^ If finite-dimensionality is demanded, the results is the $(m, n)$ representations, see Tung (1985, Problem 10.8.) If neither is demanded, then a broader classification of all irreducible representations is obtained, including the finite-dimensional and the unitary ones. This approach is taken in Harish-Chandra (1947).

Notes

Freely available online references

References

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Relativity

Special
relativity

Background	Principle of relativity (Galilean relativity Galilean transformation) Special relativity Doubly special relativity
Fundamental concepts	Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations Proper length Proper time Proper acceleration Relativistic mass
Formulation	Lorentz transformation
Phenomena	Time dilation Mass–energy equivalence (E=mc²) Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Ladder paradox Twin paradox Terrell rotation
Spacetime	Light cone World line Minkowski diagram Biquaternions Minkowski space

General
relativity

Background	Introduction Mathematical formulation
Fundamental concepts	Equivalence principle Riemannian geometry Penrose diagram Geodesics Mach's principle
Formulation	ADM formalism BSSN formalism Einstein field equations Linearized gravity Post-Newtonian formalism Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst equation
Phenomena	Black hole Event horizon Singularity Two-body problem Gravitational waves: astronomy detectors (LIGO and collaboration Virgo LISA Pathfinder GEO) Hulse–Taylor binary Other tests: precession of Mercury lensing (together with Einstein cross and Einstein rings) redshift Shapiro delay frame-dragging / geodetic effect (Lense–Thirring precession) pulsar timing arrays
Advanced theories	Brans–Dicke theory Kaluza–Klein Quantum gravity
Solutions	Cosmological: Friedmann–Lemaître–Robertson–Walker (Friedmann equations) Lemaître–Tolman Kasner BKL singularity Gödel Milne Spherical: Schwarzschild (interior Tolman–Oppenheimer–Volkoff equation) Reissner–Nordström Axisymmetric: Kerr (Kerr–Newman) Weyl−Lewis−Papapetrou Taub–NUT van Stockum dust discs Others: pp-wave Ozsváth–Schücking Alcubierre In computational physics: Numerical relativity

Scientists

Development

Applications

Classical field theory

Relativistic quantum mechanics

Quantum field theory

Speculative theories

Finite-dimensional representations

History

The Lie algebra

The unitarian trick

The (μ, ν)-representations of sl(2, C)

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

Common representations

Off-diagonal direct sums

The group

Surjectiveness of exponential map for SO(3, 1)

Fundamental group

Projective representations

The covering group SL(2, C)

A geometric view[edit]

An algebraic view[edit]

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

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

Complex linear representations

Real linear representations

Properties of the (m, n) representations

Dimension

Faithfulness

Non-unitarity

Restriction to SO(3)

Spinors

Dual representations

Complex conjugate representations

The adjoint representation, the Clifford algebra, and the Dirac spinor representation

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

Reducible representations

Space inversion and time reversal

Space parity inversion

Time reversal

Action on function spaces

Euclidean rotations

The Möbius group

The Riemann P-functions

Infinite-dimensional unitary representations

History

Principal series for SL(2, C)

Complementary series for SL(2, C)

Plancherel theorem for SL(2, C)

Classification of representations of SO(3, 1)

Step 1

Step 2

Step 3

Step 4

Explicit formulas

Conventions and Lie algebra bases

Weyl spinors and bispinors

Open problems

See also

Remarks

Notes

Freely available online references

References

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

The $(1 / 2, 0) \oplus (0, 1 / 2)$ spin representation

Complementary series for $SL(2, C)$

Classification of representations of $SO(3, 1)$