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{{Lie groups |Physics}}
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The [[Lorentz group]], a [[Lie group]] on which [[special relativity]] is based, has a variety of [[Representations of Lie groups|representations]]. Many of these representations are important in [[theoretical physics]] in the description of [[particle]]s in [[relativistic quantum mechanics]], as well as fields in [[quantum field theory]]. This representation theory provides a theoretical ground for the concept of [[spin (physics)|spin]] (which can be either [[integer]] or [[half-integer]] in the unit of the [[reduced Planck constant]] ℏ); representations with half-integer weights are normally constructed out of [[spinor]]s.
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The group may also be represented in terms of a set of functions defined on the [[Riemann sphere]]. These are the [[Riemann P-function]]s, which are expressible as [[hypergeometric function]]s. The [[identity component]] {{math|SO(3;1)<sup>+</sup>}} of the Lorentz group is [[isomorphism|isomorphic]] to the [[Möbius group]], and hence any representation of the Lorentz group is necessarily a representation of the Möbius group.
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The subgroup [[Rotation group SO(3)|SO(3)]] with [[representation theory of SU(2)|its representation theory]] form a simpler theory, but the two are related (see details [[#The Lie algebra|below]]) and both are prominent in [[theoretical physics]] as descriptions of spin, [[angular momentum]], and other things related to [[rotation]].
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== Finite-dimensional representations ==
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Representation theory of groups in general, and Lie groups is particular, is a very rich subject. The full Lorentz group makes no exception. The Lorentz group has some properties that makes it "agreeable" and other that makes it "not very agreeable" within the context of representation theory. The group is [[Semisimple Lie algebra|semisimple]], and also [[simple (abstract algebra)|simple]]. It is not [[Connected set|connected]], and none of its components are [[simply connected space|simply connected]]. Perhaps most importantly, the Lorentz group is not [[compact space|compact]].


For finite-dimensional representations, the presence of semisimplicity means the Lorentz group can be much be dealt with the same way as simple groups - all representations are built from the irreducible ones. But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, can not be dealt with in all the aspects of the simple framework that applies to simply connected, compact groups. Non-compactness implies that no nontrivial finite-dimensional unitary representations exist. Lack of simple connectedness gives rise to [[spin representation]]s of the group. The non-connectedness means that representations of the full Lorentz group (including time reversal and space parity inversion) has to dealt with separately.
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=== History ===
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The development of the finite-dimensional representation theory of the Lorentz group mostly follows that of the subject in general. Lie groups originated with [[Sophus Lie]] in 1873. By 1888 the [[list of simple lie groups|classification of simple Lie algebras]] was essentially completed by [[Wilhelm Killing]]. In 1913 the [[Cartan's theorem|theorem of highest weight]] for representations of simple Lie algebras, the path that will be followed here, was completed by [[Élie Cartan]]. [[Richard Brauer]] was 1935-1938 largely responsible for the development of the connection between spin representations and [[Clifford algebra]]s. The Lorentz group has also historically received special attention in representation theory, see [[Representation theory of the Lorentz group#infinite-dimensional unitary representations#history|History of infinite-dimensional unitary representations]] below, due to its exceptional importance in physics. [[Hermann Weyl]], a mathematician who also made major contributions to the general theory, and the physicist [[Eugene Wigner|E.P Wigner]] made substantial contributions over the years. Physicist [[Paul Dirac|P.A.M Dirac]] was perhaps the first to manifestly knit everything together in a practical application of major lasting importance with the [[Dirac equation]] in 1928.


===The Lie algebra===
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According to the general [[representation theory]] of [[Lie group]]s, one first looks for the representations of the [[complexification]], '''so'''(3;1)<sub>'''C'''</sub> of the [[Lie algebra]] '''so'''(3;1) of the [[Lorentz group]]. A convenient basis for '''so'''(3;1) is given by the three [[Explicit formulas|generators]] {{math|''J''<sub>''i''</sub>}} of [[rotation]]s and the three generators {{math|''K''<sub>''i''</sub>}} of [[Lorentz boost|boost]]s as described in [[#Conventions and Lie algebra bases|Conventions]] below. Now [[Complexification (Lie group)|complexify the Lie algebra]], and then change basis to the components of:
 
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:<math>\mathbf{A} = \frac{\mathbf{J} + i \mathbf{K}}{2}\,,\quad \mathbf{B} = \frac{\mathbf{J} - i \mathbf{K}}{2}\,.</math>
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In this new basis, one checks that the components of {{math|'''A''' {{=}} (''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>)}} and {{math|'''B''' {{=}} (''B''<sub>1</sub>, ''B''<sub>2</sub>,''B''<sub>3</sub>)}} separately satisfy the [[commutation relation]]s of the Lie algebra '''[[su(2)]]''' and moreover that they commute with each other<ref>{{harvnb|Weinberg|2003}}, Chapter 5</ref>
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:<math>\left[A_i ,A_j\right] = i\varepsilon_{ijk}A_k\,,\quad \left[B_i ,B_j\right] = i\varepsilon_{ijk}B_k\,,\quad \left[A_i ,B_j\right] = 0\,,</math>
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where {{math|''i'', ''j'', ''k''}} are indices which each take values {{math|1, 2, 3}}, and {{math|''ε<sub>ijk</sub>''}} is the three-dimensional [[Levi-Civita symbol]]. Let {{math|'''A'''<sub>'''C'''</sub>}} and {{math|'''B'''<sub>'''C'''</sub>}} denote the complex [[linear span]] of {{math|A}} and {{math|B}} respectively. One has the isomorphisms
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{{NumBlk|:|<math>\mathfrak{so}(3,1)_C \cong \mathbf{A}_C \oplus \mathbf{B}_C \cong \mathfrak{sl}(2,C) \oplus \mathfrak{sl}(2,C) \cong \mathfrak{sl}(2,C) \oplus i\mathfrak{sl}(2,C) = \mathfrak{sl}(2,C)_C \,,</math><ref name="Hall 2003">{{harvnb|Hall|2003}}, Chapter 6</ref>|{{EquationRef|A1}}}}
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where [[SL(2,C)|sl(2, '''C''')]] is the complexification of '''su'''(2) ≈ {{math|A}} ≈ {{math|B}}. The utility of these isomorphisms comes from the fact that all irreducible [[Representation theory of SU(2)|representations of '''su'''(2)]] are known. Every irreducible representation of '''su'''(2) is isomorphic to one of the [[highest weight representation]]s. Moreover, there is a one-to-one correspondence between linear representations of '''su'''(2) and complex linear representations of '''sl'''(2, '''C''').<ref>{{harvnb|Hall|2003}}, Chapter 4</ref>
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====The unitarian trick====
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Now, the representations of {{math|sl(2;C) ⊕ sl(2;C)}}, which is the Lie algebra of {{math|SL(2;'''C''') × SL(2;'''C''')}}, are supposed to be irreducible. This means that they must be tensor products of complex linear representations of {{math|sl(2;C)}}, as can be seen by restriction to the subgroup {{math|SU(2) × SU(2) ⊂ SL(2;C) × SL(2;C)}}, a compact group to which the [[Peter-Weyl theorem]] applies.<ref name="Knapp 2001">{{harvnb|Knapp|2001}}</ref> The irreducible unitary representations of {{math|SU(2) × SU(2)}} are precisely the tensor products of irreducible unitary representations of {{math|SU(2)}} by the Peter-Weyl theorem. These stand in one-to-one correspondence with the [[holomorphic]] representations of {{math|SL(2;'''C''') × SL(2;'''C''')}}<ref name="Knapp 2001"/> and these, in turn, are in one-to-one correspondence with the complex linear representations of {{math|sl(2;C) ⊕ sl(2;C)}} because {{math|SL(2;'''C''') × SL(2;'''C''')}} is simply connected.<ref name="Knapp 2001"/><ref>A somewhat simpler argument, or at least an analogy, related to the [[Stone Weierstrass theorem]] is that polynomials in {{math|''x'' + ''iy''}} and {{math|''x'' - ''iy''}} (but not {{math|''z'' {{=}} ''x'' + ''iy''}} alone) can uniformly approximate any continuous real valued function on the group manifold.</ref> For {{math|sl(2;C)}}, there exists the highest weight representations, here indexed by {{math|μ}} for {{math|μ}} = {{math|0,1,...}}. The tensor products of two complex linear factors then form the irreducible complex linear representations of {{math|sl(2;C) ⊕ sl(2;C)}}. For reference, if {{math|(π<sub>1</sub>, ''U'')}} and {{math|(π<sub>2</sub>, ''V'')}} are representations of a Lie algebra {{math|'''g'''}}, then their tensor product {{math|(π<sub>1</sub> ⊗ π<sub>2</sub>, ''U'' ⊗ ''V'')}} is given by either of


