Representation theory of the Lorentz group#A geometric view

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

The development of the representation theory has historically followed the development of the more general theory of representation theory of semisimple groups, largely due to Élie Cartan and Hermann Weyl, but the Lorentz group has also received special attention due to its importance in physics. Notable contributors are physicist E. P. Wigner and mathematician Valentine Bargmann with their Bargmann–Wigner program,{{harvnb|Bargmann|Wigner|1948}} one conclusion of which is, roughly, a classification of all unitary representations of the inhomogeneous Lorentz group amounts to a classification of all possible relativistic wave equations.{{harvnb|Bekaert|Boulanger|2006}} The classification of the irreducible infinite-dimensional representations of the Lorentz group was established by Paul Dirac's doctoral student in theoretical physics, Harish-Chandra, later turned mathematician,In 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 {{harvnb|Dalitz|Peierls|1986}} in 1947. The corresponding classification for $\backslash mathrm\{SL\}(2,\; \backslash Complex)$ was published independently by Bargmann and Israel Gelfand together with Mark Naimark in the same year.

Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string theory and beyond. The representation theory also provides the theoretical ground for the concept of spin. The theory enters into general relativity in the sense that in small enough regions of spacetime, physics is that of special relativity.{{harvnb|Misner|Thorne|Wheeler|1973}}

The finite-dimensional irreducible non-unitary representations together with the irreducible infinite-dimensional unitary representations of the inhomogeneous Lorentz group, the Poincare group, are the representations that have direct physical relevance.{{harvnb|Weinberg|2002|loc=Section 2.5, Chapter 5.}}{{harvnb|Tung|1985|loc=Sections 10.3, 10.5.}}

Infinite-dimensional unitary representations of the Lorentz group appear by restriction of the irreducible infinite-dimensional unitary representations of the Poincaré group acting on the Hilbert spaces of relativistic quantum mechanics and quantum field theory. But these are also of mathematical interest and of potential direct physical relevance in other roles than that of a mere restriction.{{harvnb|Tung|1985|loc=Section 10.4.}} There were speculative theories,{{harvnb|Dirac|1945}} (tensors and spinors have infinite counterparts in the expansors of Dirac and the expinors of Harish-Chandra) consistent with relativity and quantum mechanics, but they have found no proven physical application. Modern speculative theories potentially have similar ingredients per below.

While the electromagnetic field together with the gravitational field are the only classical fields providing accurate descriptions of nature, other types of classical fields are important too. In the approach to quantum field theory (QFT) referred to as second quantization, the starting point is one or more classical fields, where e.g. the wave functions solving the Dirac equation are considered as classical fields prior to (second) quantization.{{harvnb|Greiner|Reinhardt|1996|loc=Chapter 2.}} While second quantization and the Lagrangian formalism associated with it is not a fundamental aspect of QFT,{{harvnb|Weinberg|2002|loc=Foreword and introduction to chapter 7.}} it is the case that so far all quantum field theories can be approached this way, including the standard model.{{harvnb|Weinberg|2002|loc=Introduction to chapter 7.}} In these cases, there are classical versions of the field equations following from the Euler–Lagrange equations derived from the Lagrangian using the principle of least action. These field equations must be relativistically invariant, and their solutions (which will qualify as relativistic wave functions according to the definition below) must transform under some representation of the Lorentz group.

The action of the Lorentz group on the space of field configurations (a field configuration is the spacetime history of a particular solution, e.g. the electromagnetic field in all of space over all time is one field configuration) resembles the action on the Hilbert spaces of quantum mechanics, except that the commutator brackets are replaced by field theoretical Poisson brackets.

For the present purposes the following definition is made:{{harvnb|Tung|1985|loc=Definition 10.11.}} A relativistic wave function is a set of {{mvar|n}} functions {{math|ψ^α}} on spacetime which transforms under an arbitrary proper Lorentz transformation {{math|Λ}} as

$$\backslash psi\text{'}^\backslash alpha(x)\; =\; D\{[\backslash Lambda]^\backslash alpha\}\_\backslash beta\; \backslash psi^\backslash beta\; \backslash left(\backslash Lambda^\{-1\}\; x\backslash right),$$

where {{math|D[Λ]}} is an {{math|n}}-dimensional matrix representative of {{math|Λ}} belonging to some direct sum of the {{math|(m, n)}} representations to be introduced below.

The most useful relativistic quantum mechanics one-particle theories (there are no fully consistent such theories) are the Klein–Gordon equation{{harvtxt|Greiner|Müller|1994|loc=Chapter 1}} and the Dirac equation{{harvtxt|Greiner|Müller|1994|loc=Chapter 2}} in their original setting. They are relativistically invariant and their solutions transform under the Lorentz group as Lorentz scalars ({{math|1=(m, n) = (0, 0)}}) and bispinors ({{math|(0, {{sfrac|1|2}}) ⊕ ({{sfrac|1|2}}, 0)}}) respectively. The electromagnetic field is a relativistic wave function according to this definition, transforming under {{math|(1, 0) ⊕ (0, 1)}}.{{harvnb|Tung|1985|p=203.}}

The infinite-dimensional representations may be used in the analysis of scattering.{{harvnb|Delbourgo|Salam|Strathdee|1967}}

In quantum field theory, the demand for relativistic invariance enters, among other ways in that the S-matrix necessarily must be Poincaré invariant.{{harvtxt|Weinberg|2002|loc=Section 3.3}} This has the implication that there is one or more infinite-dimensional representation of the Lorentz group acting on Fock space.See formula (1) in S-matrix#From free particle states for how free multi-particle states transform. One way to guarantee the existence of such representations is the existence of a Lagrangian description (with modest requirements imposed, see the reference) of the system using the canonical formalism, from which a realization of the generators of the Lorentz group may be deduced.{{harvtxt|Weinberg|2002|loc=Section 7.4.}}

The transformations of field operators illustrate the complementary role played by the finite-dimensional representations of the Lorentz group and the infinite-dimensional unitary representations of the Poincare group, witnessing the deep unity between mathematics and physics.{{harvnb|Tung |1985| loc=Introduction to chapter 10.}} For illustration, consider the definition an {{mvar|n}}-component field operator:{{harvnb|Tung|1985|loc=Definition 10.12.}} A relativistic field operator is a set of {{mvar|n}} operator valued functions on spacetime which transforms under proper Poincaré transformations {{math|(Λ, a)}} according to{{harvnb|Tung|1985|loc=Equation 10.5-2.}}{{harvnb|Weinberg|2002|loc=Equations 5.1.6–7.}}

\Psi^\alpha(x) \to \Psi'^\alpha(x) =

U[\Lambda, a]\Psi^\alpha(x) U^{-1} \left[\Lambda, a\right] =

D{\left[\Lambda^{-1}\right]^\alpha}_\beta \Psi^\beta (\Lambda x + a)

Here {{math|U[Λ, a]}} is the unitary operator representing {{math|(Λ, a)}} on the Hilbert space on which {{math|Ψ}} is defined and {{mvar|D}} is an {{mvar|n}}-dimensional representation of the Lorentz group. The transformation rule is the second Wightman axiom of quantum field theory.

By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass {{mvar|m}} and spin {{mvar|s}} (or helicity), it is deduced that{{harvnb|Tung|1985|loc=Equation 10.5–18.}}{{harvnb|Weinberg|2002|loc=Equations 5.1.4–5.}} Weinberg deduces the necessity of creation and annihilation operators from another consideration, the cluster decomposition principle, {{harvtxt|Weinberg|2002|loc=Chapter 4.}}

{{NumBlk||$$\backslash Psi^\backslash alpha(x)\; =\; \backslash sum\_\backslash sigma\; \backslash int\; dp\; \backslash left(a(\backslash mathbf\{p\},\; \backslash sigma)\; u^\backslash alpha(\backslash mathbf\{p\},\; \backslash sigma)\; e^\{ip\; \backslash cdot\; x\}\; +\; a^\backslash dagger(\backslash mathbf\{p\},\; \backslash sigma)\; v^\backslash alpha(\backslash mathbf\{p\},\; \backslash sigma)\; e^\{-ip\; \backslash cdot\; x\}\; \backslash right),$$|{{EquationRef|X1|X1}}}}

where {{math|a^†, a}} are interpreted as creation and annihilation operators respectively. The creation operator {{math|a^†}} transforms according to{{harvnb|Weinberg|2002|loc=Equations 5.1.11–12.}}

a^\dagger(\mathbf{p}, \sigma) \rightarrow

a'^\dagger \left(\mathbf{p}, \sigma\right) =

U[\Lambda]a^\dagger(\mathbf{p}, \sigma) U \left[\Lambda^{-1}\right] =

a^\dagger(\Lambda \mathbf{p}, \rho) D^{(s)}{\left[R(\Lambda, \mathbf{p})^{-1}\right]^\rho}_\sigma,

and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin {{math|(m, s)}} of the particle. The connection between the two are the wave functions, also called coefficient functions

$$u^\backslash alpha(\backslash mathbf\{p\},\; \backslash sigma)\; e^\{ip\; \backslash cdot\; x\},\backslash quad\; v^\backslash alpha(\backslash mathbf\{p\},\; \backslash sigma)\; e^\{-ip\; \backslash cdot\; x\}$$

that carry both the indices {{math|(x, α)}} operated on by Lorentz transformations and the indices {{math|(p, σ)}} operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection.{{harvnb|Tung|1985|loc=Section 10.5.3.}} To exhibit the connection, subject both sides of equation {{EquationNote|X1|(X1)}} to a Lorentz transformation resulting in for e.g. {{mvar|u}},

{D[\Lambda]^\alpha}_{\alpha'} u^{\alpha'}(\mathbf{p}, \lambda) =

{D^{(s)}[R(\Lambda, \mathbf{p})]^{\lambda'}}_\lambda u^\alpha \left(\Lambda \mathbf{p}, \lambda'\right),

where {{mvar|D}} is the non-unitary Lorentz group representative of {{math|Λ}} and {{math|D^(s)}} is a unitary representative of the so-called Wigner rotation {{mvar|R}} associated to {{math|Λ}} and {{math|p}} that derives from the representation of the Poincaré group, and {{mvar|s}} is the spin of the particle.

All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the {{math|(m, n)}} representation under which it is supposed to transform,A prescription for how the particle should behave under CPT symmetry may be required as well. and also that of the wave function, can be derived from group theoretical considerations alone once the frameworks of quantum mechanics and special relativity is given.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 {{harvtxt|Weinberg|2002|loc=Chapter 5}}, {{harvtxt|Tung|1985|loc=Section 10.5.2}} and references given in these works.

It should be remarked that high spin theories ({{math|s > 1}}) encounter difficulties. See {{harvtxt|Weinberg|2002|loc=Section 5.8}}, on general {{math|(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.

In theories in which spacetime can have more than {{math|1=D = 4}} dimensions, the generalized Lorentz groups {{math|O(D − 1; 1)}} of the appropriate dimension take the place of {{math|O(3; 1)}}.For part of their representation theory, see {{harvtxt|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.

The requirement of Lorentz invariance takes on perhaps its most dramatic effect in string theory. Classical relativistic strings can be handled in the Lagrangian framework by using the Nambu–Goto action.{{harvnb|Zwiebach|2004|loc=Section 6.4.}} This results in a relativistically invariant theory in any spacetime dimension.{{harvnb|Zwiebach|2004|loc=Chapter 7.}} But as it turns out, the theory of open and closed bosonic strings (the simplest string theory) is impossible to quantize in such a way that the Lorentz group is represented on the space of states (a Hilbert space) unless the dimension of spacetime is 26.{{harvnb|Zwiebach|2004|loc=Section 12.5.}} The corresponding result for superstring theory is again deduced demanding Lorentz invariance, but now with supersymmetry. In these theories the Poincaré algebra is replaced by a supersymmetry algebra which is a graded Lie algebra extending the Poincaré algebra. The structure of such an algebra is to a large degree fixed by the demands of Lorentz invariance. In particular, the fermionic operators (grade {{math|1}}) belong to a {{math|(0, {{sfrac|1|2}})}} or {{math|({{sfrac|1|2}}, 0)}} representation space of the (ordinary) Lorentz Lie algebra.{{harvnb|Weinberg|2000|loc=Section 25.2.}} The only possible dimension of spacetime in such theories is 10.{{harvnb|Zwiebach|2004|loc=Last paragraph, section 12.6.}}

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.These facts can be found in most introductory mathematics and physics texts. See e.g. {{harvtxt|Rossmann|2002}}, {{harvtxt|Hall|2015}} and {{harvtxt|Tung|1985}}.