:<math>\pi_1\otimes\pi_2(X) = \pi_1(X) \otimes \mathrm{Id}_V + \mathrm{Id}_U \otimes \pi_2(X), \quad X \in \mathbf{g}, \qquad \pi_1\otimes\pi_2(X, Y) = \pi_1(X) \otimes \mathrm{Id}_V + \mathrm{Id}_U \otimes \pi_2(Y), \quad X,Y \in \mathbf{g},</math><ref name="Hall 2003"/><ref>Tensor products of representations, {{math|π<sub>g</sub>⊗π<sub>h</sub>}} of {{math|'''g''' ⊕ '''h'''}} can, when both factors come from the same Lie algebra {{math|1=('''h''' = '''g''')}}, either be thought of as a representation of {{math|'''g'''}} or {{math|'''g''' ⊕ '''g'''}}.</ref>
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where {{math|Id}} is the identity operator. Here, the latter interpretation is intended. These manipulations to obtain representations of a non-compact algebra (here '''so'''(3;1)), and subsequently the non-compact group itself, from qualitative knowledge about unitary representations of a compact group (here '''SU'''(2)) is a variant of Weyls so-called [[unitarian trick]]. The not necessarily complex linear representations of {{math|sl(2;C)}} come using another variant of the same trick as is shown in the last Lie algebra isomorphism in {{EquationNote|A1}}. The complex linear representations of the complexification of {{math|sl(2;C)}}, {{math|sl(2;C)<sub>C</sub>}} stand in one-to-one correspondence with the real linear representations of {{math|sl(2;C)}}.<ref name="Knapp 2001"/> The set of all, ''at least real linear'', irreducible representations of {{math|sl(2;C)}} are thus indexed by a pair {{math|(μ, ν)}}. The complex linear ones, corresponding precisely to the complexification of the real linear {{math|su(2)}} representations, are of the form {{math|(μ, 0)}}, while the conjugate linear ones are the {{math|(0, ν)}}.<ref name="Knapp 2001"/> Representations on the form {{math|(ν, ν)}} or {{math|(μ, ν)}} ⊕ {{math|(ν, μ)}} are given by ''real'' matrices (the latter is not irreducible).
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Via the displayed isomorphisms and knowledge of the complex linear irreducible representations of {{math|sl(2, C) ⊕ sl(2; C}}, upon solving for {{math|J}} and {{math|K}}, all irreducible representations of '''so'''(3;1)<sub>'''C'''</sub>, and, by restriction, those of '''so'''(3;1) are known. It's worth noting that the representations of '''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.<ref name="Hall 2003"/> Since '''so'''(3;1) is [[semisimple Lie algebra|semisimple]],<ref name="Hall 2003"/> all its representations, not necessarily irreducible, can be built up as [[direct sum]]s of the irreducible ones.
 
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Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers {{math|m}} = {{math|μ/2}} and {{math|n}} = {{math|ν/2}}, conventionally written as one of
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:<math>(m,n) \equiv D^{(m,n)} \equiv D^{(m)} \otimes D^{(n)} \equiv \pi_{m} \otimes \pi_{n} \equiv \pi_{m,n},</math>
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where {{math|&otimes;}} denotes the [[tensor product]] Let {{math|''π''<sub>(''m'',''n'')</sub>: '''so'''(3;1) → '''gl'''(''V'')}}, where {{mvar|V}} is a vector space, denote the irreducible representations of {{math|'''so'''(3;1)}} according to the this classification. These are, up to a [[Matrix similarity|similarity transformation]], uniquely given by
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{{NumBlk|:|<math>\begin{align}
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\pi_{m,n}(J_i) & = 1_{(2m+1)}\otimes J^{(n)}_i + J^{(m)}_i\otimes 1_{(2n+1)}\\
 
\pi_{m,n}(K_i) & = i(1_{(2m+1)}\otimes J^{(n)}_i - J^{(m)}_i \otimes 1_{(2n+1)}),
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\end{align}</math><ref name="Weinberg 2002"/>|{{EquationRef|A2}}}}
 
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where the {{math|'''''J'''''<sup>(n)</sup> {{=}} (''J''<sup>(n)</sup><sub>1</sub>,''J''<sup>(n)</sup><sub>2</sub>,''J''<sup>(n)</sup><sub>3</sub>)}} are the {{math|(2''n'' + 1)}}-dimensional irreducible spin {{mvar|n}} [[Representation theory of SU(2)|representations of {{math|'''so'''(3)}}]] ≈ {{math|'''su'''(2)}} and {{math|1<sub>''n''</sub>}} is the {{mvar|n}}-dimensional [[unit matrix]]. [[#Explicit formulas|Explicit formulas]] on component form are given at the end of the article.
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The highest weight for a '''sl'''(2,'''C''') representation coincides with the highest [[eigenvalue]] of the representative of the third [[Pauli matrix]] {{math|''σ''<sub>3</sub> ∈ '''sl'''(2,'''C''')}} and is in quantum mechanics thus related to the maximum value the spin ''z''-component can take on for a particle whose state vector transform under the representation in question.
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[[File:Lorentz group commutative diagram 2.svg|300px|right|thumb|Here {{mvar|V}} is a finite-dimensional vector space, GL({{mvar|V}}) is the set of all invertible linear transformations on {{mvar|V}} and {{nowrap|'''{{math|gl}}'''({{mvar|V}})}} is its Lie algebra. The maps {{math|π}} and {{math|Π}} are Lie algebra and group representations respectively, and exp is the exponential mapping. The diagram commutes only up to a sign if {{math|Π}} is projective.]]Given a '''so'''(3;1) representation, one may try to construct a representation of SO(3;1)<sup>+</sup>, the identity component of the Lorentz group, by using the [[exponential mapping]]. If {{mvar|X}} is an element of '''so'''(3;1) in the standard representation, then


{{NumBlk|:|<math>\Lambda = e^{iX} \equiv \sum_{n=0}^{\infty} \frac{(iX)^n}{n!}</math>|{{EquationRef|G1}}}}
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is a Lorentz transformation by general properties of Lie algebras. Motivated by this, let {{math|''π'': '''so'''(3;1) → '''gl'''(''V'')}} for some vector space {{mvar|V}} be a representation and attempt to define a representation Π of SO(3)<sup>+</sup> by first setting
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{{NumBlk|:|<math>\Pi_U(e^{iX}) = e^{i\pi(X)}, \quad X \in \mathfrak{so}(3;1).</math>|{{EquationRef|G2}}}}
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There are at least two potential problems with this definition. The first is that it is not obvious that this yields a [[group homomorphism]], or even a well defined map at all. But a theorem<ref name="harvnb|Hall|2003">{{harvnb|Hall|2003}}, Chapter 3</ref> based on the [[inverse function theorem]] states that the map {{math|exp: '''so'''(3;1) → SO(3;1)<sup>+</sup>}} is one-to-one for {{mvar|X}} small enough (A). The qualitative form of the [[Baker-Campbell-Hausdorff formula]] then guarantees that it ''is'' a group homomorphism, still for {{mvar|X}} small enough (B).<ref name="harvnb|Hall|2003"/> Let {{math|U⊂SO(3;1)<sup>+</sup>}} denote image under the exponential mapping of the open set in {{math|so(3;1)}} where conditions (A) and (B) both hold.
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The second problem is that for a given {{math|''g'' ∈ SO(3;1)<sup>+</sup>}} there may not be exactly one {{math|''X'' ∈ '''so'''(3;1)}} such that {{math|1=''g'' =  
''e''<sup>''iX''</sup>}}. Technically, formula {{EquationNote|(G2)}} is used to ''define'' Π near the identity. For other elements {{math|''g'' ∉ ''U''}} one chooses a [[path (topology)|path]] from the identity to {{mvar|g}} and defines Π along that path by partitioning it finely enough so that formula {{EquationNote|(G2)}} can be used again on the resulting factors in the partition. In detail, one sets


{{NumBlk|:|<math>g = g_n = (g_n g_{n-1}^{-1})(g_{n-1}g_{n-2}^{-1})\cdots(g_{2}g_{1}^{-1})(g_{1}g_{0}), \qquad \Pi(g) \equiv \Pi_U(g_{n}g_{n-1}^{-1})\Pi_U(g_{n-1}g_{n-2}^{-1})\cdots\Pi_U(g_{2}g_{1}^{-1})\Pi_U(g_{1}g_{0}), \quad g_0 = 1</math><ref name="harvnb|Hall|2003"/>|{{EquationRef|G3}}}}
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where the {{math|g<sub>i</sub>}} are on the path and the factors on the far right are uniquely defined by (G2) provided that all {{math|''g''<sub>''i''</sub> ''g''<sub>''i'' + 1</sub><sup>−1</sup> ∈ ''U''}}. For each {{math|i}} take, by the inverse function theorem, the unique {{math|X<sub>i</sub>}} such that {{math| exp(X<sub>i</sub>)}} = {{math|g<sub>i</sub>g<sub>i-1</sub><sup>-1</sup>}} and obtain
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{{NumBlk|:|<math>\Pi(g) = \Pi_U(e^{iX_n})\Pi_U(e^{iX_{n - 1}})\cdots\Pi_U(e^{iX_2})\Pi_U(e^{iX_1}) = \Pi_U(e^{iX_n}e^{iX_{n - 1}}\cdots e^{iX_2}e^{iX_1}),</math>|{{EquationRef|G4}}}}
where the final step is by the Baker-Campbell-Hausdorff formula. It shows that {{math|Π}} is a group homomorphism for elements close to the identity. By [[compactness]] of the path there is an {{mvar|n}} large enough so that {{math|Π(''g'')}} is well defined, possibly depending on the partition and/or the path, whether {{math|g}} is close to the identity or not. Likewise, the formula yields a homomorphism for all elements, whether small or not, as seen using expressions similar to (G2). One simply drops the subscript U in formula {{EquationNote|(G2)}}.