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.{{harvtxt|Hall|2015|loc=Section 4.4.}}

{{Paragraph break}}

One says that a group has the complete reducibility property if every representation decomposes as a direct sum of irreducible representations.{{harvtxt|Hall|2015|loc=Theorem 4.34 and following discussion.}} 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.{{harvnb|Wigner|1939}} Lack of simple connectedness gives rise to spin representations of the group.{{harvnb|Hall|2015|loc=Appendix D2.}} 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.{{harvnb|Greiner|Reinhardt|1996}}{{harvnb|Weinberg|2002|loc=Section 2.6 and Chapter 5.}}

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.{{harvnb|Coleman|1989|p=30.}}{{harvnb|Lie|1888}}, 1890, 1893. Primary source. By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing.{{harvnb|Coleman|1989|p=34.}}{{harvnb|Killing|1888}} Primary source. 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.{{harvnb|Rossmann|2002|loc=Historical tidbits scattered across the text.}}{{harvnb|Cartan|1913}} Primary source. 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.{{harvnb|Green|1998|loc=p=76.}}{{harvnb|Brauer|Weyl|1935}} Primary source. 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{{harvnb|Tung|1985|loc=Introduction.}}{{harvnb|Weyl|1931}} Primary source.{{harvnb|Weyl|1939}} Primary source. and Harish-Chandra{{harvnb|Langlands|1985|pp=203–205}}{{harvnb|Harish-Chandra|1947}} Primary source. and physicists Eugene Wigner{{harvnb|Tung|1985|loc=Introduction}}{{harvnb|Wigner|1939}} Primary source. and Valentine Bargmann{{harvnb|Klauder|1999}}{{harvnb|Bargmann|1947}} Primary source.Bargmann was also a mathematician. He worked as Albert Einsteins assistant at the Institute for Advanced Study in Princeton ({{harvtxt|Klauder|1999}}). made substantial contributions both to general representation theory and in particular to the Lorentz group.{{harvnb|Bargmann|Wigner|1948}} Primary source. 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.{{harvnb|Dalitz|Peierls|1986}}{{harvnb|Dirac|1928}} Primary source.Dirac suggested the topic of {{harvtxt|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 {{harvtxt|Dirac|1945}} ({{harvnb|Langlands|1985}}), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations ({{harvnb|Dalitz|Peierls|1986}}).

File:Wilhelm Karl Joseph Killing.jpeg, Independent discoverer of Lie algebras. The simple Lie algebras were first classified by him in 1888.]]

This section addresses the irreducible complex linear representations of the complexification $\backslash mathfrak\{so\}(3;\; 1)\_\backslash Complex$ of the Lie algebra $\backslash mathfrak\{so\}(3;\; 1)$ of the Lorentz group. A convenient basis for $\backslash mathfrak\{so\}(3;\; 1)$ is given by the three generators {{math|J_i}} of rotations and the three generators {{math|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{{harvnb|Weinberg|2002|loc=Equations 5.6.7–8.}}

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

The components of {{math|1=A = (A₁, A₂, A₃)}} and {{math|1=B = (B₁, B₂, B₃)}} separately satisfy the commutation relations of the Lie algebra $\backslash mathfrak\{su\}(2)$ and, moreover, they commute with each other,{{harvnb|Weinberg|2002|loc=Equations 5.6.9–11.}}

$$\backslash left[A\_i,\; A\_j\backslash right]\; =\; i\backslash varepsilon\_\{ijk\}\; A\_k,\backslash quad\; \backslash left[B\_i,\; B\_j\backslash right]\; =\; i\backslash varepsilon\_\{ijk\}\; B\_k,\backslash quad\; \backslash left[A\_i,\; B\_j\backslash right]\; =\; 0,$$

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

One has the isomorphisms{{harvnb|Hall|2003|loc=Chapter 6.}}{{harvnb|Knapp|2001}} The rather mysterious looking third isomorphism is proved in chapter 2, paragraph 4.

{{NumBlk|| $$\backslash begin\{align\}$$

\mathfrak{so}(3; 1) \hookrightarrow

\mathfrak{so}(3; 1)_\Complex &\cong

\mathbf{A}_\Complex \oplus \mathbf{B}_\Complex

\cong \mathfrak{su}(2)_\Complex \oplus \mathfrak{su}(2)_\Complex \\[5pt]

&\cong \mathfrak{sl}(2, \Complex) \oplus \mathfrak{sl}(2, \Complex) \\[5pt]

&\cong \mathfrak{sl}(2, \Complex) \oplus i\mathfrak{sl}(2, \Complex) =

\mathfrak{sl}(2, \Complex)_\Complex \hookleftarrow \mathfrak{sl}(2, \Complex),

\end{align}

| {{EquationRef|A1|A1}}

}}

where $\backslash mathfrak\{sl\}(2,\; \backslash Complex)$ is the complexification of $\backslash mathfrak\{su\}(2)\; \backslash cong\; \backslash mathbf\{A\}\; \backslash cong\; \backslash mathbf\{B\}.$

The utility of these isomorphisms comes from the fact that all irreducible representations of $\backslash mathfrak\{su\}(2)$, and hence all irreducible complex linear representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex),$ are known. The irreducible complex linear representation of $\backslash mathfrak\{sl\}(2,\; \backslash Complex)$ is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex).$

File:Hermann Weyl ETH-Bib Portr 00890.jpg, inventor of the unitarian trick. There are several concepts and formulas in representation theory named after Weyl, e.g. the Weyl group and the Weyl character formula. ]]

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

By compactness, the Peter–Weyl theorem applies to {{math|SU(2) × SU(2)}},{{harvnb|Knapp|2001}} and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of {{math|SU(2) × SU(2)}} are precisely the tensor products of irreducible unitary representations of {{math|SU(2)}}.This is an application of {{harvnb|Rossmann|2002|loc=Section 6.3, Proposition 10.}}

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 $\backslash text\{SL\}(2,\; \backslash Complex)\; \backslash times\; \backslash text\{SL\}(2,\; \backslash Complex)$
• Smooth representations of {{math|SU(2) × SU(2)}}
• Real linear representations of $\backslash mathfrak\{su\}(2)\; \backslash oplus\; \backslash mathfrak\{su\}(2)$
• Complex linear representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex)\; \backslash oplus\; \backslash mathfrak\{sl\}(2,\; \backslash Complex)$

Tensor products of representations appear at the Lie algebra level as either ofTensor products of representations, {{math|π_g ⊗ π_h}} of $\backslash mathfrak\{g\}\; \backslash oplus\; \backslash mathfrak\{h\}$ can, when both factors come from the same Lie algebra $\backslash mathfrak\{h\}\; =\; \backslash mathfrak\{g\},$ either be thought of as a representation of $\backslash mathfrak\{g\}$ or $\backslash mathfrak\{g\}\; \backslash oplus\; \backslash mathfrak\{g\}$.

{{NumBlk|| $$\backslash begin\{align\}$$

\pi_1\otimes\pi_2(X) &= \pi_1(X) \otimes \mathrm{Id}_V + \mathrm{Id}_U \otimes \pi_2(X) && X \in \mathfrak{g} \\

\pi_1\otimes\pi_2(X, Y) &= \pi_1(X) \otimes \mathrm{Id}_V + \mathrm{Id}_U \otimes \pi_2(Y) && (X, Y) \in \mathfrak{g} \oplus \mathfrak{g}

\end{align}

| {{EquationRef|A0|A0}}

}}

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

Finally, the $\backslash R$-linear representations of the real forms of the far left, $\backslash mathfrak\{so\}(3;\; 1)$, and the far right, $\backslash mathfrak\{sl\}(2,\; \backslash Complex),$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. in {{EquationNote|A1|(A1)}} are obtained from the $\backslash Complex$-linear representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex)\; \backslash oplus\; \backslash mathfrak\{sl\}(2,\; \backslash Complex)$ characterized in the previous paragraph.

The complex linear representations of the complexification of $\backslash mathfrak\{sl\}(2,\; \backslash Complex),\; \backslash mathfrak\{sl\}(2,\; \backslash Complex)\_\backslash Complex,$ obtained via isomorphisms in {{EquationNote|A1|(A1)}}, stand in one-to-one correspondence with the real linear representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex).${{harvnb|Knapp|2001|p=32.}} The set of all real linear irreducible representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex)$ are thus indexed by a pair {{math|(μ, ν)}}. The complex linear ones, corresponding precisely to the complexification of the real linear $\backslash mathfrak\{su\}(2)$ representations, are of the form {{math|(μ, 0)}}, while the conjugate linear ones are the {{math|(0, ν)}}. All others are real linear only. The linearity properties follow from the canonical injection, the far right in {{EquationNote|A1|(A1)}}, of $\backslash mathfrak\{sl\}(2,\; \backslash Complex)$ into its complexification. Representations on the form {{math|(ν, ν)}} or {{math|(μ, ν) ⊕ (ν, μ)}} are given by real matrices (the latter are not irreducible). Explicitly, the real linear {{math|(μ, ν)}}-representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex)$ are

$$\backslash varphi\_\{\backslash mu,\; \backslash nu\}(X)\; =$$

\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, \Complex)

where $\backslash varphi\_\backslash mu,\; \backslash mu\; =\; 0,\; \backslash tfrac\{1\}\{2\},\; 1,\; \backslash tfrac\{3\}\{2\},\; \backslash ldots$ are the complex linear irreducible representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex)$ and $\backslash overline\{\backslash varphi\_\backslash nu\},\; \backslash nu\; =\; 0,\; \backslash tfrac\{1\}\{2\},\; 1,\; \backslash tfrac\{3\}\{2\},\; \backslash ldots$ their complex conjugate representations. (The labeling is usually in the mathematics literature {{math|0, 1, 2, ...}}, but half-integers are chosen here to conform with the labeling for the $\backslash mathfrak\{so\}(3,\; 1)$ Lie algebra.) Here the tensor product is interpreted in the former sense of {{EquationNote|A0|(A0)}}. These representations are concretely realized below.

Via the displayed isomorphisms in {{EquationNote|A1|(A1)}} and knowledge of the complex linear irreducible representations of $\backslash mathfrak\{sl\}(2,\; \backslash Complex)\; \backslash oplus\; \backslash mathfrak\{sl\}(2,\; \backslash Complex)$ upon solving for {{math|J}} and {{math|K}}, all irreducible representations of $\backslash mathfrak\{so\}(3;\; 1)\_\backslash Complex,$ and, by restriction, those of $\backslash mathfrak\{so\}(3;\; 1)$ are obtained. The representations of $\backslash mathfrak\{so\}(3;\; 1)$ 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. Since $\backslash mathfrak\{so\}(3;\; 1)$ is semisimple, all its representations can be built up as direct sums of the irreducible ones.

Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers {{math|1=m = μ}} and {{math|1=n = ν}}, conventionally written as one of

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

where {{mvar|V}} is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given byCombine {{harvtxt|Weinberg|2002|loc=Equations 5.6.7–8, 5.6.14–15}} with {{harvtxt|Hall|2015|loc=Proposition 4.18}} about Lie algebra representations of group tensor product representations.

{{NumBlk||

$$\backslash pi\_\{(m,n)\}(J\_i)\; =\; J^\{(m)\}\_i\; \backslash otimes\; 1\_\{(2n+1)\}+1\_\{(2m+1)\}\; \backslash otimes\; J^\{(n)\}\_i$$

$$\backslash pi\_\{(m,n)\}(K\_i)\; =\; -i\; \backslash left(J^\{(m)\}\_i\; \backslash otimes\; 1\_\{(2n+1)\}\; -\; 1\_\{(2m+1)\}\; \backslash otimes\; J^\{(n)\}\_i\backslash right),$$

| {{EquationRef|A2|A2}} }}

where {{math|1_n}} is the {{mvar|n}}-dimensional unit matrix and

$$\backslash mathbf\{J\}^\{(n)\}\; =\; \backslash left(J^\{(n)\}\_1,\; J^\{(n)\}\_2,\; J^\{(n)\}\_3\backslash right)$$

are the {{math|(2n + 1)}}-dimensional irreducible representations of $\backslash mathfrak\{so\}(3)\; \backslash cong\; \backslash mathfrak\{su\}(2)$ also termed spin matrices or angular momentum matrices. These are explicitly given as{{harvnb|Weinberg|2002|loc=Equations 5.6.16–17.}}

$$\backslash begin\{align\}$$

\left(J_1^{(j)}\right)_{a'a} &= \frac{1}{2} \left(\sqrt{(j - a)(j + a + 1)}\delta_{a',a + 1} + \sqrt{(j + a)(j - a + 1)}\delta_{a',a - 1}\right) \\

\left(J_2^{(j)}\right)_{a'a} &= \frac{1}{2i}\left(\sqrt{(j - a)(j + a + 1)}\delta_{a',a + 1} - \sqrt{(j + a)(j - a + 1)}\delta_{a',a - 1}\right) \\

\left(J_3^{(j)}\right)_{a'a} &= a\delta_{a',a}

\end{align}

where {{math|δ}} denotes the Kronecker delta. In components, with {{math|−m ≤ a, a′ ≤ m}}, {{math|−n ≤ b, b′ ≤ n}}, the representations are given by{{harvnb|Weinberg|2002|loc=Section 5.6.}} The equations follow from equations 5.6.7–8 and 5.6.14–15.

$$\backslash begin\{align\}$$

\left(\pi_{(m,n)}\left(J_i\right)\right)_{a'b', ab} &= \delta_{b'b} \left(J_i^{(m)}\right)_{a'a} + \delta_{a'a} \left(J_i^{(n)}\right)_{b'b}\\

\left(\pi_{(m,n)}\left(K_i\right)\right)_{a'b', ab} &= -i \left( \delta_{b'b} \left(J_i^{(m)}\right)_{a'a} - \delta_{a'a} \left(J_i^{(n)}\right)_{b'b}\right)

\end{align}

• The {{math|(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 {{math|(0, {{sfrac|1|2}})}} or {{math|({{sfrac|1|2}}, 0)}} representations (Weyl spinors).
• The four-momentum of a particle (either massless or massive) transforms under the {{math|({{sfrac|1|2}}, {{sfrac|1|2}})}} representation, a four-vector.
• A physical example of a (1,1) traceless symmetric tensor field is the tracelessThe "traceless" property can be expressed as {{math|1=S_αβg^αβ = 0}}, or {{math|1=S_α^α = 0}}, or {{math|1=S^αβg_αβ = 0}} depending on the presentation of the field: covariant, mixed, and contravariant respectively. part of the energy–momentum tensor {{mvar|T^μν}}.{{harvnb|Tung|1985}}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.