It turns out that the result is always independent of the partitioning of the path.<ref name="harvnb|Hall|2003"/> For [[simply connected]] groups, the construction will be independent of the path as well, yielding a well defined representation.<ref name="harvnb|Hall|2003"/> In that case formula {{EquationNote|(G2)}} can unambiguously be used directly. For a group that is connected but not simply connected, such as SO(3;1)<sup>+</sup>, the result ''may'' depend on the [[homotopy class]] of the chosen path.<ref>{{harvnb|Weinberg|2002}}, Appendix B, Chapter 2</ref> The result, when using {{EquationNote|(G2)}}, will then depend on ''which'' {{mvar|X}} in the Lie algebra is used to obtain the representative matrix for {{mvar|g}}.
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The Lorentz group is [[doubly connected]] so that its [[fundamental group]] {{math|π<sub>1</sub>(SO(3,1)<sup>+</sup>)}}, whose elements are the path homotopy classes, has two members. Thus not all representations of the Lie algebra will yield representations of the group, but some will instead yield [[projective representation]]s.<ref name="harvnb|Hall|2003"/> Once these conclusions has been reached, and once one knows whether a representation is projective, there is no need not be concerned about paths and partitions. '''''Formula {{EquationNote|(G2)}} applies to all group elements and all representations, including the projective ones.'''''
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{{NumBlk|:|<math>
 
[\Pi(\Lambda_1)\Pi(\Lambda_2)\Pi^{-1}(\Lambda_1\Lambda_2)]^2 = 1\Rightarrow \Pi(\Lambda_1\Lambda_2) = \pm \Pi(\Lambda_1)\Pi(\Lambda_2), \qquad \Lambda_1,\Lambda_2 \in SO(3;1),
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</math><ref>{{harvnb|Weinberg|2002}}, Section 2.7, Chapter 2</ref>|{{EquationRef|G5}}}}
 
since any loop in SO(3;1)<sup>+</sup> traversed twice, due to the double connectedness, is [[contractible]] to a point so that its homotopy class is that of a constant map. It follows that Π is a double-valued function. One cannot consistently chose sign to obtain a continuous representation of all of {{math|SO(3;1)<sup>+</sup>}}, but this is possible to do ''locally'' around any point.<ref>{{harvnb|Wigner|1937}}</ref>
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=== The covering group ===
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Let {{math|π<sub>g</sub>}} denote the set of path homotopy classes {{math|[''p''<sub>''g''</sub>]}} of paths {{math|''p''<sub>''g''</sub>(t), 0 ≤ ''t'' ≤ 1}}, from {{math|1 ∈ SO<sup>+</sup>(3;1)}} to {{math|''g'' ∈ SO<sup>+</sup>(3;1)}} and define the set


:<math>G = \{(g,[p_g]): g\in SO^+(3,1),[p_g]\in \pi_g\}</math>
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and endow it with the multiplication operation
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:<math>(g_1,[p_1])(g_2,[p_2]) = (g_1g_2,[p_{12}]),\quad g_1,g_2\in SO(3;1)^+,\quad [p_1]\in\pi_{g_1}, [p_2]\in \pi_{g_2}, [p_{12}]\in \pi_{g_{12}},\quad p_{12}(t) = p_1(t)\cdot p_2(t).</math>
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The dot on the far right denotes [[Fundamental group#Definition|path multiplication]]. With this multiplication, {{mvar|''G''}} is a [[group (mathematics)|group]] and {{math|''G'' ≈}} [[SL(2,C)|{{math|SL(2,C)}}]],<ref>{{harvnb|Wigner|1937, p.27}}</ref> the [[universal covering group]] of {{math|SO<sup>+</sup>(3;1)}}. By the above construction, there is, since each {{math|π<sub>g</sub>}} has two elements, a [[covering map|2:1 covering map]] {{math|p:SL(2.C) → SO(3,1)<sup>+</sup>}}. According to [[covering group]] theory, the Lie algebras {{math|'''so'''(3;1)}} and {{math|'''sl'''(2,C)}} are isomorphic and the {{math|Π<sub>(''m'',''n'')</sub>}} representations (whether projective or not) correspond to non-projective representations of the simply connected group {{math|SL(2,C)}}.
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For an algebraic view of the universal covering group, let {{math|'''SL'''(2,'''C''')}} act on the set of all Hermitean {{math|2 × 2}} matrices {{math|'''h'''}} by the operation<ref>{{harvnb|Weinberg|2003}}, Section 2.7</ref>
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:<math>\mathbf{P}(A): \mathbf{h} \rightarrow \mathbf{h}; \quad X \rightarrow A^\dagger XA, \quad X \in \mathbf{h}, A \in SL(2,C).</math>
 
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Since {{math|X ∈ '''h'''}} is Hermitean, {{math|A<sup>†</sup>XA}} is again Hemitean because {{math|1=(A<sup>†</sup>XA)<sup>†</sup> = A<sup>†</sup>X<sup>†</sup>A<sup>††</sup> = A<sup>†</sup>XA}}, and also {{math|1=A<sup>†</sup>(αX + βY)A = αA<sup>†</sup>XA + βA<sup>†</sup>YA}}, so the action is linear as well. An element of {{math|'''h'''}} may generally be written in the form
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:<math>X = \bigl(\begin{smallmatrix}
\xi_4 + \xi_3&\xi_1 + i\xi_2\\ \xi_1 - i\xi_2&\xi_4 - \xi_3\\
\end{smallmatrix}\bigr) </math>
for {{math|ξ<sub>i</sub>}} real, showing that {{math|'''h'''}} is a 4-dimensional real vector space. Moreover, {{math|1=(AB)<sup>†</sup>X(AB) = B<sup>†</sup>A<sup>†</sup>XAB}} meaning that {{math|'''P'''}} is a group homomorphism into {{math|End('''h''')}}. Thus {{math|'''P''':'''SL'''(2,'''C''') → End('''h''')}} is a 4-dimensional representation of {{math|'''SL'''(2,'''C''')}}. Now map {{math|'''h'''}} to spacetime {{math|'''R'''<sup>4</sup>}}, endowed with the Lorentz metric, via
:<math>X = (\xi_1,\xi_2,\xi_3,\xi_4) \leftrightarrow \overrightarrow{(\xi_1,\xi_2,\xi_3,\xi_4)} = (x,y,z,t) = \overrightarrow{X}.</math>
The action of {{math|'''P'''(A)}} on {{math|'''h'''}} preserves determinants since {{math|1=det(A<sup>†</sup>XA) = (det A)(det A<sup>†</sup>)(det X) = det X}}. The induced representation {{math|'''p'''}} of {{math|'''SL'''(2,'''C''')}} on {{math|'''R'''<sup>4</sup>}}, via the above isomorphism, given by
:<math>\mathbf{p}(A)\overrightarrow{X} = \overrightarrow{AXA^\dagger}</math>
will preserve the Lorentz inner product since {{math|1=−det X = ξ<sub>1</sub><sup>2</sup> + ξ<sub>2</sub><sup>2</sup> + ξ<sub>3</sub><sup>2</sup> - ξ<sub>4</sub><sup>2</sup> = x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> - t<sup>2</sup>}}. This means that {{math|'''p'''(A)}} belongs to the full Lorentz group {{math|'''O'''(3;1)}}. By the [[main theorem of connectedness]], since {{math|'''SL'''(2,'''C''')}} is connected, its image under {{math|'''p'''}} in {{math|'''O'''(3;1)}} is connected as well, and hence is contained in {{math|'''SO'''(3;1)<sup>+</sup>}}. It can be shown that the Lie map of {{math|'''p''':'''SL'''(2,'''C''') → '''SO'''(3;1)<sup>+</sup>}}, {{math|π:'''sl'''(2,'''C''') → '''so'''(3;1)}} is a Lie algebra isomorphism (it's kernel is {∅} and must therefore be an isomorphism for dimensional reasons) and thus {{math|'''SL'''(2,'''C''')}}, since it is simply connected, is the universal covering group of {{math|'''SO'''(3;1)<sup>+</sup>}}, isomorphic to the group {{math|G}} of above.


==== Representations of {{math|SL(2,C)}} and {{math|sl(2,C)}} ====
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The complex linear representations of {{math|'''sl'''(2,'''C''')}} and {{math|'''SL'''(2,'''C''')}} are more straightforward to obtain than the {{math|'''SO'''(3;1)<sup>+</sup>}} representations. If {{math|π<sub>m</sub>}} is a representation of {{math|'''su'''(2)}} with highest weight {{mvar|m}}, then the complexification of {{mvar|π}} is a complex linear representation of {{math|'''sl'''(2,'''C''')}}. All complex linear representation of {{math|'''sl'''(2,'''C''')}} are of this form. The group representations are obtained by exponentiation. By simple connectedness of {{math|'''SL'''(2,'''C''')}}, this always yields a representation of the group as opposed to in the {{math|'''SO'''(3;1)<sup>+</sup>}} case.
 
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The kernel of the mapping {{math|'''p'''}} of above is {{math|{I, -I}}}. By the [[first isomorphism theorem]], a representation {{math|Π}} of {{math|'''SL'''(2,'''C''')}} descends to a representation of {{math|'''SO'''(3;1)<sup>+</sup>}} if and only if {{math|ker '''p''' ⊂ ker Π}}. In particular, if {{math|Π}} is [[Faithful representation|faithful]], then there is no corresponding proper representation of {{math|'''SO'''(3;1)<sup>+</sup>}}, but there is a projective one as was shown in a previous section.
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=== Common representations ===
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{| class="wikitable" align=right valign=center style="margin-left:1em"
 
|
  <li>[http://cqlhzx.com/forum.php?mod=viewthread&tid=208359 http://cqlhzx.com/forum.php?mod=viewthread&tid=208359]</li>
! {{mvar|m}}=0
 
! {{sfrac|1|2}}
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! 1
 
|-
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!{{mvar|n}}=0
 
| style="padding:8px; color:#CC00FF" |'''scalar'''
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|<div style="color:#CC00FF">Weyl spinor</div>bispinor
 
| style="padding:8px" |<div style="color:#CC00FF">self-dual 2-form</div>2-form field
</ul>
|-
!{{sfrac|1|2}}
| style="color:#CC00FF" |Weyl spinor<br/>(right-handed)
| style="color:#CC00FF" align=center |'''4-vector'''
|Rarita–Schwinger<br/>field
|-
!1
| style="padding:8px; color:#CC00FF" |anti-self-dual<br/>2-form
|
| style="padding:8px; color:#CC00FF" |'''traceless<br/>symmetric tensor'''
|-
| colspan=4 style="padding-left:20px; font-size:80%" |
<span style="color:#CC00FF;">Purple</span>: {{math|(''m'', ''n'')}} complex irreps &nbsp;
Black: {{math|(''m'', ''n'') ⊕ (''n'', ''m'')}}<br/>
<span style="color:#CC00FF;">'''Bold'''</span>: {{math|(''m'', ''m'')}}
|}
Since for any irrep where {{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'')}} has a particular relevance to physics, since it permits to use [[linear operator]]s over [[real numbers]].
* (0, 0) is the Lorentz [[scalar (mathematics)|scalar]]<!-- do not forget here is mathematics! --> representation. This representation is carried by relativistic [[scalar field]] theories.
* ({{sfrac|1|2}}, 0) is the left-handed [[Weyl spinor]] and (0, {{sfrac|1|2}}) is the right-handed Weyl spinor representation. <!-- where is a Majorana spinor? -->
* ({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}}) is the [[bispinor]] representation. (See also [[Dirac spinor]] and [[#Weyl spinors and bispinors|Weyl spinors and bispinors]] below.)
* ({{sfrac|1|2}}, {{sfrac|1|2}}) is the [[four-vector]] representation. The [[four-momentum]] of a [[particle]] (either massless or [[invariant mass|massive]]) transforms under this representation.
* (1, 0) is the [[Hodge dual#Four dimensions|self-dual 2-form field]] representation and (0, 1) is the anti-self-dual 2-form field representation.
* (1, 0) ⊕ (0, 1) is the representation of a [[P-symmetry|parity]]-invariant [[2-form]] field (a.k.a. [[curvature form]]). The [[electromagnetic field tensor]] transforms under this representation.
* (1, {{sfrac|1|2}}) ⊕ ({{sfrac|1|2}}, 1) is the [[Rarita–Schwinger equation|Rarita–Schwinger]] field representation.
* (1, 1) is the spin&nbsp;2 representation of a traceless [[symmetric bilinear form|symmetric]] [[tensor field]].<ref>The "traceless" property can be expressed as {{math|1=''S''<sub>''αβ''</sub>[[metric tensor (general relativity)|''g'']]<sup>''αβ''</sup> = 0}}, or {{math|1=''S''<sub>''α''</sub><sup>''α''</sup> = 0}}, or {{math|1=''S''<sup>''αβ''</sup>''g''<sub>''αβ''</sub> = 0}} depending on the presentation of the field: covariant, mixed, and contravariant respectively.</ref>
 
=== Properties of the (''m'', ''n'') representations ===
The irreducible {{math|(''m'', ''n'')}} representations are {{math|(2''m'' + 1)(2''n'' + 1)}}-dimensional, and they are the only irreducible representations.
 
Since the [[angular momentum]] operator is given by {{math|'''J''' {{=}} '''A''' + '''B'''}}, the highest weight (or spin modulo ℏ in quantum mechanics) of the rotation subrepresentation will be {{math|''m'' + ''n''}}.
 
The {{math|(''m'', ''n'')}} Lie algebra representation is not [[Hermitian]]. Accordingly, the corresponding (projective) representation of the group is never [[unitary matrix|unitary]]. This is due to the non-compactness of the Lorentz group. It 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]]. 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.<ref name="Weinberg 2002">{{harvnb|Weinberg|2002}}, Chapter 5</ref>
 
The {{math|(''m'', ''n'')}} representation is, however, unitary when restricted to the rotation subgroup SO(3), but these representations are not irreducible as representations of SO(3). A [[Clebsch–Gordan coefficients|Clebsch–Gordan decomposition]] can be applied showing that an {{math|(''m'', ''n'')}} representation have SO(3)-invariant subspaces of highest weight (spin) {{math|''m'' + ''n'', ''m'' + ''n'' − 1, …, {{!}} ''m'' − ''n'' {{!}}}},<ref name="Weinberg 2002"/> where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) ''j'' is {{math|(2''j'' + 1)-dimensional}}. So for example, the ({{sfrac|1|2}}, {{sfrac|1|2}}) representation has spin&nbsp;1 and spin&nbsp;0 subspaces of dimension 3 and 1 respectively.
 
The {{math|(''m'', 0)}} representation is [[dual representation|self-dual]]. This means that it is [[Group representation#Definitions|equivalent]] to its dual. The {{math|(0,''n'')}} representation is '''anti-self-dual'''.
 
=== Induced representations ===
In general representation theory, if {{math|(''π'', ''V'')}} is a representation of a Lie algebra '''g''', then there is an associated representation of '''g''' on {{math|[[Endomorphism|End]] (''V'')}}, also denoted {{mvar|π}}, given by
 
:<math>\pi(X)(A) = [\pi(X),A], \quad A\in \mathrm{End}\,(V),\ X\in\mathfrak{g}.</math>
 
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 Π, given by
 
:<math>\Pi(g)(A) = \Pi(g)A\Pi(g)^{-1}, \quad A\in \mathrm{End}\,(V),\ g\in G.</math>
 
Applying this to the Lorentz group, if {{math|(''Π'', ''V'')}} is a projective representation, then direct calculation using ([[#equation G4|G4]]) shows that the induced representation on {{math|End ''(V)''}} is, in fact, 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 {{math|''H''}}, then the corresponding induced representation acts on the set of linear operators on {{math|''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.<ref name="Weinberg 2002"/>
 
For simplicity, consider now only the "discrete part" of {{math|End(''H'')}}, that is, given a basis for {{math|''H''}}, the set of constant matrices of various dimension, including possibly infinite dimensions. A general element of the full {{math|End(''H'')}} is the sum of tensor products of a matrix from the simplified {{math|End(''H'')}} and an operator from the left out part. The left out part consists of functions of spacetime, differential and integral operators and the like. See [[Dirac operator]] for an illustrative example. Also left out are operators corresponding to other degrees of freedom not related to spacetime, such as gauge degrees of freedom in gauge theories.
 