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

• {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}} is the bispinor representation. See also Dirac spinor and Weyl spinors and bispinors below.
• {{math|(1, {{sfrac|1|2}}) ⊕ ({{sfrac|1|2}}, 1)}} is the Rarita–Schwinger field representation.
• {{math|({{sfrac|3|2}}, 0) ⊕ (0, {{sfrac|3|2}})}} would be the symmetry of the hypothesized gravitino.This is provided parity is a symmetry. Else there would be two flavors, {{math|({{sfrac|3|2}}, 0)}} and {{math|(0, {{sfrac|3|2}})}} in analogy with neutrinos. It can be obtained from the {{math|(1, {{sfrac|1|2}}) ⊕ ({{sfrac|1|2}}, 1)}} representation.
• {{math|(1, 0) ⊕ (0, 1)}} is the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation.

The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence.{{harvnb|Lie|1888}} The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras.{{harvnb|Rossmann|2002|loc=Section 2.5.}} The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted $\backslash exp\; :\; \backslash mathfrak\{g\}\; \backslash to\; G.$

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

{{NumBlk||$$\backslash begin\{align\}$$

\Pi(g = e^{iX}) &\equiv e^{i\pi(X)}, && X \in \mathfrak g, \quad g = e^{iX} \in \mathrm{im}(\exp),\\

\Pi(g = g_1g_2\cdots g_n) &\equiv \Pi(g_1)\Pi(g_2)\cdots \Pi(g_n), && g \notin \mathrm{im}(\exp), \quad g_1 , g_2, \ldots, g_n \in \mathrm{im}(\exp).

\end{align}|{{EquationRef|G2|G2}}}}

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

From a practical point of view, it is important whether the first formula in {{EquationNote|G2|(G2)}} can be used for all elements of the group. It holds for all $X\; \backslash in\; \backslash mathfrak\{g\}$, however, in the general case, e.g. for $\backslash text\{SL\}(2,\backslash Complex)$, not all {{math|g ∈ G}} are in the image of {{math|exp}}.

But $\backslash exp\; :\; \backslash mathfrak\{so\}(3;1)\; \backslash to\; \backslash text\{SO\}(3;1)^+$ is surjective. One way to show this is to make use of the isomorphism $\backslash text\{SO\}(3;\; 1)^+\; \backslash cong\; \backslash text\{PGL\}(2,\backslash Complex),$ the latter being the Möbius group. It is a quotient of $\backslash text\{GL\}(n,\backslash Complex)$ (see the linked article). The quotient map is denoted with $p\; :\; \backslash text\{GL\}(n,\backslash Complex)\; \backslash to\; \backslash text\{PGL\}(2,\backslash Complex).$ The map $\backslash exp\; :\; \backslash mathfrak\{gl\}(n,\; \backslash Complex)\; \backslash to\; \backslash text\{GL\}(n,\; \backslash Complex)$ is onto.{{harvnb|Hall|2015|loc=Theorem 2.10.}} Apply {{EquationNote|Lie|(Lie)}} with {{mvar|π}} being the differential of {{mvar|p}} at the identity. Then

$$\backslash forall\; X\; \backslash in\; \backslash mathfrak\{gl\}(n,\; \backslash Complex):\; \backslash quad\; p\; (\; \backslash exp\; (iX))\; =\backslash exp\; (\; i\; \backslash pi\; (X)).$$

Since the left hand side is surjective (both {{math|exp}} and {{mvar|p}} are), the right hand side is surjective and hence $\backslash exp\; :\; \backslash mathfrak\{pgl\}(2,\; \backslash Complex)\; \backslash to\; \backslash text\{PGL\}(2,\; \backslash Complex)$ is surjective.{{harvnb|Bourbaki|1998|p=424.}} Finally, recycle the argument once more, but now with the known isomorphism between {{math|SO(3; 1)⁺}} and $\backslash text\{PGL\}(2,\; \backslash Complex)$ to find that {{math|exp}} is onto for the connected component of the Lorentz group.

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

{{math proof | proof =

To exhibit the fundamental group of {{math|SO(3; 1)⁺}}, the topology of its covering group $\backslash text\{SL\}(2,\backslash Complex)$ is considered. By the polar decomposition theorem, any matrix $\backslash lambda\; \backslash in\; \backslash text\{SL\}(2,\backslash Complex)$ may be uniquely expressed as{{harvnb|Weinberg|2002|loc=Section 2.7 p.88.}}

$$\backslash lambda\; =\; ue^h,$$

where {{mvar|u}} is unitary with determinant one, hence in {{math|SU(2)}}, and {{mvar|h}} is Hermitian with trace zero. The trace and determinant conditions imply:{{harvnb|Weinberg|2002|loc=Section 2.7.}}

$$\backslash begin\{align\}$$

h &= \begin{pmatrix}c&a-ib\\a+ib&-c\end{pmatrix} && (a,b,c) \in \R^3 \\[4pt]

u &= \begin{pmatrix}d+ie&f+ig\\-f+ig&d-ie\end{pmatrix} && (d,e,f,g) \in \R^4 \text{ subject to } d^2 + e^2 + f^2 + g^2 = 1.

\end{align}

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

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

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

$$\backslash text\{SO\}(3;\; 1)\; \backslash cong\; \backslash R^3\; \backslash times\; (\; \backslash mathbb\{S\}^3/\backslash Z\_2),$$

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

}}

Since {{math|π₁(SO(3; 1)⁺)}} has two elements, some representations of the Lie algebra will yield projective representations.{{harvnb|Hall|2015|loc=Appendix C.3.}}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. Once it is known whether a representation is projective, formula {{EquationNote|G2|(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 {{mvar|X}} in {{EquationNote|G2|(G2)}}) is used to represent the group element in the standard representation.

For the Lorentz group, the {{math|(m, n)}}-representation is projective when {{math|m + n}} is a half-integer. See {{section link||Spinors}}.

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

{{NumBlk||$$$$

\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),

|{{EquationRef|G5|G5}}

}}

since any loop in {{math|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 {{math|Π}} is a double-valued function. It is not possible to consistently choose a sign to obtain a continuous representation of all of {{math|SO(3; 1)⁺}}, but this is possible locally around any point.

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

$$\backslash left(\backslash frac\{1\}\{2\}\backslash sigma\_1,\; \backslash frac\{1\}\{2\}\backslash sigma\_2,\; \backslash frac\{1\}\{2\}\backslash sigma\_3,\; \backslash frac\{i\}\{2\}\backslash sigma\_1,\; \backslash frac\{i\}\{2\}\backslash sigma\_2,\; \backslash frac\{i\}\{2\}\backslash sigma\_3\backslash right)\backslash equiv(j\_1,\; j\_2,\; j\_3,\; k\_1,\; k\_2,\; k\_3),$$

where the sigmas are the Pauli matrices. From the relations

{{NumBlk||$$[\backslash sigma\_i,\; \backslash sigma\_j]\; =\; 2i\backslash epsilon\_\{ijk\}\backslash sigma\_k$$|{{EquationRef|J1|J1}}}}

is obtained

{{NumBlk||$$[j\_i,\; j\_j]\; =\; i\backslash epsilon\_\{ijk\}j\_k,\; \backslash quad\; [j\_i,\; k\_j]\; =\; i\backslash epsilon\_\{ijk\}k\_k,\; \backslash quad\; [k\_i,\; k\_j]\; =\; -i\backslash epsilon\_\{ijk\}j\_k,$$|{{EquationRef|J2|J2}}}}

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

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=More on covering groups in general and the $\backslash text\{SL\}(2,\backslash Complex)$ covering of the Lorentz group in particular}}

File:Wigner.jpg 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 {{math|p_g(t), 0 ≤ t ≤ 1}} be a path from {{math|1 ∈ SO(3; 1)⁺}} to {{math|g ∈ SO(3; 1)⁺}}, denote its homotopy class by {{math|[p_g]}} and let {{mvar|π_g}} be the set of all such homotopy classes. Define the set

{{NumBlk||$$G\; =\; \backslash \{(g,[p\_g]):\; g\backslash in\; \backslash mathrm\{SO\}(3;\; 1)^+,[p\_g]\backslash in\; \backslash pi\_g\backslash \}$$|{{EquationRef|C1|C1}}}}

and endow it with the multiplication operation

{{NumBlk||$$(g\_1,[p\_1])(g\_2,[p\_2])\; =\; (g\_1g\_2,[p\_\{12\}]),$$|{{EquationRef|C2|C2}}}}

where $p\_\{12\}$ is the path multiplication of $p\_1$ and $p\_2$:

With this multiplication, {{mvar|G}} becomes a group isomorphic to $\backslash text\{SL\}(2,\backslash Complex),${{harvnb|Wigner|1939|p=27.}} the universal covering group of {{math|SO(3; 1)⁺}}. Since each {{mvar|π_g}} has two elements, by the above construction, there is a 2:1 covering map {{math|p : G → SO(3; 1)⁺}}. According to covering group theory, the Lie algebras $\backslash mathfrak\{so\}(3;\; 1),\; \backslash mathfrak\{sl\}(2,\backslash Complex)$ and $\backslash mathfrak\{g\}$ of {{mvar|G}} are all isomorphic. The covering map {{math|p : G → SO(3; 1)⁺}} is simply given by {{math|1=p(g, [p_g]) = g}}.

For an algebraic view of the universal covering group, let $\backslash text\{SL\}(2,\backslash Complex)$ act on the set of all Hermitian {{gaps|2|×|2}} matrices $\backslash mathfrak\{h\}$ by the operation

{{NumBlk||$$\backslash begin\{cases\}$$

\mathbf{P}(A): \mathfrak{h} \to \mathfrak{h} \\

X \mapsto A^\dagger XA

\end{cases} \qquad A \in \mathrm{SL}(2,\Complex)|{{EquationRef|C3|C3}}}}

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

{{NumBlk|||{{EquationRef|C4|C4}}}}

The map {{math|P}} is a group homomorphism into $\backslash text\{GL\}(\backslash mathfrak\{h\})\; \backslash subset\; \backslash text\{End\}(\backslash mathfrak\{h\}).$ Thus $\backslash mathbf\{P\}\; :\; \backslash text\{SL\}(2,\backslash Complex)\; \backslash to\; \backslash text\{GL\}(\backslash mathfrak\{h\})$ is a 4-dimensional representation of $\backslash text\{SL\}(2,\backslash Complex)$. Its kernel must in particular take the identity matrix to itself, {{math|1=A^†IA = A^†A = I}} and therefore {{math|1=A^† = A⁻¹}}. Thus {{math|1=AX = XA}} for {{mvar|A}} in the kernel so, by Schur's lemma,In particular, {{math|A}} commutes with the Pauli matrices, hence with all of {{math|SU(2)}} making Schur's lemma applicable. {{mvar|A}} is a multiple of the identity, which must be {{math|±I}} since {{math|1=det A = 1}}.{{harvnb|Gelfand|Minlos|Shapiro|1963}} This construction of the covering group is treated in paragraph 4, section 1, chapter 1 in Part II. The space $\backslash mathfrak\{h\}$ is mapped to Minkowski space {{math|M⁴}}, via

{{NumBlk||$$X\; =\; (\backslash xi\_1,\backslash xi\_2,\backslash xi\_3,\backslash xi\_4)\; \backslash leftrightarrow\; \backslash overrightarrow\{(\backslash xi\_1,\backslash xi\_2,\backslash xi\_3,\backslash xi\_4)\}\; =\; (x,y,z,t)\; =\; \backslash overrightarrow\{X\}.$$|{{EquationRef|C5|C5}}}}

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

{{NumBlk||$$\backslash mathbf\{p\}(A)\backslash overrightarrow\{X\}\; =\; \backslash overrightarrow\{AXA^\backslash dagger\}$$|{{EquationRef|C6|C6}}}}

preserves the Lorentz inner product since

$$-\; \backslash det\; X\; =\; \backslash xi\_1^2\; +\; \backslash xi\_2^2\; +\backslash xi\_3^2\; -\backslash xi\_4^2\; =\; x^2\; +\; y^2\; +z^2\; -\; t^2.$$

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

It can be shown that the Lie map of $\backslash mathbf\{p\}\; :\; \backslash text\{SL\}(2,\backslash Complex)\; \backslash to\; \backslash text\{SO\}(3;\; 1)^+,$ is a Lie algebra isomorphism: $\backslash pi\; :\; \backslash mathfrak\{sl\}(2,\backslash Complex)\; \backslash to\; \backslash mathfrak\{so\}(3;\; 1).$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 {{math|p}} is {{math|2:1}} and both $\backslash text\{SL\}(2,\backslash Complex)$ and {{math|SO(3; 1)⁺}} are {{nowrap|{{math|6}}-dimensional}}, the kernel must be {{nowrap|{{math|0}}-dimensional}}, hence {{math|{0}.}} The map {{math|P}} is also onto.The exponential map is one-to-one in a neighborhood of the identity in $\backslash text\{SL\}(2,\backslash Complex),$ hence the composition $\backslash exp\; \backslash circ\; \backslash sigma\; \backslash circ\; \backslash log\; :\; \backslash text\{SL\}(2,\backslash Complex)\; \backslash to\; \backslash text\{SO\}(3;\; 1)^+,$ where {{mvar|σ}} is the Lie algebra isomorphism, is onto an open neighborhood {{math|U ⊂ SO(3; 1)⁺}} containing the identity. Such a neighborhood generates the connected component.