The induced 4-vector representation of above on this simplified {{math|End(''H'')}} has an invariant 4-dimensional subspace that is spanned by the four [[gamma matrices]].<ref name="Weinberg 2002"/> (Note the different metric convention in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, {{math|Cℓ<sub>3,1</sub>('''R''')}}, whose complexification is {{math|M<sub>4</sub>('''C''')}}, generated by the gamma matrices decomposes as a direct sum of [[representation space]]s of a scalar irreducible representation (irrep), a pseudoscalar irrep, the already mentioned vector irrep, a pseudovector irrep, and a tensor irrep.<ref name="Weinberg 2002"/> The dimensions add up to {{math|1 + 1 + 4 + 4 + 6}} = {{math|16}}. This is, in fact, a reasonably convenient way to show that the algebra spanned by the gammas is 16-dimensional.<ref name="Weinberg 2002"/> An algebraic proof of this fact is fairly lengthy.<ref name="Greiner">{{citation|last=Greiner|first=W|year=2000|title=Relativistic Quantum Mechanics|isbn=3-540-67457-8}}</ref> For details, see [[Dirac algebra]].
 
The conclusion is that every element of the complexified {{math|Cℓ<sub>3,1</sub>('''R''')}} in {{math|End(''H'')}} (i.e. every complex 4×4 matrix) has well defined Lorentz transformation properties. The "left out" part transforms under the [[#Infinite dimensional unitary representations|infinite-dimensional representations]] or, if not related to spacetime, not at all under Lorentz transformations, but instead they transform under the corresponding gauge group. The gauge transformations in turn leave the spacetime parts unaffected.
 
There is also a multitude of other representations that can be said being "induced" by the irreducible ones, such as those obtained in a standard manner by taking direct sums, tensor products, duals, quotients, etc. of the irreducible representations. These are not discussed here.
 
=== The full Lorentz group ===
The (possibly projective) {{math|(''m'', ''n'')}} representation is irreducible as a representation {{math|SO<sup>+</sup>(3;1)}}, the identity component of the Lorentz group, in physics terminology the [[orthochronous|proper orthochronous]] Lorentz group. If {{math|1=''m'' = ''n''}} it can be extended to a representation of all of O(3;1), the full Lorentz group, including [[P-symmetry|space parity inversion]] and [[T-symmetry|time reversal]].<ref name="Weinberg 2002"/> This follows from considering the [[adjoint representation|adjoint action]] {{math|Ad<sub>''P''</sub>}} of {{math|''P'' ∈ O(3;1)}} on '''so'''(3;1), where {{mvar|P}} is the standard representative of space parity inversion, {{math|1=''P'' = diag(1, −1, −1, −1)}}, given by
 
{{NumBlk|:|<math>\mathrm{Ad}_P(J_i) = PJ_iP^{-1} = J_i, \qquad \mathrm{Ad}_P(K_i) = PK_iP^{-1} = -K_i.</math>|{{EquationRef|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 {{mvar|π}} is any representation of '''so'''(3;1) and {{math|Π}} is its associated group representation, then {{math|Π(SO(3;1)<sup>+</sup>)}} acts on the representation of {{mvar|π}} by the adjoint action, {{math|π(''X'') ↦ Π(''g'') ''π''(''X'') Π(''g'')<sup>−1</sup>}} for {{math|''X'' ∈ '''so'''(3;1)}}, {{math|g ∈ SO(3;1)<sup>+</sup>}}. If {{mvar|P}} is to be included in {{math|Π}}, then consistency with {{EquationNote|F1|(F1)}} requires that
 
{{NumBlk|:|<math>\Pi(P)\pi(B_i)\Pi(P)^{-1} = \pi(A_i)</math>|{{EquationRef|F2}}}}
 
holds, where {{mvar|'''A'''}} and {{mvar|'''B'''}} are defined as in the first section. This can hold only if {{math|''A''<sub>''i''</sub>}} and {{math|''B''<sub>''i''</sub>}} 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|O<sup>+</sup>(3;1)}}, the orthocronous 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=[[Gamma matrices|''β'' = ''i'' ''γ''<sup>0</sup>]]}} (or a multiple of modulus {{math|-1}} times it) may be used in the {{nowrap|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}}<ref name="Weinberg 2002"/> representation. If parity is included with a minus-sign in the {{math|(0,0)}} representation, it is called a [[pseudoscalar]] representation.
 
==== Time-reversal ====
Time-reversal {{math|1=''T'' = diag(−1, 1, 1, 1)}}, acts similarly on {{math|'''so'''(3;1)}} by
 
{{NumBlk|:|<math>\mathrm{Ad}_T(J_i) = TJ_iT^{-1} = -J_i,  \qquad \mathrm{Ad}_P(K_i) = TK_iT^{-1} = K_i.</math>|{{EquationRef|F3}}}}
 
By explicitly including a representative for {{math|''T''}}, as well as one for {{math|''P''}}, one obtains a representation of the full Lorentz group {{math|O(3;1)}}. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full [[Representation theory of the Poincare group|Poincaré group]], four more generators, the {{math|P<sup>μ</sup>}}, in addition to the {{math|J<sup>i</sup>}} and {{math|K<sup>i</sup>}} generate the group. These are interpreted as generators of translations. The time-component {{math|P<sup>0</sup>}} is the Hamiltonian {{math|H}}. The operator {{math|''T''}} satisfies the relation
 
{{NumBlk|:|<math>\mathrm{Ad}_{T}(iH) = TiHT^{-1} = -iH</math>|{{EquationRef|F4}}}}
 
in analogy to the relations above with {{math|'''so'''(3;1)}} replaced by the full [[Representation theory of the Poincare group|Poincaré algebra]]. By just cancelling the {{math|i}}'s, the result {{math|THT<sup>-1</sup>}} = {{math|-H}} would imply that for every state {{math|Ψ}} with positive energy {{math|E}} in a Hilbert space of quantum states with time-reversal invariance, there would be a state {{math|Π(T<sup>-1</sup>)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 [[anticommutator|anticommutes]] with {{math|i}}, resulting in {{math|THT<sup>-1</sup>}} = {{math|+H}}, and its action on Hilbert space likewise becomes antilinear and antiunitary.<ref>{{harvnb|Weinberg|2002}} Chapter 2</ref> It may be expressed as the composition of [[complex conjugation]] with multiplication by a unitary matrix.<ref name="Greiner"/> This is mathematically sound, see [[Wigner's theorem]], but if one is very strict with terminology, {{math|Π}} is not a ''representation''.
 
When constructing theories such as [[quantum electrodynamics|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.<ref>{{harvnb|Weinberg|2002}}, Chapter 3</ref>
 
==Infinite-dimensional representations==
 
===History===
 
The Lorentz group SO(3,1){{clarify|date=January 2013|orthochronic?}} and its double cover SL(2, '''C''') also have infinite dimensional unitary representations, first studied independently by {{harvtxt|Bargmann|1947}}, {{harvtxt|Gelfand|Naimark|1947}} and {{harvtxt|Harish-Chandra|1947}} (at the instigation of [[Paul Dirac]]). The [[Plancherel formula]] for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by {{harvtxt|Harish-Chandra|1951}} and {{harvtxt|Gelfand|Graev|1953}}, based on an analogue for SL(2, '''C''')  of the integration formula of [[Hermann Weyl]] for [[compact Lie group]]s. Elementary accounts of this approach can be found in {{harvtxt|Rühl|1970}} and {{harvtxt|Knapp|2001}}.
 
The theory of [[zonal spherical function|spherical functions]] for the Lorentz group, required for [[harmonic analysis]] on the 3-dimensional [[hyperboloid]] in Minkowski space, or equivalently 3-dimensional [[hyperbolic space]], is considerably easier than the general theory. It only involves representations from the spherical [[principal series]] and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on '''R'''. This theory is discussed in {{harvtxt|Takahashi|1963}}, {{harvtxt|Helgason|1968}}, {{harvtxt|Helgason|2000}} and the posthumous text of {{harvtxt|Jorgenson|Lang|2008}}.
 