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

{{Hidden end}}

[[File:Commutative diagram SO(3, 1) latex.svg|350px|thumb|left|This diagram shows the web of maps discussed in the text. Here {{mvar|V}} is a finite-dimensional vector space carrying representations of $\backslash mathfrak\{sl\}(2,\backslash Complex),\; \backslash mathfrak\{so\}(3;\; 1),$ $\backslash text\{SL\}(2,\backslash Complex)$ and $\backslash text\{SO\}(3;\; 1)^+.$ $\backslash exp$ is the exponential mapping, {{math|p}} is the covering map from $\backslash text\{SL\}(2,\backslash Complex),$ onto {{math|SO(3; 1)⁺}} and {{mvar|σ}} is the Lie algebra isomorphism induced by it.

The maps {{math|Π, π}} and the two {{math|Φ}} are representations. The picture is only partially true when {{math|Π}} is projective.]]

The exponential mapping $\backslash exp\; :\; \backslash mathfrak\{sl\}(2,\backslash Complex)\; \backslash to\; \backslash text\{SL\}(2,\backslash Complex)$ is not onto.{{harvnb|Rossmann|2002|loc=Section 2.1.}} The matrix

{{NumBlk|||{{EquationRef|S6|S6}}}}

is in $\backslash text\{SL\}(2,\backslash Complex),$ but there is no $Q\backslash in\; \backslash mathfrak\{sl\}(2,\backslash Complex)$ such that {{math|1=q = exp(Q)}}.{{harvnb|Rossmann|2002}} From Example 4 in section 2.1 : This can be seen as follows. The matrix {{mvar|q}} has eigenvalues {{math|{−1, −1}}}, but it is not diagonalizable. If {{math|1=q = exp(Q)}}, then {{mvar|Q}} has eigenvalues {{math|λ, −λ}} with {{math|1=λ = iπ + 2πik}} for some {{mvar|k}} because elements of $\backslash mathfrak\{sl\}(2,\backslash Complex)$ are traceless. But then {{mvar|Q}} is diagonalizable, hence {{mvar|q}} is diagonalizable, which is a contradiction.

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

{{NumBlk||$$g\; =\; \backslash exp(X\_1)\; \backslash cdots\; \backslash exp(X\_n),\; \backslash qquad\; X\_1,\; \backslash ldots\; X\_n\; \backslash in\; \backslash mathfrak\{g\}.$$|{{EquationRef|S7|S7}}}}

The matrix {{mvar|q}} can be written

{{NumBlk||$$\backslash begin\{align\}$$

&\exp(-X)\exp(i\pi H) \\

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

{}={} &\begin{pmatrix} 1&-1\\ 0&1 \\ \end{pmatrix} \begin{pmatrix} -1&0\\ 0&-1\\ \end{pmatrix}\\[6pt]

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

{}={} &q.

\end{align}

| {{EquationRef|S8|S8}}

}}

The complex linear representations of $\backslash mathfrak\{sl\}(2,\backslash Complex)$ and $\backslash text\{SL\}(2,\backslash Complex)$ are more straightforward to obtain than the $\backslash mathfrak\{so\}(3;\; 1)^+$ 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 $\backslash mathfrak\{sl\}(2,\backslash Complex)$ are exactly the {{math|(μ, ν)}}-representations. They can be exponentiated too. The {{math|(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 {{math|i}} and there is no factor of {{math|i}} in the exponential mapping compared to the physics convention used elsewhere. Let the basis of $\backslash mathfrak\{sl\}(2,\backslash Complex)$ be{{harvnb|Hall|2015|loc=First displayed equations in section 4.6.}}

{{NumBlk||

|{{EquationRef|S1|S1}}}}

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

The irreducible holomorphic {{math|(n + 1)}}-dimensional representations $\backslash text\{SL\}(2,\backslash Complex),\; n\; \backslash geqslant\; 2,$ can be realized on the space of homogeneous polynomial of degree {{math|n}} in 2 variables $\backslash mathbf\{P\}^2\_n,${{harvnb|Hall|2015|loc=Example 4.10.}}{{harvnb|Knapp|2001|loc=Chapter 2.}} the elements of which are

$$P\backslash begin\{pmatrix\}\; z\_1\backslash \backslash \; z\_2\backslash \backslash \; \backslash end\{pmatrix\}\; =\; c\_n\; z\_1^n\; +\; c\_\{n-1\}\; z\_1^\{n-1\}z\_2\; +\; \backslash cdots\; +\; c\_0\; z\_2^n,\; \backslash quad\; c\_0,\; c\_1,\; \backslash ldots,\; c\_n\; \backslash in\; \backslash mathbb\; Z.$$

The action of $\backslash text\{SL\}(2,\backslash Complex)$ is given by{{harvnb|Knapp|2001}} Equation 2.1.{{harvnb|Hall|2015|loc=Equation 4.2.}}

{{NumBlk||

P\left( \begin{pmatrix} a&b\\ c&d\\ \end{pmatrix}^{-1} \begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} \right ), \qquad P \in \mathbf{P}^2_n.|{{EquationRef|S2|S2}}}}

The associated $\backslash mathfrak\{sl\}(2,\backslash Complex)$-action is, using {{EquationNote|G6|(G6)}} and the definition above, for the basis elements of $\backslash mathfrak\{sl\}(2,\backslash Complex),${{harvnb|Hall|2015|loc=Equation before 4.5.}}

{{NumBlk||$$\backslash phi\_n(H)\; =\; -z\_1\backslash frac\{\backslash partial\}\{\backslash partial\; z\_1\}\; +\; z\_2\backslash frac\{\backslash partial\}\{\backslash partial\; z\_2\},\; \backslash quad\; \backslash phi\_n(X)\; =\; -z\_2\backslash frac\{\backslash partial\}\{\backslash partial\; z\_1\},\; \backslash quad\; \backslash phi\_n(Y)\; =\; -z\_1\backslash frac\{\backslash partial\}\{\backslash partial\; z\_2\}.$$|{{EquationRef|S5|S5}}}}

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

The {{math|(μ, ν)}}-representations are realized on a space of polynomials $\backslash mathbf\{P\}^2\_\{\backslash mu,\backslash nu\}$ in $z\_1,\; \backslash overline\{z\_1\},\; z\_2,\; \backslash overline\{z\_2\},$ homogeneous of degree {{mvar|μ}} in $z\_1,\; z\_2$ and homogeneous of degree {{mvar|ν}} in $\backslash overline\{z\_1\},\; \backslash overline\{z\_2\}.$ The representations are given by{{harvnb|Knapp|2001}} Equation 2.4.

{{NumBlk||

P \left( \begin{pmatrix} a&b\\ c&d\\ \end{pmatrix}^{-1} \begin{pmatrix} z_1\\ z_2\\ \end{pmatrix} \right ), \quad P \in \mathbf{P}^2_{\mu,\nu}.|{{EquationRef|S6|S6}}}}

By employing {{EquationNote|G6|(G6)}} again it is found that

{{NumBlk||$$\backslash begin\{align\}$$

\phi_{\mu,\nu}(E)P = &- \frac{\partial P}{\partial z_1} \left (E_{11}z_1 + E_{12}z_2 \right ) - \frac{\partial P}{\partial z_2} \left (E_{21}z_1 + E_{22}z_2 \right) \\

&- \frac{\partial P}{\partial \overline{z_1}}\left (\overline{E_{11}}\overline{z_1} + \overline{E_{12}}\overline{z_2} \right ) -\frac{\partial P}{\partial \overline{z_2}} \left (\overline{E_{21}}\overline{z_1} + \overline{E_{22}}\overline{z_2} \right )

\end{align}, \quad E \in \mathfrak{sl}(2, \mathbf{C}).|{{EquationRef|S7|S7}}}}

In particular for the basis elements,

{{NumBlk||$$\backslash begin\{align\}$$

\phi_{\mu,\nu}(H) &= -z_1\frac{\partial}{\partial z_1} + z_2\frac{\partial}{\partial z_2}-\overline{z_1}\frac{\partial}{\partial \overline{z_1}} + \overline{z_2}\frac{\partial}{\partial \overline{z_2}} \\

\phi_{\mu,\nu}(X) &= -z_2\frac{\partial}{\partial z_1} - \overline{z_2}\frac{\partial}{\partial \overline{z_1}} \\

\phi_{\mu,\nu}(Y) &= -z_1\frac{\partial}{\partial z_2} - \overline{z_1}\frac{\partial}{\partial \overline{z_2}}

\end{align}|{{EquationRef|S8|S8}}}}

The {{math|(m, n)}} representations, defined above via {{EquationNote|A1|(A1)}} (as restrictions to the real form $\backslash mathfrak\{sl\}(3,\; 1)$) of tensor products of irreducible complex linear representations {{math|1=π_{m = μ}}} and {{math|1=π_{n = ν}}} of $\backslash mathfrak\{sl\}(2,\backslash Complex),$ are irreducible, and they are the only irreducible representations.

• Irreducibility follows from the unitarian trick{{harvnb|Knapp|2001|loc=Section 2.3.}} and that a representation {{math|Π}} of {{math|SU(2) × SU(2)}} is irreducible if and only if {{math|1=Π = Π_μ ⊗ Π_ν}},{{harvnb|Rossmann|2002|loc=Proposition 10, paragraph 6.3.}} This is easiest proved using character theory. where {{math|Π_μ, Π_ν}} are irreducible representations of {{math|SU(2)}}.
• Uniqueness follows from that the {{math|Π_m}} are the only irreducible representations of {{math|SU(2)}}, which is one of the conclusions of the theorem of the highest weight.{{harvnb|Hall|2015|loc=Theorems 9.4–5.}}

The {{math|(m, n)}} representations are {{math|(2m + 1)(2n + 1)}}-dimensional.{{harvnb|Weinberg|2002|loc=Chapter 5.}} This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of $\backslash text\{SL\}(2,\backslash Complex)$ and $\backslash mathfrak\{sl\}(2,\; \backslash Complex)$. For a Lie general algebra $\backslash mathfrak\{g\}$ the Weyl dimension formula,{{harvnb|Hall|2015|loc= Theorem 10.18.}}

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

applies, where {{math|R⁺}} is the set of positive roots, {{math|ρ}} is the highest weight, and {{math|δ}} is half the sum of the positive roots. The inner product $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ is that of the Lie algebra $\backslash mathfrak\{g\},$ invariant under the action of the Weyl group on $\backslash mathfrak\{h\}\; \backslash subset\; \backslash mathfrak\{g\},$ the Cartan subalgebra. The roots (really elements of $\backslash mathfrak\{h\}^*$) are via this inner product identified with elements of $\backslash mathfrak\{h\}.$ For $\backslash mathfrak\{sl\}(2,\backslash Complex),$ the formula reduces to {{math|1=dim π_μ = 2μ + 1 = 2m + 1}}, where the present notation must be taken into account. The highest weight is {{math|2μ}}.{{harvnb|Hall|2003|p=235.}} By taking tensor products, the result follows.

If a representation {{math|Π}} of a Lie group {{math|G}} is not faithful, then {{math|1=N = ker Π}} is a nontrivial normal subgroup.See any text on basic group theory. There are three relevant cases.

1. {{math|N}} is non-discrete and abelian.
2. {{math|N}} is non-discrete and non-abelian.
3. {{math|N}} is discrete. In this case {{math|N ⊂ Z}}, where {{math|Z}} is the center of {{math|G}}.Any discrete normal subgroup of a path connected group {{math|G}} is contained in the center {{math|Z}} of {{math|G}}.

{{Paragraph break}}

{{harvnb|Hall|2015|loc=Exercise 11, chapter 1.}}

In the case of {{math|SO(3; 1)⁺}}, the first case is excluded since {{math|SO(3; 1)⁺}} is semi-simple.A semisimple Lie group does not have any non-discrete normal abelian subgroups. This can be taken as the definition of semisimplicity. The second case (and the first case) is excluded because {{math|SO(3; 1)⁺}} is simple.A simple group does not have any non-discrete normal subgroups. For the third case, {{math|SO(3; 1)⁺}} is isomorphic to the quotient $\backslash text\{SL\}(2,\backslash Complex)/\backslash \{\backslash pm\; I\backslash \}.$ But $\backslash \{\backslash pm\; I\backslash \}$ is the center of $\backslash text\{SL\}(2,\backslash Complex).$ It follows that the center of {{math|SO(3; 1)⁺}} is trivial, and this excludes the third case. The conclusion is that every representation {{math|Π : SO(3; 1)⁺ → GL(V)}} and every projective representation {{math|Π : SO(3; 1)⁺ → PGL(W)}} for {{math|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,{{harvnb|Rossmann|2002}} Propositions 3 and 6 paragraph 2.5. and the center of {{math|SO(3; 1)⁺}} replaced by the center of $\backslash mathfrak\{sl\}(3;\; 1)^+$The center of any semisimple Lie algebra is trivial{{harvnb|Hall|2003}} See exercise 1, Chapter 6. and $\backslash mathfrak\{so\}(3;\; 1)$ is semi-simple and simple, and hence has no non-trivial ideals.

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

The {{math|(m, n)}} Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary.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 {{math|(Π, V)}} is a finite-dimensional representation of a compact Lie group {{mvar|G}} and if {{math|(·, ·)}} is any inner product on {{mvar|V}}, define a new inner product {{math|(·, ·)_Π}} by {{math|1=(x, y)_Π = ∫_G(Π(g)x, Π(g)y) dμ(g)}}, where {{mvar|μ}} is Haar measure on {{mvar|G}}. Then {{math|Π}} is unitary with respect to {{math|(·, ·)_Π}}. See {{harvtxt|Hall|2015|loc=Theorem 4.28.}}

{{Paragraph break}}

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. {{harvtxt|Hall|2015|loc=Definition 4.24., Theorem 4.28.}}

{{Paragraph break}}

It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in {{harvtxt|Greiner|Müller|1994|loc=Section 15.2.}}. This is due to the non-compactness of the Lorentz group. In fact, a connected simple non-compact Lie group cannot have any nontrivial unitary finite-dimensional representations. There is a topological proof of this.{{harvnb|Bekaert|Boulanger|2006}} p.4. Let {{math|u : G → GL(V)}}, where {{math|V}} is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group {{mvar|G}}. Then {{math|u(G) ⊂ U(V) ⊂ GL(V)}} where {{math|U(V)}} is the compact subgroup of {{math|GL(V)}} consisting of unitary transformations of {{mvar|V}}. The kernel of {{math|u}} is a normal subgroup of {{mvar|G}}. Since {{mvar|G}} is simple, {{math|ker u}} is either all of {{mvar|G}}, in which case {{math|u}} is trivial, or {{math|ker u}} is trivial, in which case {{math|u}} is faithful. In the latter case {{math|u}} is a diffeomorphism onto its image,{{harvnb|Hall|2003}} Proposition 1.20. {{math|u(G) ≅ G}} and {{math|u(G)}} is a Lie group. This would mean that {{math|u(G)}} is an embedded non-compact Lie subgroup of the compact group {{math|U(V)}}. This is impossible with the subspace topology on {{math|u(G) ⊂ U(V)}} since all embedded Lie subgroups of a Lie group are closed{{harvnb|Lee|2003|loc=Theorem 8.30.}} If {{math|u(G)}} were closed, it would be compact,{{harvnb|Lee|2003}} Lemma A.17 (c). Closed subsets of compact sets are compact. and then {{mvar|G}} would be compact,{{harvnb|Lee|2003}} Lemma A.17 (a). If {{math|f : X → Y}} is continuous, {{mvar|X}} is compact, then {{math|f(X)}} is compact. contrary to assumption.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 {{harvtxt|Weinberg|2000|loch=Chapter 24.}}

In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of {{math|A}} and {{math|B}} used in the construction are Hermitian. This means that {{math|J}} is Hermitian, but {{math|K}} is anti-Hermitian.{{harvnb|Weinberg|2002|loc=Section 5.6, p. 231.}} 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.{{harvnb|Weinberg|2002|loc=Section 5.6.}}

The {{math|(m, n)}} representation is, however, unitary when restricted to the rotation subgroup {{math|SO(3)}}, but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an {{math|(m, n)}} representation have {{math|SO(3)}}-invariant subspaces of highest weight (spin) {{math|m + n, m + n − 1, ..., {{abs| m − n}}}},{{harvnb|Weinberg|2002|p=231.}} where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) {{math|j}} is {{math|(2j + 1)}}-dimensional. So for example, the ({{sfrac|1|2}}, {{sfrac|1|2}}) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.

Since the angular momentum operator is given by {{math|1=J = A + B}}, the highest spin in quantum mechanics of the rotation sub-representation will be {{math|(m + n)ℏ}} and the "usual" rules of addition of angular momenta and the formalism of 3-j symbols, 6-j symbols, etc. applies.{{harvnb|Weinberg|2002|loc=Sections 2.5, 5.7.}}

It is the {{math|SO(3)}}-invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the {{math|(m, n)}} representation has spin if {{math|m + n}} is half-integer. The simplest are {{math|({{sfrac|1|2}}, 0)}} and {{math|(0, {{sfrac|1|2}})}}, the Weyl-spinors of dimension {{math|2}}. Then, for example, {{math|(0, {{sfrac|3|2}})}} and {{math|(1, {{sfrac|1|2}})}} are a spin representations of dimensions {{math|1=2⋅{{sfrac|3|2}} + 1 = 4}} and {{math|1=(2 + 1)(2⋅{{sfrac|1|2}} + 1) = 6}} respectively. According to the above paragraph, there are subspaces with spin both {{math|{{sfrac|3|2}}}} and {{math|{{sfrac|1|2}}}} in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under {{math|SO(3)}}. It cannot be ruled out in general, however, that representations with multiple {{math|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.{{harvnb|Tung|1985|loc=Section 10.5.}}

Construction of pure spin {{math|{{sfrac|n|2}}}} representations for any {{math|n}} (under {{math|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.{{harvnb|Weinberg|2002}} This is outlined (very briefly) on page 232, hardly more than a footnote.

File:Root system A1xA1.svg {{math|A₁ × A₁}} of $\backslash mathfrak\{sl\}(2,\backslash Complex)\; \backslash oplus\; \backslash mathfrak\{sl\}(2,\backslash Complex).$]]

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

1. The set of weights of the dual representation of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation.{{harvnb|Hall|2003|loc=Proposition 7.39.}}
2. Two irreducible representations are isomorphic if and only if they have the same highest weight.This is one of the conclusions of Cartan's theorem, the theorem of the highest weight.{{Paragraph break}}{{harvtxt|Hall|2015|loc=Theorems 9.4–5.}}
3. For each semisimple Lie algebra there exists a unique element {{math|w₀}} of the Weyl group such that if {{math|μ}} is a dominant integral weight, then {{math|w₀ ⋅ (−μ)}} is again a dominant integral weight.{{harvnb|Hall|2003|loc=Theorem 7.40.}}
4. If $\backslash pi\_\{\backslash mu\_0\}$ is an irreducible representation with highest weight {{math|μ₀}}, then $\backslash pi^*\_\{\backslash mu\_0\}$ has highest weight {{math|w₀ ⋅ (−μ)}}.

Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If {{math|−I}} is an element of the Weyl group of a semisimple Lie algebra, then {{math|1=w₀ = −I}}. In the case of $\backslash mathfrak\{sl\}(2,\backslash Complex),$ the Weyl group is {{math|1=W = {I, −I}}}.{{harvnb|Hall|2003|loc=Section 6.6.}} It follows that each {{math|1=π_μ, μ = 0, 1, ...}} is isomorphic to its dual $\backslash pi^*\_\{\backslash mu\}.$ The root system of $\backslash mathfrak\{sl\}(2,\backslash Complex)\; \backslash oplus\; \backslash mathfrak\{sl\}(2,\backslash Complex)$ is shown in the figure to the right.{{harvnb|Hall|2015|loc=Section 8.2}} The root system is the union of two copies of {{math|A₁}}, where each copy resides in its own dimensions in the embedding vector space. The Weyl group is generated by $\backslash \{w\_\{\backslash gamma\}\backslash \}$ where $w\_\backslash gamma$ is reflection in the plane orthogonal to {{math|γ}} as {{math|γ}} ranges over all roots.{{harvnb|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. Inspection shows that {{math|1=w_α ⋅ w_β = −I}} so {{math|−I ∈ W}}. Using the fact that if {{math|π, σ}} are Lie algebra representations and {{math|π ≅ σ}}, then {{math|Π ≅ Σ}},{{harvnb|Hall|2003|loc=Second item in proposition 4.5.}} the conclusion for {{math|SO(3; 1)⁺}} is

$$\backslash pi\_\{m,\; n\}^\{*\}\; \backslash cong\; \backslash pi\_\{m,\; n\},\; \backslash quad\; \backslash Pi\_\{m,\; n\}^\{*\}\; \backslash cong\; \backslash Pi\_\{m,\; n\},\; \backslash quad\; 2m,\; 2n\; \backslash in\; \backslash mathbf\{N\}.$$

If {{math|π}} is a representation of a Lie algebra, then $\backslash overline\{\backslash pi\}$ 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.{{harvnb|Hall|2003|p=219.}} In general, every irreducible representation {{math|π}} of $\backslash mathfrak\{sl\}(n,\backslash Complex)$ can be written uniquely as {{math|1=π = π⁺ + π⁻}}, where{{harvnb|Rossmann|2002|loc=Exercise 3 in paragraph 6.5.}}

$$\backslash pi^\backslash pm(X)\; =\; \backslash frac\{1\}\{2\}\backslash left(\backslash pi(X)\; \backslash pm\; i\backslash pi\backslash left(i^\{-1\}X\backslash right)\backslash right),$$

with $\backslash pi^+$ holomorphic (complex linear) and $\backslash pi^-$ anti-holomorphic (conjugate linear). For $\backslash mathfrak\{sl\}(2,\backslash Complex),$ since $\backslash pi\_\backslash mu$ is holomorphic, $\backslash overline\{\backslash pi\_\backslash mu\}$ is anti-holomorphic. Direct examination of the explicit expressions for $\backslash pi\_\{\backslash mu,\; 0\}$ and $\backslash pi\_\{0,\; \backslash nu\}$ in equation {{EquationNote|S8|(S8)}} below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression {{EquationNote|S8|(S8)}} also allows for identification of $\backslash pi^+$ and $\backslash pi^-$ for $\backslash pi\_\{\backslash mu,\; \backslash nu\}$ as

$$\backslash pi^+\_\{\backslash mu,\; \backslash nu\}\; =\; \backslash pi\_\backslash mu^\{\backslash oplus\_\{\backslash nu+1\}\},\backslash qquad\; \backslash pi^-\_\{\backslash mu,\; \backslash nu\}\; =\; \backslash overline\{\backslash pi\_\backslash nu^\{\backslash oplus\_\{\backslash mu+1\}\}\}.$$

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

$$\backslash begin\{align\}$$

\overline{\pi_{m, n}} &= \overline{\pi_{m, n}^+ + \pi_{m, n}^-}=\overline{\pi_m^{\oplus_{2n + 1}}} + \overline{\overline{\pi_n}^{\oplus_{2m + 1}}} \\

&=\pi_n^{\oplus_{2m + 1}} + \overline{\pi_m}^{\oplus_{2n + 1}} =

\pi_{n, m}^+ + \pi_{n, m}^- = \pi_{n, m} \\

& &&2m, 2n \in \mathbb{N} \\

\overline{\Pi_{m, n}} &= \Pi_{n, m}

\end{align}

where the statement for the group representations follow from {{math|1=exp({{overline|X}}) = {{overline|exp(X)}}}}. It follows that the irreducible representations {{math|(m, n)}} have real matrix representatives if and only if {{math|1=m = n}}. Reducible representations on the form {{math|(m, n) ⊕ (n, m)}} have real matrices too.

File:Richard Brauer.jpg and wife Ilse 1970. Brauer generalized the spin representations of Lie algebras sitting inside Clifford algebras to spin higher than {{sfrac|1|2}}. {{Paragraph break}}Photo courtesy of MFO.]]