===Action on function spaces===
In the classification of the irreducible finite-dimensional representations of above it was never specified precisely ''how'' a representative of a group or Lie algebra element acts on vectors in the representation space. The action can be anything long as it is linear. If {{math|V}} is a vector space of functions of a finite number of variables {{math|n}}, then the action on a scalar function {{math|f ∈ V}} given by
{{NumBlk|:|<math>(\Pi_f(g)f)(x) = f(\Pi_x(g)^{-1}x),\qquad x \in \mathbb{R}^n, f \in V</math>|{{EquationRef|H1}}}}
produces another function {{math|Πf ∈ V}}. Here {{math|Π<sub>x</sub>}} is an {{math|n}}-dimensional representation, and {{math|Π<sub>f</sub>}} is a possibly infinite-dimensional representation. The rotation subgroup {{math|SO(3)}} has an infinite-dimensional representation on the vector space {{math|1=V = {Y<sup>l</sup><sub>m</sub>, l ∈ N<sup>+</sup>, −l ≤ m ≤ l} }} of all [[spherical harmonics]]. If {{math|f}} is an arbitrary sufficiently smooth function defined on the unit sphere {{math|S<sup>2</sup>}}, then it can be expressed as
{{NumBlk|:|<math>f = \sum_{l = 1}^\infty\sum_{m = -l}^{m = l}f_{lm}Y^l_m.</math>|{{EquationRef|H2}}}}
The Lorentz group action restricts to that of {{math|SO(3)}} and is expressed as
{{NumBlk|:|<math>(\Pi(R)f)(x) = \sum_{l = 1}^\infty\sum_{m = -l}^{m = l}\sum_{m' = -l}^{m' = l}D^{(l)}_{mm'}(R)f_{lm'}Y^l_m(R^{-1}x), \qquad R \in SO(3).</math>|{{EquationRef|H3}}}}
The {{math|D<sup>(l)</sup>}} can be obtained from the {{math|D<sup>(m,n)</sup>}} of above using Clebsch-Gordan-decomposition, but they are more easily directly expressed as an exponential of an odd-dimensional {{math|su(2)}}-representation (the 3-dimensional one is exactly {{math|so(3)}}). In this case the space {{math|V}} decomposes neatly into an infinite sum of irreducible finite-dimensional representations according to
{{NumBlk|:|<math>V = \sum_{i = 1}^{\infty} \oplus V_i</math>.|{{EquationRef|H4}}}}
 
===The Möbius group===
{{Main|Möbius transformation}}
The identity component of the Lorentz group is isomorphic to the [[Möbius transformation|Möbius group]] {{math|M}} as is described in detail in [[Lorentz group#Relation to the Möbius group|Lorentz group]]. This group can be thought of as [[conformal mapping]]s 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
{{NumBlk|:|<math>f(z) = \frac{a z + b}{c z + d}, \qquad ad - bc \neq 0</math>.|{{EquationRef|S1}}}}
and can be represented by complex matrices
{{NumBlk|:|<math>\Pi_f = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad \mathrm{det} \Pi_f = 1.</math>|{{EquationRef|S2}}}}
These are elements of {{math|SL(2;C)}} and are unique up to a sign and {{math|M ≈ SL(2;C)/{I, -I} ≈ SO(3;1)}}. The conformal mappings of the Riemann sphere are thoroughly described in [[Möbius transformation]]s.
 
===The Riemann P-functions===
{{Main|Riemann's differential equation}}
The [[Riemann's differential equation|Riemann P-functions]] are an example of a set of functions that transform among themselves under the action of the Lorentz (Möbius) group. The Riemann P-functions are expressed as
 
{{NumBlk|
:|<math>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\}
</math>,|{{EquationRef|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]] giving
 
{{NumBlk|
:|<math>w(z)=
\left(\frac{z-a}{z-b}\right)^\alpha
\left(\frac{z-c}{z-b}\right)^\gamma
\;_2F_1 \left(
\alpha+\beta +\gamma,
\alpha+\beta'+\gamma;
1+\alpha-\alpha';
\frac{(z-a)(c-b)}{(z-b)(c-a)} \right)
.</math>|{{EquationRef|T2}}}}
 
Now define an action of the Lorentz group on the set of all Riemann P-functions by
{{NumBlk|
:|<math>u=\frac{Az+B}{Cz+D}
\quad \text{ and } \quad
\eta=\frac{Aa+B}{Ca+D}</math>|{{EquationRef|T3}}}}
 
and
 
{{NumBlk|
:|<math>\zeta=\frac{Ab+B}{Cb+D}
\quad \text{ and } \quad
\theta=\frac{Ac+B}{Cc+D},</math>
|{{EquationRef|T4}}}}
 
where {{math|A, B, C, D}} are the entries in
 
{{NumBlk|:|<math>\pi_{f_\Lambda}^{-1} = \begin{pmatrix} A & B \\ C & D \end{pmatrix},</math>|{{EquationRef|T5}}}}
 
where, {{math|Λ}} is a Lorentz transformation and {{math|f<sub>Λ</sub>}} is the corresponding Möbius transformation, and, finally, {{math|Π<sub>f<sub>Λ</sub></sub>}} is one of the two possible {{math|SL(2;C)}}-matrices corresponding to it, then one has the relation
 
{{NumBlk|
:|<math>P  \left\{ \begin{matrix} a & b & c & \; \\
\alpha & \beta & \gamma & z \\
\alpha' & \beta' & \gamma' & \;
\end{matrix} \right\}
=P  \left\{ \begin{matrix}
\eta & \zeta & \theta & \; \\
\alpha & \beta & \gamma & u \\
\alpha' & \beta' & \gamma' & \;
\end{matrix} \right\}</math>
|{{EquationRef|T6}}}}
 
expressing the symmetry. The inverse in {{EquationNote|T5}} is needed to obtain a (local) homomorphism.
 
===Principal series===
 
The '''principal series''', or '''unitary principal series''', are the unitary representations [[induced representation|induced]] from the one-dimensional representations of the lower triangular subgroup&nbsp;{{mvar|B}} of {{math|1=''G'' = SL(2, '''C''')}}. 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''<sup>−1</sup>}}, they thus have the form
 
:<math>\chi_{\nu,k}\begin{pmatrix}z& 0\\ c& z^{-1}\end{pmatrix}=r^{i\nu} e^{ik\theta},</math>
 
for {{mvar|k}} an integer, {{mvar|ν}} real and with {{mvar|z}} = {{math|re<sup>iθ</sup>}}. The representations are [[irreducible representation|irreducible]]; the only repetitions occur when {{mvar|k}} is replaced by {{math|−''k''}}.  By definition the representations are realized on L<sup>2</sup> sections of [[line bundle]]s on {{math|1=''G'' / ''B'' = ''S''<sup>2</sup>}}, 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&nbsp;{{mvar|G}} can also be realized as an induced representation of&nbsp;{{mvar|K}} using the identification {{math|1=''G'' / ''B'' = ''K'' / ''T''}}, where {{math|1=''T'' = ''B'' ∩ ''K''}} is the [[maximal torus]] in&nbsp;{{mvar|K}} consisting of diagonal matrices with {{math|1={{!}} ''z'' {{!}} = 1}}. It is the representation induced from the 1-dimensional representation {{math|''z''<sup>k</sup> ''T''}}, and is independent of&nbsp;{{mvar|ν}}. By [[Frobenius reciprocity]], on&nbsp;{{mvar|K}} they decompose as a direct sum of the irreducible representations of&nbsp;{{mvar|K}} with dimensions {{math|1={{!}} ''k'' {{!}} + 2''m'' + 1}} with {{mvar|m}} a non-negative integer.
 
Using the identification between the Riemann sphere minus a point and {{math|'''C'''}}, the principal series can be defined directly on {{math|L<sup>2</sup>('''C''')}} by the formula
 
:<math>\pi_{\nu,k}\begin{pmatrix}a& b\\ c& d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-i\nu} \left({cz+d\over |cz+d|}\right)^{-k}f\left({az+b\over cz+d}\right).</math>
 
Irreducibility can be checked in a variety of ways:
 
* The representation is already irreducible on&nbsp;{{mvar|B}}. This can be seen directly, but is also a special case of general results on ireducibility of induced representations due to [[François Bruhat]] and [[George Mackey]], relying on the [[Bruhat decomposition]] {{math|1=''G'' = ''B'' ∪ ''B'' ''s'' ''B''}} where {{mvar|s}} is the [[Weyl group]] element <math>\begin{pmatrix}0& -1\\ 1& 0\end{pmatrix}</math>.<ref>{{harvnb|Knapp|2001}}, Chapter II</ref>
 
* The action of the Lie algebra <math>\mathfrak{g}</math> of&nbsp;{{mvar|G}} can be computed on the algebraic direct sum of the irreducible subspaces of&nbsp;{{mvar|K}} can be computed explicitly and the it can be verified directly that the lowest dimensional subspace generates this direct sum as a <math>\mathfrak{g}</math>-module.<ref name="Harish-Chandra 1947">{{harvnb|Harish-Chandra|1947}}</ref><ref name="Taylor 1986">{{harvnb|Taylor|1986}}</ref>
 
===Complementary series===
 
The for {{math|0 < t < 2}}, the '''complementary series''' is defined on [[Lp space|{{math|L<sup>2</sup>}}-functions]] {{mvar|f}} on {{math|'''C'''}} for the inner product
 