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

{{NumBlk||$$\backslash pi(X)(A)\; =\; [\backslash pi(X),\; A],\backslash qquad\; A\; \backslash in\; \backslash operatorname\{End\}(V),\backslash \; X\; \backslash in\; \backslash mathfrak\{g\}.$$|{{EquationRef|I1|I1}}}}

Likewise, a representation {{math|(Π, V)}} of a group {{mvar|G}} yields a representation {{math|Π}} on {{math|End(V)}} of {{mvar|G}}, still denoted {{math|Π}}, given by{{harvnb|Hall|2003}} See appendix D.3

{{NumBlk||$$\backslash Pi(g)(A)\; =\; \backslash Pi(g)A\backslash Pi(g)^\{-1\},\backslash qquad\; A\; \backslash in\; \backslash operatorname\{End\}(V),\backslash \; g\; \backslash in\; G.$$|{{EquationRef|I2|I2}}}}

If {{mvar|π}} and {{math|Π}} are the standard representations on $\backslash R^4$ and if the action is restricted to $\backslash mathfrak\{so\}(3,\; 1)\; \backslash subset\; \backslash text\{End\}(\backslash R^4),$ 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 $\backslash R^n$ or $\backslash Complex^n$) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular.

Applying this to the Lorentz group, if {{math|(Π, V)}} is a projective representation, then direct calculation using {{EquationNote|G5|(G5)}} shows that the induced representation on {{math|End(V)}} is a proper representation, i.e. a representation without phase factors.

In quantum mechanics this means that if {{math|(π, H)}} or {{math|(Π, H)}} is a representation acting on some Hilbert space {{mvar|H}}, then the corresponding induced representation acts on the set of linear operators on {{mvar|H}}. As an example, the induced representation of the projective spin {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}} representation on {{math|End(H)}} is the non-projective 4-vector ({{sfrac|1|2}}, {{sfrac|1|2}}) representation.{{harvnb|Weinberg|2002|loc=Equation 5.4.8.}}

For simplicity, consider only the "discrete part" of {{math|End(H)}}, that is, given a basis for {{mvar|H}}, the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4-vector representation of above on this simplified {{math|End(H)}} has an invariant 4-dimensional subspace that is spanned by the four gamma matrices.{{harvnb|Weinberg|2002|loc=Section 5.4.}} (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, $\backslash mathcal\{Cl\}\_\{3,1\}(\backslash R),$ whose complexification is $\backslash text\{M\}(4,\; \backslash Complex),$ generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the {{math|(0, 0)}}, a pseudoscalar irrep, also the {{math|(0, 0)}}, but with parity inversion eigenvalue {{math|−1}}, see the next section below, the already mentioned vector irrep, {{math|({{sfrac|1|2}}, {{sfrac|1|2}})}}, a pseudovector irrep, {{math|({{sfrac|1|2}}, {{sfrac|1|2}})}} with parity inversion eigenvalue +1 (not −1), and a tensor irrep, {{math|(1, 0) ⊕ (0, 1)}}.{{harvnb|Weinberg|2002|pp=215–216.}} The dimensions add up to {{math|1=1 + 1 + 4 + 4 + 6 = 16}}. In other words,

{{NumBlk||$$\backslash mathcal\{Cl\}\_\{3,1\}(\backslash R)\; =\; (0,0)\; \backslash oplus\; \backslash left(\backslash frac\{1\}\{2\},\; \backslash frac\{1\}\{2\}\backslash right)\; \backslash oplus\; [(1,\; 0)\; \backslash oplus\; (0,\; 1)]\; \backslash oplus\; \backslash left(\backslash frac\{1\}\{2\},\; \backslash frac\{1\}\{2\}\backslash right)\_p\; \backslash oplus\; (0,\; 0)\_p,$$|{{EquationRef|I3|I3}}}}

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

The six-dimensional representation space of the tensor {{math|(1, 0) ⊕ (0, 1)}}-representation inside $\backslash mathcal\{Cl\}\_\{3,1\}(\backslash R)$ has two roles. The{{harvnb|Weinberg|2002|loc=Equation 5.4.6.}}

{{NumBlk||$$\backslash sigma^\{\backslash mu\backslash nu\}\; =\; -\backslash frac\{i\}\{4\}\; \backslash left[\backslash gamma^\backslash mu,\; \backslash gamma^\backslash nu\backslash right],$$|{{EquationRef|I4|I4}}}}

where $\backslash gamma^0,\; \backslash ldots,\; \backslash gamma^3\; \backslash in\; \backslash mathcal\{Cl\}\_\{3,1\}(\backslash R)$ are the gamma matrices, the sigmas, only {{math|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,{{harvnb|Weinberg|2002}} Section 5.4.

{{NumBlk||$$\backslash left[\backslash sigma^\{\backslash mu\backslash nu\},\; \backslash sigma^\{\backslash rho\backslash tau\}\backslash right]\; =\; i\; \backslash left(\backslash eta^\{\backslash tau\backslash mu\}\backslash sigma^\{\backslash rho\backslash nu\}\; +\; \backslash eta^\{\backslash nu\backslash tau\}\backslash sigma^\{\backslash mu\backslash rho\}\; -\; \backslash eta^\{\backslash rho\backslash mu\}\backslash sigma^\{\backslash tau\backslash nu\}\; -\; \backslash eta^\{\backslash nu\backslash rho\}\backslash sigma^\{\backslash mu\backslash tau\}\backslash right),$$|{{EquationRef|I5|I5}}}}

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

The conclusion is that every element of the complexified $\backslash mathcal\{Cl\}\_\{3,1\}(\backslash R)$ in {{math|End(H)}} (i.e. every complex {{gaps|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 $\backslash Complex^4,$ making it a space of bispinors.

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. $\backslash text\{GL\}(n,\backslash R)$ 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.

The (possibly projective) {{math|(m, n)}} representation is irreducible as a representation {{math|SO(3; 1)⁺}}, the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If {{math|1=m = n}} it can be extended to a representation of all of {{math|O(3; 1)}}, the full Lorentz group, including space parity inversion and time reversal. The representations {{math|(m, n) ⊕ (n, m)}} can be extended likewise.{{harvnb|Weinberg|2002| loc=Section 5.7, pp. 232–233.}}

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

{{NumBlk||$$\backslash mathrm\{Ad\}\_P(J\_i)\; =\; PJ\_iP^\{-1\}\; =\; J\_i,\; \backslash qquad\; \backslash mathrm\{Ad\}\_P(K\_i)\; =\; PK\_iP^\{-1\}\; =\; -K\_i.$$|{{EquationRef|F1|F1}}}}

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

{{NumBlk||$$\backslash Pi(P)\backslash pi(B\_i)\backslash Pi(P)^\{-1\}\; =\; \backslash pi(A\_i)$$|{{EquationRef|F2|F2}}}}

holds, where {{math|A}} and {{math|B}} are defined as in the first section. This can hold only if {{math|A_i}} and {{math|B_i}} have the same dimensions, i.e. only if {{math|1=m = n}}. When {{math|m ≠ n}} then {{math|(m, n) ⊕ (n, m)}} can be extended to an irreducible representation of {{math|SO(3; 1)⁺}}, the orthochronous Lorentz group. The parity reversal representative {{math|Π(P)}} does not come automatically with the general construction of the {{math|(m, n)}} representations. It must be specified separately. The matrix {{math|1=β = i γ⁰}} (or a multiple of modulus −1 times it) may be used in the {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}}{{harvnb|Weinberg|2002|loc=Section 5.7, p. 233.}} representation.

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

Time reversal {{math|1=T = diag(−1, 1, 1, 1)}}, acts similarly on $\backslash mathfrak\{so\}(3;\; 1)$ by{{harvnb|Weinberg|2002}} Equation 2.6.5.

{{NumBlk||$$\backslash mathrm\{Ad\}\_T(J\_i)\; =\; TJ\_iT^\{-1\}\; =\; -J\_i,\; \backslash qquad\; \backslash mathrm\{Ad\}\_T(K\_i)\; =\; TK\_iT^\{-1\}\; =\; K\_i.$$|{{EquationRef|F3|F3}}}}

By explicitly including a representative for {{math|T}}, as well as one for {{math|P}}, a representation of the full Lorentz group {{math|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 {{mvar|P^μ}}, in addition to the {{mvar|Jⁱ}} and {{mvar|Kⁱ}} generate the group. These are interpreted as generators of translations. The time-component {{math|P⁰}} is the Hamiltonian {{math|H}}. The operator {{math|T}} satisfies the relation{{harvnb|Weinberg|2002}} Equation following 2.6.6.

{{NumBlk||$$\backslash mathrm\{Ad\}\_\{T\}(iH)\; =\; TiHT^\{-1\}\; =\; -iH$$|{{EquationRef|F4|F4}}}}

in analogy to the relations above with $\backslash mathfrak\{so\}(3;\; 1)$ replaced by the full Poincaré algebra. By just cancelling the {{mvar|i}}'s, the result {{math|1=THT⁻¹ = −H}} would imply that for every state {{math|Ψ}} with positive energy {{mvar|E}} in a Hilbert space of quantum states with time-reversal invariance, there would be a state {{math|Π(T⁻¹)Ψ}} with negative energy {{math|−E}}. Such states do not exist. The operator {{math|Π(T)}} is therefore chosen antilinear and antiunitary, so that it anticommutes with {{mvar|i}}, resulting in {{math|1=THT⁻¹ = H}}, and its action on Hilbert space likewise becomes antilinear and antiunitary.{{harvnb|Weinberg|2002|loc=Section 2.6.}} It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix.For a detailed discussion of the spin 0, {{sfrac|1|2}} and 1 cases, see {{harvnb|Greiner|Reinhardt|1996}}. This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, {{math|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, {{nowrap|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|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 {{math|P}} and {{math|T}}, charge conjugation symmetry {{math|C}}, has nothing directly to do with Lorentz invariance.{{harvnb|Weinberg|2002|loc=Chapter 3.}}

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

{{NumBlk||$$(\backslash Pi(g)f)(x)\; =\; f\backslash left(\backslash Pi\_x(g)^\{-1\}\; x\backslash right),\backslash qquad\; x\; \backslash in\; \backslash R^n,\; f\; \backslash in\; V$$|{{EquationRef|H1|H1}}}}

produces another function {{math|Πf ∈ V}}. Here {{math|Π_x}} is an {{mvar|n}}-dimensional representation, and {{math|Π}} is a possibly infinite-dimensional representation. A special case of this construction is when {{mvar|V}} is a space of functions defined on the a linear group {{mvar|G}} itself, viewed as a {{mvar|n}}-dimensional manifold embedded in $\backslash R^\{m^2\}$ (with {{mvar|m}} the dimension of the matrices).{{harvnb|Rossmann|2002}} See section 6.1 for more examples, both finite-dimensional and infinite-dimensional. This is the setting in which the Peter–Weyl theorem and the Borel–Weil theorem are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations. The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. $\backslash text\{SL\}(2,\backslash Complex).$

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

{{Main|Rotation group SO(3)#Spherical harmonics|Spherical harmonics}}

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

$$L^2\; \backslash left(\backslash mathbb\{S\}^2\backslash right)\; =\; \backslash operatorname\{span\}\; \backslash left\backslash \{\; Y^l\_m,\; l\; \backslash in\; \backslash mathbb\{N\}^+,\; -l\; \backslash leqslant\; m\; \backslash leqslant\; l\; \backslash right\backslash \},$$

where $Y^l\_m$ are the spherical harmonics. An arbitrary square integrable function {{mvar|f}} on the unit sphere can be expressed as{{harvnb|Gelfand|Minlos|Shapiro|1963}}

{{NumBlk||$$f(\backslash theta,\; \backslash varphi)\; =\; \backslash sum\_\{l\; =\; 1\}^\backslash infty\backslash sum\_\{m\; =\; -l\}^l\; f\_\{lm\}\; Y^l\_m(\backslash theta,\; \backslash varphi),$$|{{EquationRef|H2|H2}}}}

where the {{math|f_lm}} are generalized Fourier coefficients.

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

{{NumBlk||$$\backslash begin\{align\}$$

(\Pi(R)f)(\theta(x), \varphi(x)) &=

\sum_{l = 1}^\infty\sum_{m, = -l}^l\sum_{m' = -l}^lD^{(l)}_{mm'}(R) f_{lm'} Y^l_m \left(\theta\left(R^{-1} x\right), \varphi\left(R^{-1}x\right) \right), \\[5pt]

& R \in \mathrm{SO}(3),

x \in \mathbb{S}^2,

\end{align}

|{{EquationRef|H4|H4}}}}

where the {{math|D^l}} are obtained from the representatives of odd dimension of the generators of rotation.