:<math> (f,g)=\int \int {f(z) \overline{g(w)}\, dz\, dw\over |z-w|^{2-t}}.</math>
 
with the action given by
 
:<math>\pi_{t}\begin{pmatrix}a& b\\ c& d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-t} f\left({az+b\over cz+d}\right).</math>
 
The complementary series are irreducible and inequivalent. As a representation of&nbsp;{{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 analysing the action of <math>\mathfrak{g}</math> on the
algebraic sum of these subspaces<ref name="Harish-Chandra 1947"/><ref name="Taylor 1986"/> or directly without using the Lie algebra.<ref>{{harvnb|Gelfand|Naimark|1947}}</ref><ref>{{harvnb|Takahashi|1963|page=343}}</ref>
 
===Plancherel theorem===
 
The only irreducible unitary representations of {{math|SL(2, '''C''')}} are the principal series, the complementary series and the trivial representation.
Since {{math|−''I''}} acts {{math|(−1)<sup>''k''</sup>}} 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&nbsp;{{mvar|G}} on {{math|L<sup>2</sup>(''G'')}}, only the principal series are required. This immediately yields the decomposition on the subrepresentations {{math|L<sup>2</sup>(''G''/±I)}}, the left regular representation of the Lorentz group, and {{math|L<sup>2</sup>(''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 λ and ρ are defined on {{math|L<sup>2</sup>(''G'')}} by
 
:<math>\lambda(g)f(x)=f(g^{-1}x),\,\,\rho(g)f(x)=f(xg).</math>
 
Now if {{mvar|f}} is an element of {{math|C<sub>c</sub>(''G'')}}, the operator {{math|''π''<sub>''&nu;'',''k''</sub>(''f'')}} defined by
 
:<math>\pi_{\nu,k}(f)=\int_G f(g)\pi(g)\, dg</math>
 
is [[Hilbert–Schmidt operator|Hilbert–Schmidt]]. We define a Hilbert space&nbsp;{{mvar|H}} by
 
:<math> H=\bigoplus_{k\ge 0} HS(L^2(C)) \otimes L^2(R, c_k(\nu^2 + k^2)^{1/2} d\nu),</math>
 
where
 
:<math>c_0=1/4\pi^{3/2}, \,\, c_k=1/(2\pi)^{3/2}\,\,(k\ne 0)</math>
 
and {{math|HS(L<sup>2</sup>('''C'''))}} denotes the Hilbert space of Hilbert–Schmidt operators on {{math|L<sup>2</sup>('''C''')}}.<ref>Note that for a Hilbert space&nbsp;{{mvar|H}}, {{math|HS(''H'')}} may be identified canonically with the Hilbert space tensor product of {{mvar|H}} and its conjugate space.</ref>  Then the map&nbsp;{{mvar|U}} defined on {{math|C<sub>c</sub>(''G'')}} by
 
:<math>U(f)(\nu,k) = \pi_{\nu,k}(f)</math>
 
extends to a unitary of {{math|L<sup>2</sup>(''G'')}} onto {{mvar|H}}.
 
The map&nbsp;{{mvar|U}} satisfies
 
:<math> U(\lambda(x)\rho(y)f)(\nu,k) = \pi_{\nu,k}(x)^{-1} \pi_{\nu,k}(f)\pi_{\nu,k}(y).</math>
 
If {{math|''f''<sub>1</sub>}}, {{math|''f''<sub>2</sub>}} are in {{math|C<sub>c</sub>(''G'')}} then
 
:<math> (f_1,f_2) = \sum_{k\ge 0} c_k^2 \int_{-\infty}^\infty {\rm Tr}(\pi_{\nu,k}(f_1)\pi_{\nu,k}(f_2)^*) (\nu^2 +k^2) \, d\nu.</math>
 
Thus if {{math|1=''f'' = ''f''<sub>1</sub> ∗ ''f''<sub>2</sub>*<!-- these asterisks are different! do not touch anything unless you can think about it, please -->}} denotes the [[convolution]] of {{math|''f''<sub>1</sub>}} and {{math|''f''<sub>2</sub>*}}, and
<math>f_2^*(g)=\overline{f_2(g^{-1})}</math>, then
 
:<math> f(1) = \sum_{k\ge 0} c_k^2 \int_{-\infty}^\infty  {\rm Tr}(\pi_{\nu,k}(f)) (\nu^2+k^2)\, d\nu.</math>
 
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 {{math|''f''<sub>''i''</sub>}} in {{math|L<sub>2</sub>(''G'')}}. By a theorem of [[Jacques Dixmier]] and [[Paul Malliavin]],
every function&nbsp;{{mvar|f}} in <math>C^\infty_c(G)</math> 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.<ref name="Knapp 2001"/>
 
== Explicit formulas ==
 
=== Conventions and Lie algebra bases ===
The metric of choice is given by {{math|η}} = {{math|diag(−1, 1, 1, 1)}}, and the physics convention for Lie algebras and the exponential mapping is used in this article. These choices are arbitrary, but once they are made, fixed. The rationale is to allow the use of a single reference<ref name="Weinberg 2002"/> for several related formulas. One possible choice of '''basis''' for the Lie algebra (which is not fixed by the reference) is, in the 4-vector representation, given by
 
:<math>\begin{align}
J_1 &= J^{23} = -J^{32} = i\biggl(\begin{smallmatrix}
0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&0&1&0\\
\end{smallmatrix}\biggr),\\
J_2 &= J^{31} = -J^{13} = i\biggl(\begin{smallmatrix}
0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&-1&0&0\\
\end{smallmatrix}\biggr),\\
J_3 &= J^{12} = -J^{21} = i\biggl(\begin{smallmatrix}
0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\\
\end{smallmatrix}\biggr),\\
K_1 &= J^{01} = J^{10} = i\biggl(\begin{smallmatrix}
0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\
\end{smallmatrix}\biggr),\\
K_2 &= J^{02} = J^{20} = i\biggl(\begin{smallmatrix}
0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\
\end{smallmatrix}\biggr),\\
K_3 &= J^{03} = J^{30} = i\biggl(\begin{smallmatrix}
0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\
\end{smallmatrix}\biggr).
\end{align}</math>
 
The commutation relations of the Lie algebra '''so'''(3;1) are
 
:<math>[J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\sigma\mu}J^{\rho\nu} + \eta^{\nu\sigma}J^{\mu\rho} - \eta^{\rho\mu}J^{\sigma\nu} -\eta^{\nu\rho}J^{\mu\sigma}).</math><ref name="Weinberg 2002"/>
 
In three-dimensional notation, these are
 
:<math>[J_i,J_j] = i\epsilon_{ijk}J_k, \quad [J_i,K_j] = i\epsilon_{ijk}K_k, \quad [K_i,K_j] = -i\epsilon_{ijk}J_k.</math><ref name="Weinberg 2002"/>
 
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.
 
Let {{math|''π''<sub>(''m'',''n'')</sub>: '''so'''(3;1) → '''gl'''(''V'')}}, where {{mvar|V}} is a vector space, denote the irreducible representations of {{math|'''so'''(3;1)}} according to the {{math|(''m'',''n'')}} classification. In components, with {{math|-''m'' ≤ ''a'',''a′'' ≤ ''m''}}, {{math|-''n'' ≤ ''b'',''b′'' ≤ ''n''}}, the representations are given by
:<math>\begin{align}
(\pi_{m,n}(J_i))_{a'b' , ab} &= \delta_{b'b}(J_i^{(m)})_{a'a} + \delta_{a'a}(J_i^{(n)})_{b'b},\\
(\pi_{m,n}(K_i))_{a'b' , ab} &= i(\delta_{a'a}(J_i^{(n)})_{b'b} - \delta_{b'b}(J_i^{(m)})_{a'a}),
\end{align}</math><ref name="Weinberg 2002"/>
 
where {{math|δ}} is the [[Kronecker delta]] and the {{math|''J''<sub>i</sub><sup>(n)</sup>}} are the {{math|(2''n'' + 1)}}-dimensional irreducible representations of {{math|'''so'''(3)}}, also termed '''spin matrices''' or '''angular momentum matrices'''. These are explicitly given by
 
:<math>\begin{align}
(J_3^{(j)})_{a'a} &= a\delta_{a'a},\\
(J_1^{(j)} \pm iJ_2^{(j)})_{a'a} &= \sqrt{(j \mp a)(j \pm a + 1)}\delta_{a',a \pm 1}.
\end{align}</math><ref name="Weinberg 2002"/>
 
=== Weyl spinors and bispinors ===
By taking, in turn, {{math|''m'' {{=}} {{sfrac|1|2}}}}, {{math|''n'' {{=}} 0}} and {{math|''m'' {{=}} 0}}, {{math|''n'' {{=}} {{sfrac|1|2}}}} and by setting
 