{{Main|Möbius transformation|Lorentz group#Relation to the Möbius group}}

The identity component of the Lorentz group is isomorphic to the Möbius group {{math|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 {{math|a, b, c, d}} acts on the plane according to{{harvnb|Churchill|Brown|2014|loc=Chapter 8 pp. 307–310.}}

{{NumBlk||$$f(z)\; =\; \backslash frac\{a\; z\; +\; b\}\{c\; z\; +\; d\},\; \backslash qquad\; ad\; -\; bc\; \backslash neq\; 0$$.|{{EquationRef|M1|M1}}}}

and can be represented by complex matrices

{{NumBlk||$$\backslash Pi\_f\; =$$

\begin{pmatrix} A & B \\ C & D \end{pmatrix} =

\lambda \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad

\lambda \in \Complex - \{0\}, \operatorname{det} \Pi_f = 1,

|{{EquationRef|M2|M2}}}}

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

{{Main|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{{cite journal|last1=Gonzalez|first1=P. A.|last2=Vasquez|first2=Y.|title=Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity|year=2014|journal= Eur. Phys. J. C |issn=1434-6044|volume=74:2969| issue=7|arxiv=1404.5371| doi=10.1140/epjc/s10052-014-2969-1|page=3|bibcode=2014EPJC...74.2969G|s2cid=118725565}}

{{NumBlk||$$\backslash begin\{align\}$$

w(z) &= P \left\{ \begin{matrix}

a & b & c & \\

\alpha & \beta & \gamma & \; z \\

\alpha' & \beta' & \gamma' &

\end{matrix} \right\} \\ &=

\left(\frac{z - a}{z - b}\right)^\alpha \left(\frac{z - c}{z - b}\right)^\gamma

P \left\{ \begin{matrix}

0 & \infty & 1 & \\

0 & \alpha + \beta + \gamma & 0 & \;\frac{(z - a)(c - b)}{(z - b)(c - a)} \\

\alpha' - \alpha & \alpha + \beta' + \gamma & \gamma' - \gamma &

\end{matrix} \right\}\end{align},

|{{EquationRef|T1|T1}}}}

where the {{math|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{{harvnb|Abramowitz|Stegun|1965|loc=Equation 15.6.5.}}

{{NumBlk||$$P\; \backslash left\backslash \{\; \backslash begin\{matrix\}$$

0 & \infty & 1 & \\

0 & a & 0 & \;z \\

1 - c & b & c - a - b &

\end{matrix} \right\} =

{}_2 F_1(a,\, b;\, c;\, z).

|{{EquationRef|T2|T2}}}}

The set of constants {{math|0, ∞, 1}} in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation.{{harvnb|Simmons|1972|loc=Sections 30, 31.}} Its exponents, i. e. solutions of the indicial equation, for expansion around the singular point {{math|0}} are {{math|0}} and {{math|1 − c}} ,corresponding to the two linearly independent solutions,See {{harvtxt|Simmons|1972|loc=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. and for expansion around the singular point {{math|1}} they are {{math|0}} and {{math|c − a − b}}.{{harvnb|Simmons|1972|loc=Sections 30.}} Similarly, the exponents for {{math|∞}} are {{mvar|a}} and {{mvar|b}} for the two solutions.{{harvnb|Simmons|1972|loc=Section 31.}}

One has thus

{{NumBlk||$$w(z)\; =\; \backslash left(\backslash frac\{z\; -\; a\}\{z\; -\; b\}\backslash right)^\backslash alpha\; \backslash left(\backslash frac\{z\; -\; c\}\{z\; -\; b\}\backslash right)^\backslash gamma\; \{\}\_2F\_1\; \backslash left(\backslash alpha\; +\; \backslash beta\; +\; \backslash gamma,\backslash ,\; \backslash alpha\; +\; \backslash beta\text{'}\; +\; \backslash gamma;\backslash ,\; 1\; +\; \backslash alpha\; -\; \backslash alpha\text{'};\backslash ,\; \backslash frac\{(z\; -\; a)(c\; -\; b)\}\{(z\; -\; b)(c\; -\; a)\}\backslash right),$$|{{EquationRef|T3|T3}}}}

where the condition (sometimes called Riemann's identity){{harvnb|Simmons|1972|loc=Equation 11 in appendix E, chapter 5.}}

$$\backslash alpha\; +\; \backslash alpha\text{'}\; +\; \backslash beta\; +\; \backslash beta\text{'}\; +\; \backslash gamma\; +\; \backslash gamma\text{'}\; =\; 1$$

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

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

{{NumBlk||$$u(\backslash Lambda)(z)\; =\; \backslash frac\{Az\; +\; B\}\{Cz\; +\; D\},$$|{{EquationRef|T4|T4}}}}

where {{math|A,  B,  C,  D}} are the entries in

{{NumBlk|||{{EquationRef|T5|T5}}}}

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

Define

{{NumBlk||$$[\backslash Pi(\backslash Lambda)\; P](z)\; =\; P[u(\backslash Lambda)(z)],$$|{{EquationRef|T6|T6}}}}

where {{mvar|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

{{NumBlk||$$[\backslash Pi(\backslash Lambda)\; P](u)\; =$$

P \left\{ \begin{matrix}

\eta & \zeta & \theta & \\

\alpha & \beta & \gamma & \;u \\

\alpha' & \beta' & \gamma' &

\end{matrix} \right\},

|{{EquationRef|T6|T6}}}}

where

{{NumBlk||$$\backslash eta\; =\; \backslash frac\{Aa\; +\; B\}\{Ca\; +\; D\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash zeta\; =\; \backslash frac\{Ab\; +\; B\}\{Cb\; +\; D\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash theta\; =\; \backslash frac\{Ac\; +\; B\}\{Cc\; +\; D\}.$$|{{EquationRef|T7|T7}}}}

==Infinite-dimensional unitary representations==

The Lorentz group {{math|SO(3; 1)⁺}} and its double cover $\backslash text\{SL\}(2,\backslash Complex)$ also have infinite dimensional unitary representations, studied independently by {{harvtxt|Bargmann|1947}}, {{harvtxt|Gelfand|Naimark|1947}} and {{harvtxt|Harish-Chandra|1947}} at the instigation of Paul Dirac.{{harvnb|Langlands|1985|p=205.}}{{harvnb|Varadarajan|1989|loc=Sections 3.1. 4.1.}} This trail of development begun with {{harvtxt|Dirac|1936}} where he devised matrices {{math|U}} and {{math|B}} necessary for description of higher spin (compare Dirac matrices), elaborated upon by {{harvtxt|Fierz|1939}}, see also {{harvtxt|Fierz|Pauli|1939}}, and proposed precursors of the Bargmann-Wigner equations.{{harvnb|Langlands|1985|p=203.}} In {{harvtxt|Dirac|1945}} he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors."This is as close as one comes to the source of the theory of infinite-dimensional representations of semisimple and reductive groups...", {{harvtxt|Langlands|1985|p=204.}}, referring to an introductory passage in Dirac's 1945 paper. 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 {{harvtxt|Harish-Chandra|1951}} and {{harvtxt|Gelfand|Graev|1953}}, based on an analogue for $\backslash text\{SL\}(2,\backslash Complex)$ of the integration formula of Hermann Weyl for compact Lie groups.{{harvnb|Varadarajan|1989|loc=Section 4.1.}} Elementary accounts of this approach can be found in {{harvtxt|Rühl|1970}} and {{harvtxt|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 $\backslash R.$ This theory is discussed in {{harvtxt|Takahashi|1963}}, {{harvtxt|Helgason|1968}}, {{harvtxt|Helgason|2000}} and the posthumous text of {{harvtxt|Jorgenson|Lang|2008}}.

The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup {{mvar|B}} of $G\; =\; \backslash text\{SL\}(2,\backslash Complex).$ Since the one-dimensional representations of {{mvar|B}} correspond to the representations of the diagonal matrices, with non-zero complex entries {{mvar|z}} and {{math|z⁻¹}}, they thus have the form

for {{mvar|k}} an integer, {{mvar|ν}} real and with {{mvar|1=z = re^iθ}}. The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when {{mvar|k}} is replaced by {{math|−k}}. By definition the representations are realized on {{math|L²}} sections of line bundles on $G/B\; =\; \backslash mathbb\{S\}^2,$ which is isomorphic to the Riemann sphere. When {{math|1=k = 0}}, these representations constitute the so-called spherical principal series.

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

Using the identification between the Riemann sphere minus a point and $\backslash Complex,$ the principal series can be defined directly on $L^2(\backslash Complex)$ by the formula{{harvnb|Gelfand|Graev|Pyatetskii-Shapiro|1969}}

Irreducibility can be checked in a variety of ways:

• The representation is already irreducible on {{mvar|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 {{math|1=G = B ∪ BsB}} where {{mvar|s}} is the Weyl group element{{harvnb|Knapp|2001|loc=Chapter II.}} .
• The action of the Lie algebra $\backslash mathfrak\{g\}$ of {{mvar|G}} can be computed on the algebraic direct sum of the irreducible subspaces of {{mvar|K}} can be computed explicitly and the it can be verified directly that the lowest-dimensional subspace generates this direct sum as a $\backslash mathfrak\{g\}$-module.{{harvnb|Harish-Chandra|1947}}{{harvnb|Taylor|1986}}

The for {{math|0 < t < 2}}, the complementary series is defined on $L^2(\backslash Complex)$ for the inner product{{harvnb|Knapp|2001}} Chapter 2. Equation 2.12.

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

with the action given by{{harvnb|Bargmann|1947}}{{harvnb|Gelfand|Graev|1953}}

The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of {{mvar|K}}, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of {{math|1=K = SU(2)}}. Irreducibility can be proved by analyzing the action of $\backslash mathfrak\{g\}$ on the algebraic sum of these subspaces or directly without using the Lie algebra.{{harvnb|Gelfand|Naimark|1947}}{{harvnb|Takahashi|1963|page=343.}}

The only irreducible unitary representations of $\backslash text\{SL\}(2,\backslash Complex)$ are the principal series, the complementary series and the trivial representation.

Since {{math|−I}} acts as {{math|(−1)^k}} on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided {{mvar|k}} is taken to be even.

To decompose the left regular representation of {{mvar|G}} on $L^2(G)$ only the principal series are required. This immediately yields the decomposition on the subrepresentations $L^2(G/\backslash \{\backslash pm\; I\backslash \}),$ 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 {{math|1=k = 0}}.)

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

$$\backslash begin\{align\}$$

(\lambda(g)f)(x) &= f\left(g^{-1}x\right) \\

(\rho(g)f) (x) &= f(xg)

\end{align}

Now if {{mvar|f}} is an element of {{math|C_c(G)}}, the operator $\backslash pi\_\{\backslash nu,\; k\}(f)$ defined by

$$\backslash pi\_\{\backslash nu,\; k\}(f)\; =\; \backslash int\_G\; f(g)\backslash pi(g)\backslash ,\; dg$$

is Hilbert–Schmidt. Define a Hilbert space {{mvar|H}} by

$$H\; =\; \backslash bigoplus\_\{k\backslash geqslant\; 0\}\; \backslash text\{HS\}\; \backslash left(L^2(\backslash Complex)\backslash right)\; \backslash otimes\; L^2\; \backslash left(\backslash R,\; c\_k\backslash sqrt\{\backslash nu^2\; +\; k^2\}\; d\backslash nu\; \backslash right),$$

where

$$c\_k\; =\; \backslash begin\{cases\}$$

\frac{1}{4\pi^{3/2}} & k = 0 \\

\frac{1}{(2\pi)^{3/2}} & k \neq 0

\end{cases}

and $\backslash text\{HS\}\backslash left(L^2(\backslash Complex)\backslash right)$ denotes the Hilbert space of Hilbert–Schmidt operators on $L^2(\backslash Complex).$Note that for a Hilbert space {{mvar|H}}, {{math|HS(H)}} may be identified canonically with the Hilbert space tensor product of {{mvar|H}} and its conjugate space. Then the map {{mvar|U}} defined on {{math|C_c(G)}} by

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

extends to a unitary of $L^2(G)$ onto {{mvar|H}}.

The map {{mvar|U}} satisfies the intertwining property

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

If {{math|f₁, f₂}} are in {{math|C_c(G)}} then by unitarity

$$(f\_1,\; f\_2)\; =\; \backslash sum\_\{k\backslash geqslant\; 0\}\; c\_k^2\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{Tr\}\; \backslash left(\backslash pi\_\{\backslash nu,k\}(f\_1)\backslash pi\_\{\backslash nu,k\}(f\_2)^*\backslash right)\; \backslash left(\backslash nu^2\; +\; k^2\backslash right)\; \backslash ,\; d\backslash nu.$$

Thus if $f\; =\; f\_1\; *\; f\_2^*$ denotes the convolution of $f\_1$ and $f\_2^*,$ and $f\_2^*(g)=\backslash overline\{f\_2(g^\{-1\})\},$ then{{harvnb|Knapp|2001|loc=Equation 2.24.}}

$$f(1)\; =\; \backslash sum\_\{k\backslash geqslant\; 0\}\; c\_k^2\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{Tr\}\; \backslash left(\backslash pi\_\{\backslash nu,k\}(f)\; \backslash right)\; \backslash left(\backslash nu^2\; +\; k^2\backslash right)\backslash ,\; d\backslash nu.$$

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 $f\_i\; \backslash in\; L^2(G).$ 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 {{mvar|f}}. It can be extended to much wider classes of functions satisfying mild differentiability conditions.

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 {{math|Π_H}} on a Hilbert space {{mvar|H}} of {{math|SO(3; 1)⁺}} is at hand.{{harvnb|Folland|2015|loc=Section 3.1.}} Since {{math|SO(3)}} is a subgroup, {{math|Π_H}} is a representation of it as well. Each irreducible subrepresentation of {{math|SO(3)}} is finite-dimensional, and the {{math|SO(3)}} representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of {{math|SO(3)}} if {{math|Π_H}} is unitary.{{harvnb|Folland|2015|loc=Theorem 5.2.}}

The steps are the following:{{harvnb|Tung|1985|loc=Section 10.3.3.}}

1. Choose a suitable basis of common eigenvectors of {{math|J²}} and {{math|J₃}}.
2. Compute matrix elements of {{math|J₁, J₂, J₃}} and {{math|K₁, K₂, K₃}}.
3. Enforce Lie algebra commutation relations.
4. Require unitarity together with orthonormality of the basis.If finite-dimensionality is demanded, the results is the {{math|(m, n)}} representations, see {{harvtxt|Tung|1985|loc=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 {{harvtxt|Harish-Chandra|1947}}.

One suitable choice of basis and labeling is given by

$$\backslash left\; |j\_0\backslash ,\; j\_1;j\backslash ,\; m\backslash right\backslash rangle.$$

If this were a finite-dimensional representation, then {{math|j₀}} would correspond the lowest occurring eigenvalue {{math|j(j + 1)}} of {{math|J²}} in the representation, equal to {{math|{{!}}m − n{{!}}}}, and {{math|j₁}} would correspond to the highest occurring eigenvalue, equal to {{math|m + n}}. In the infinite-dimensional case, {{math|j₀ ≥ 0}} retains this meaning, but {{math|j₁}} does not. For simplicity, it is assumed that a given {{mvar|j}} occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown{{harvnb|Harish-Chandra|1947|loc=Footnote p. 374.}} that the assumption is possible to avoid (with a slightly more complicated calculation) with the same results.

The next step is to compute the matrix elements of the operators {{math|J₁, J₂, J₃}} and {{math|K₁, K₂, K₃}} forming the basis of the Lie algebra of $\backslash mathfrak\{so\}(3;\; 1).$ The matrix elements of $J\_\backslash pm\; =\; J\_1\; \backslash pm\; iJ\_2$ and $J\_3$ (the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by{{harvnb|Tung|1985|loc=Equations 7.3–13, 7.3–14.}}{{harvnb|Harish-Chandra|1947|loc=Equation 8.}}

$$\backslash begin\{align\}$$

\left\langle j\, m \right|J_+ \left| j\, m - 1 \right\rangle = \left\langle j\, m - 1 \right|J_- \left| j\, m \right\rangle &= \sqrt{(j + m)(j - m + 1)}, \\

\left\langle j\, m \right|J_3 \left| j\, m \right\rangle &= m,

\end{align}

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

Due to the commutation relations $$[J\_i,K\_j]\; =\; i\; \backslash epsilon\_\{ijk\}\; K\_k,$$ the triple {{math|(K₁, K₂, K₃) ≡ K}} is a vector operator{{harvnb|Hall|2015|loc=Proposition C.7.}} and the Wigner–Eckart theorem{{harvnb|Hall|2015|loc=Appendix C.2.}} applies for computation of matrix elements between the states represented by the chosen basis.{{harvnb|Tung|1985|loc=Step II section 10.2.}} The matrix elements of

$$\backslash begin\{align\}$$

K^{(1)}_0 &= K_3,\\

K^{(1)}_{\pm 1} &= \mp\frac{1}{\sqrt 2}(K_1 \pm iK_2),

\end{align}

where the superscript {{math|(1)}} signifies that the defined quantities are the components of a spherical tensor operator of rank {{math|1=k = 1}} (which explains the factor {{math|{{sqrt|2}}}} as well) and the subscripts {{math|0, ±1}} are referred to as {{mvar|q}} in formulas below, are given by{{harvnb|Tung|1985|loc=Equations 10.3–5. Tung's notation for Clebsch–Gordan coefficients differ from the one used here.}}

$$\backslash begin\{align\}$$

\left\langle j' m'\left|K^{(1)}_0 \right|j\,m\right\rangle &= \left \langle j' \, m' \,k = 1 \,q = 0 | j \, m \right \rangle \left \langle j \left \| K^{(1)} \right \| j' \right \rangle,\\

\left\langle j' m'\left|K^{(1)}_{\pm 1}\right |j\,m\right\rangle &= \left \langle j' \, m' \, k= 1 \,q = \pm 1 | j \, m \right \rangle \left \langle j \left \| K^{(1)} \right \| j' \right \rangle.

\end{align}

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

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

$$[K\_\backslash pm,\; K\_3]\; =\; \backslash pm\; J\_\backslash pm,\; \backslash quad\; [K\_+,\; K\_-]\; =\; -2J\_3.$$

This results in a set of equations{{harvnb|Tung|1985|loc=Equation VII-3.}} for which the solutions are{{harvnb|Tung|1985|loc=Equations 10.3–5, 7, 8.}}

$$\backslash begin\{align\}$$

\left \langle j \left \| K^{(1)} \right \| j \right \rangle &= i\frac{j_1j_0}{\sqrt{j(j+1)}},\\

\left \langle j \left \| K^{(1)} \right \| j-1 \right \rangle &= -B_j\xi_j\sqrt{j(2j-1)},\\

\left \langle j-1 \left \| K^{(1)} \right \| j \right \rangle &= B_j\xi_j^{-1}\sqrt{j(2j+1)},

\end{align}

where

$$B\_j\; =\; \backslash sqrt\{\backslash frac\{(j^2\; -\; j\_0^2)(j^2\; -\; j\_1^2)\}\{j^2(4j^2\; -\; 1)\}\},\; \backslash quad\; j\_0=0,\; \backslash tfrac\{1\}\{2\},\; 1,\; \backslash ldots\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; j\_1,\; \backslash xi\_j\; \backslash in\; \backslash Complex.$$

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

$$K\_\backslash pm^\backslash dagger\; =\; K\_\backslash mp,\backslash quad\; K\_3^\backslash dagger\; =\; K\_3.$$

This translates to{{harvnb|Tung|1985|loc=Equation VII-9.}}

$$\backslash begin\{align\}$$

\left \langle j \left \| K^{(1)} \right \| j \right \rangle &= \overline{\left \langle j \left \| K^{(1)} \right \| j \right \rangle},\\

\left \langle j \left \| K^{(1)} \right \| j - 1 \right \rangle &= -\overline{\left \langle j - 1 \left \| K^{(1)} \right \| j \right \rangle},

\end{align}

leading to{{harvnb|Tung|1985|loc=Equations VII-10, 11.}}

$$\backslash begin\{align\}$$

j_0 \left(j_1 + \overline{j_1}\right) &= 0, \\

\left|B_j\right| \left(\left|\xi_j\right|^2 - e^{-2i\beta_j}\right) &= 0,

\end{align}

where {{math|β_j}} is the angle of {{math|B_j}} on polar form. For {{math|{{!}}B_j{{!}} ≠ 0}} follows $\backslash left|\backslash xi\_j\backslash right|^2\; =\; 1$ and $\backslash xi\_j\; =\; 1$ is chosen by convention. There are two possible cases:

• $\backslash underline\{j\_1\; +\; \backslash overline\{j\_1\}\; =\; 0.\}$ In this case {{math|1=j₁ = − iν}}, {{mvar|ν}} real,{{harvnb|Tung|1985|loc=Equations VII-12.}} $$\backslash left\; \backslash langle\; j\; \backslash left\; \backslash |\; K^\{(1)\}\; \backslash right\; \backslash |\; j\; \backslash right\; \backslash rangle\; =\; \backslash frac\{\backslash nu\; j\_0\}\{j(j\; +\; 1)\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; B\_j\; =\; \backslash sqrt\{\backslash frac\{(j^2\; -\; j\_0^2)(j^2\; +\; \backslash nu^2)\}\{4j^2\; -\; 1\}\}$$ This is the principal series. Its elements are denoted $(j\_0,\; \backslash nu),\; 2j\_0\; \backslash in\; \backslash N,\; \backslash nu\; \backslash in\; \backslash R.$
• $\backslash underline\{j\_0=0.\}$ It follows:{{harvnb|Tung|1985|loc=Equations VII-13.}} $$\backslash left\; \backslash langle\; j\; \backslash left\; \backslash |\; K^\{(1)\}\; \backslash right\; \backslash |\; j\; \backslash right\; \backslash rangle\; =\; 0\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; B\_j\; =\; \backslash sqrt\{\backslash frac\{j^2\; -\; \backslash nu^2\}\{4j^2\; -\; 1\}\}$$ Since {{math|1=B₀ = B_j0}}, {{math|B{{supsub|2|j}}}} is real and positive for {{math|1=j = 1, 2, ...}}, leading to {{math|−1 ≤ ν ≤ 1}}. This is complementary series. Its elements are denoted {{math|(0, ν), −1 ≤ ν ≤ 1}}

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

The metric of choice is given by {{math|1=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:

$$\backslash begin\{align\}$$

J_1 = J^{23} = -J^{32} &= i\begin{pmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&0&1&0 \end{pmatrix},&

K_1 = J^{01} = -J^{10} &= i\begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix},\\[8pt]

J_2 = J^{31} = -J^{13} &= i\begin{pmatrix} 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&-1&0&0 \end{pmatrix},&

K_2 = J^{02} = -J^{20} &= i\begin{pmatrix} 0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0 \end{pmatrix},\\[8pt]

J_3 = J^{12} = -J^{21} &= i\begin{pmatrix} 0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0 \end{pmatrix},&

K_3 = J^{03} = -J^{30} &= i\begin{pmatrix} 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0 \end{pmatrix}.\\[8pt]

\end{align}

The commutation relations of the Lie algebra $\backslash mathfrak\{so\}(3;\; 1)$ are:{{harvnb|Weinberg|2002|loc=Equation 2.4.12.}}

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

In three-dimensional notation, these are{{harvnb|Weinberg|2002|loc=Equations 2.4.18–2.4.20.}}

\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 {{mvar|J}} above and in the sequel should be observed.

For example, a typical boost and a typical rotation exponentiate as,

symmetric and orthogonal, respectively.

File:Paul Dirac, 1933.jpg transform under the {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}}-representation. Dirac discovered the gamma matrices in his search for a relativistically invariant equation, then already known to mathematicians.]]

By taking, in turn, {{math|1=m = {{sfrac|1|2}}, n = 0}} and {{math|1=m = 0, n = {{sfrac|1|2}}}} and by setting

$$J\_i^\{\backslash left(\; \backslash frac\{1\}\{2\}\; \backslash right)\}\; =\; \backslash frac\{1\}\{2\}\backslash sigma\_i$$

in the general expression {{EquationNote|G1|(G1)}}, and by using the trivial relations {{math|1=1₁ = 1}} and {{math|1=J⁽⁰⁾ = 0}}, it follows

{{NumBlk||$$\backslash begin\{align\}$$

\pi_{\left( \frac{1}{2}, 0 \right)}(J_i) &= \frac{1}{2} \left( \sigma_i \otimes 1_{(1)} + 1_{(2)} \otimes J^{(0)}_i \right) = \frac{1}{2}\sigma_i \\

\pi_{\left( \frac{1}{2}, 0 \right)}(K_i) &= \frac{-i}{2} \left( 1_{(2)} \otimes J^{(0)}_i - \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^{(0)}_i \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^{(0)}_i \otimes 1_{(2)} \right) = \frac{-i}{2}\sigma_i

\end{align}

| {{EquationRef|W1|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) {{math|V_L}} and {{math|V_R}}, whose elements {{math|Ψ_L}} and {{math|Ψ_R}} are called left- and right-handed Weyl spinors respectively. Given

$$\backslash left(\; \backslash pi\_\{\backslash left(\backslash frac\{1\}\{2\},\; 0\; \backslash right)\},\; V\_\backslash text\{L\}\; \backslash right)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash left(\; \backslash pi\_\{\backslash left(\; 0,\; \backslash frac\{1\}\{2\}\; \backslash right)\},\; V\_\backslash text\{R\}\; \backslash right)$$

their direct sum as representations is formed,{{harvnb|Weinberg|2002|loc=Equations 5.4.19, 5.4.20.}}

{{NumBlk||$$\backslash begin\{align\}$$

\pi_{\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)}\left(J_i\right) &= \frac{1}{2} \begin{pmatrix} \sigma_i&0\\ 0&\sigma_i \end{pmatrix} \\[8pt]

\pi_{\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)}\left(K_i\right) &= \frac{i}{2} \begin{pmatrix} \sigma_i&0\\ 0&-\sigma_i \end{pmatrix}

\end{align}

| {{EquationRef|D1|D1}}

}}

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

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.{{harvnb|Zwiebach|2004|loc=Section 12.8.}}{{harvnb|Bekaert|Boulanger|2006|p=48.}} 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.{{harvnb|Zwiebach|2004|loc=Section 18.8.}}

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

• Bargmann–Wigner equations
• Dirac algebra
• Gamma matrices
• Lorentz group
• Möbius transformation
• Poincaré group
• Representation theory of the Poincaré group
• Symmetry in quantum mechanics
• Wigner's classification

{{reflist|group=nb}}

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{{Relativity}}

{{Manifolds}}

Category:Representation theory of Lie groups

Category:Special relativity

Category:Quantum mechanics