:<math>J_i^{(\frac{1}{2})} = \frac{1}{2}\sigma_i</math>
 
in the general expression {{EquationNote|G1|(G1)}}, and by using the trivial relations {{nowrap|1<sub>1</sub> {{=}} 1}} and {{math|''J''<sup>(0)</sup> {{=}} 0}}, one obtains
 
{{NumBlk|:|<math>\begin{align}
\pi_{(\frac{1}{2},0)}(J_i) & = \frac{1}{2}(\sigma_i\otimes 1_{(1)} + 1_{(2)}\otimes J^{(0)}_i) = \frac{1}{2}\sigma_i\quad\pi_{(\frac{1}{2},0)}(K_i) = \frac{i}{2}(1_{(2)}\otimes J^{(0)}_i - \sigma_i \otimes 1_{(1)}) = -\frac{i}{2}\sigma_i,\\
\pi_{(0,\frac{1}{2})}(J_i) & = \frac{1}{2}(J^{(0)}_i\otimes 1_{(2)} + 1_{(1)}\otimes \sigma_i) = \frac{1}{2}\sigma_i\quad\pi_{(0,\frac{1}{2})}(K_i) = \frac{i}{2}(1_{(1)}\otimes\sigma_i  - J^{(0)}_i \otimes 1_{(2)}) = +\frac{i}{2}\sigma_i.
\end{align}</math>|{{EquationRef|W1}}}}
 
These are the left-handed and right-handed [[Weyl spinor]] representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) ''V''<sub>L</sub> and ''V''<sub>R</sub>, whose elements Ψ<sub>L</sub> and Ψ<sub>R</sub> are called left- and right-handed Weyl spinors respectively. Given {{math|(π({{sfrac|1|2}},0),''V''<sub>L</sub>)}} and {{math|(π(0,{{sfrac|1|2}}),''V''<sub>R</sub>)}} one may form their direct sum as representations,
 
{{NumBlk|:|<math>\begin{align}
\pi_{(\frac{1}{2},0) \oplus (0,\frac{1}{2})}(J_i) &= \frac{1}{2}\biggl(\begin{matrix}
\sigma_i&0\\ 0&\sigma_i\\
\end{matrix}\biggr),\\
\pi_{(\frac{1}{2},0) \oplus (0,\frac{1}{2})}(K_i) &= \frac{i}{2}
\biggl(\begin{matrix}
\sigma_i&0\\ 0&-\sigma_i\\
\end{matrix}\biggr)\\
\end{align}.</math><ref>{{harvnb|Weinberg|2002}}, Equations (5.4.19) and (5.4.20)</ref>|{{EquationRef|D1}}}}
 
This is, up to a similarity transformation, the {{math|({{sfrac|1|2}},0)⊕(0,{{sfrac|1|2}})}} [[Dirac spinor]] representation of {{math|'''so'''(3,1)}}. It acts on the 4-component elements {{nowrap|(Ψ<sub>L</sub>, Ψ<sub>R</sub>)}} of (''V''<sub>L</sub>⊕''V''<sub>R</sub>), called [[bispinor]]s, 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 {{math|'''so'''(3,1)}}. Expressions for the group representations are obtained by exponentiation.
 
== See also ==
* [[Bargmann–Wigner equations]]
* [[Center of mass (relativistic)]]
* [[Dirac algebra]]
* [[Gamma matrices]]
* [[Lorentz group]]
* [[Möbius transformation]]
* [[Poincaré group]]
* [[Representation theory of the Poincaré group]]
* [[Symmetry in quantum mechanics]]
* [[Wigner's classification]]
 
==Notes==
{{reflist|2}}
 
==References==
*{{citation|doi=10.2307/1969129|first=V.|last= Bargmann|title=Irreducible unitary representations of the Lorenz group|journal=Ann. Of Math.|volume= 48|issue=3| year=1947|pages=568–640|postscript=<!--none-->|jstor=1969129}} (the representation theory of SO(2,1) and SL(2, '''R'''); the second part on SO(3,1) and SL(2, '''C'''), described in the introduction, was never published).
*{{citation|first=J.|last=Dixmier|authorlink=Jacques Dixmier|first2=P.|last2=Malliavin|authorlink2=Paul Malliavin|title=Factorisations de fonctions et de vecteurs indéfiniment différentiables|journal= Bull. Sc. Math.|volume =102 |year=1978|pages= 305–330|postscript=<!--none-->}}
*{{citation|first=I. M.|last=Gelfand|authorlink=I. M. Gelfand|first2=M. A.|last2=Naimark|title=Unitary representations of the Lorentz group|journal=Izvestiya Akad. Nauk SSSR. Ser. Mat.|volume= 11|year=1947|pages=411–504|postscript=<!--none-->}}
*{{Citation|first=I. M.|last=Gelfand|coauthors=M. I. Graev|title= [On a general method of decomposition of the regular representation of a Lie group into irreducible representations]|journal=[[Doklady Akademii Nauk SSSR]]|volume= 92|year=1953| pages=221–224|postscript=<!--none-->}}
*{{citation|doi=10.1098/rspa.1947.0047|last=Harish-Chandra|authorlink=Harish-Chandra|title=Infinite irreducible representations of the Lorentz group|journal=[[Proceedings of the Royal Society A]]|volume= 189|year=1947|issue=1018|pages=372–401|postscript=<!--none-->|bibcode = 1947RSPSA.189..372H }}
*{{citation|doi=10.1073/pnas.37.12.813|last=Harish-Chandra|authorlink=Harish-Chandra|title=Plancherel formula for complex semi-simple Lie groups|journal=Proc. Nat. Acad. Sci. U. S. A.|volume= 37|year=1951|issue=12|pages=813–818|postscript=<!--none-->|bibcode = 1951PNAS...37..813H }}
*{{citation|year=2003|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations An Elementary Introduction|publisher=Springer|isbn= 0-387-40122-9}}
*{{citation|last=Helgason|first= S.|title=Lie groups and symmetric spaces|series=Battelle Rencontres|year=1968|pages= 1–71|publisher= Benjamin|postscript=<!--none-->}} (a general introduction for physicists)
*{{citation|last=Helgason|first= S.|title=Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions (corrected reprint of the 1984 original)|series=Mathematical Surveys and Monographs|volume=83|publisher= American Mathematical Society|year= 2000|isbn= 0-8218-2673-5|postscript=<!--none-->}}
*{{citation|last2=Lang|first2=S.|authorlink2=Serge Lang|last=Jorgenson|first=J.|title=The heat kernel and theta inversion on SL(2,'''C''')|
series=Springer Monographs in Mathematics|publisher= Springer|year= 2008|isbn=978-0-387-38031-5|postscript=<!--none-->}}
*{{citation|first=A.|last= Knapp| title=Representation theory of semisimple groups. An overview based on examples. |series=Princeton Landmarks in Mathematics|publisher=Princeton University Press|year=2001|isbn=0-691-09089-0|postscript=<!--none-->}} (elementary treatment for SL(2,'''C'''))
*{{citation|first=M.A.|last=Naimark|title=Linear representations of the Lorentz group (translated from the Russian original by Ann Swinfen and O. J. Marstrand) |publisher=Macmillan|year= 1964|postscript=<!--none-->}}
* Paërl, E.R. (1969) ''Representations of the Lorentz group and projective geometry'', Mathematical Centre Tract #25, Amsterdam.
*{{citation|last=Rühl|first= W.|title=The Lorentz group and harmonic analysis|publisher=Benjamin|year= 1970|postscript=<!--none-->}} (a detailed account for physicists)
*{{citation|last=Takahashi|first= R.|title=Sur les représentations unitaires des groupes de Lorentz généralisés|journal=Bull. Soc. Math. France|
volume= 91| year=1963|pages= 289–433|postscript=<!--none-->}}
*{{citation|last=Taylor|first=M. E.|authorlink=Michael E. Taylor|title=Noncommutative harmonic analyis|series=Mathematical Surveys and Monographs|volume= 22|publisher= American Mathematical Society|year=1986|isbn=0-8218-1523-7|postscript=<!--none-->}}, Chapter 9, SL(2, '''C''') and more general Lorentz groups
*{{citation|last=Weinberg|first=S|year=2002|title=The Quantum Theory of Fields, vol I|isbn=0-521-55001-7}}
*{{citation|first=E. P.|last=Wigner|authorlink=Eugene Wigner|title=On unitary representations of the inhomogeneous Lorentz group|journal=[[Annals of Mathematics]]|issue=1|volume=40|pages=149–204|year=1939|doi=10.2307/1968551|mr=1503456}}.
 
[[Category:Representation theory of Lie groups|Lorentz Group]]
[[Category:Special relativity]]
[[Category:Quantum mechanics]]

Latest revision as of 20:06, 10 October 2014

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