Representation theory of the Lorentz group
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. In any relativistically invariant physical theory, these representations must enter in some fashion;^{[nb 1]} physics itself must be made out of them. Indeed, special relativity together with quantum mechanics are the two physical theories that are most thoroughly established,^{[nb 2]} and the conjunction of these two theories is the study of the infinitedimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative presentday theories.
The full theory of the finitedimensional representations of the Lie algebra of the Lorentz group is deduced using the general framework of the representation theory of semisimple Lie algebras. The finitedimensional representations of the connected component of the full Lorentz group O(3; 1) are obtained by employing the Lie correspondence and the matrix exponential. The full finitedimensional representation theory of the universal covering group (and also the spin group, a double cover) of is obtained, and explicitly given in terms of action on a function space in representations of and . The representatives of time reversal and space inversion are given in space inversion and time reversal, completing the finitedimensional 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 Pfunctions appearing as examples. The infinitedimensional case of irreducible unitary representations are realized for the principal series and the complementary series. Finally, the Plancherel formula for is given, and representations of 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,^{[1]} 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.^{[2]} The classification of the irreducible infinitedimensional representations of the Lorentz group was established by Paul Dirac's doctoral student in theoretical physics, HarishChandra, later turned mathematician,^{[nb 3]} in 1947. The corresponding classification for was published independently by Bargmann and Israel Gelfand together with Mark Naimark in the same year.
The nontechnical introduction contains some prerequisite material for readers not familiar with representation theory. Standard results used here from general finitedimensional representation theory are outlined in technical introduction to finitedimensional representation theory. The Lie algebra basis and other adopted conventions are given in conventions and Lie algebra bases. There is a substantially detailed article on the present subject to be found at Wikiversity.
Contents
Nontechnical introduction to representation theoryEdit
The present purpose is to illustrate the role of representation theory of groups in mathematics and in physics. Rigor and detail take the back seat, as the main objective is to fix the notion of finitedimensional and infinitedimensional representations of the Lorentz group. The reader familiar with these concepts should skip by.
Symmetry of space and timeEdit
Space itself possesses symmetry. It looks the same no matter how one rotates it, and the resulting rotational symmetry is referred to as isotropy of space. In the present case it is common to use passive rotations, meaning that the observer^{[nb 4]} rotates themself. Mathematically, the active operation of a rotation is performed by multiplying position vectors by a rotation matrix. A passive rotation is accomplished by rotating only the basis vectors of the coordinate system. (The coordinate system can be thought of as being fixed in the rotated observer. The observer is physically rotated.) In this way, every point in space obtains new coordinates as if it was somehow physically rigidly rotated.
The Lorentz group contains all rotation matrices, extended to four dimensions with zeros in the first row and the first column except for the upper left element which is one, as elements.
There are, in addition, matrices that effect Lorentz boosts. These can be thought of, in the passive view, as (instantly!) giving the coordinate system (and with it the observer) a velocity in a chosen direction.
Finally, two special transformations are used to invert the coordinate system in space, space inversion, and in time, time reversal. In the first case, the space coordinate axes are reversed. The latter is reversal of the time direction. This is thought of, in the passive view, as having the observer set their clock at minus what it shows and then have the clock's hands move counterclockwise. Physical time progresses forward.
Mathematically the Lorentz group is defined as the set of transformations preserving the bilinear form
where the lefthand side is the Minkowski inner product of two events in spacetime, and the righthand side is the spacetime interval, see classical group for mathematical detail.
Lorentz transformationsEdit
In the spacetime of special relativity, called Minkowski space, space and time are interwoven. Thus the four coordinates of points in spacetime, called events, change in ways unexpected before the advent of special relativity, with time dilation and length contraction as two immediate consequences. The fourdimensional matrices of Lorentz transformations compose the Lorentz group. Its elements represent symmetries, and just like physical objects can be rotated using rotation matrices, the same physical objects (whose coordinates now include the time coordinate) can be transformed using the matrices representing Lorentz transformations. In particular, the fourvector representing an event in Lorentz frame transforms as
or on short form
Multiplication table and representationsEdit
The basic feature of every finite group is its multiplication table, also called Cayley table, that records the result of multiplying any two elements. A representation of a group can be thought of as a new set of elements, finitedimensional or infinitedimensional matrices, giving the same multiplication table after mapping the old elements to the new elements in a onetoone fashion.^{[nb 5]} The same holds true in the case of an infinite group like the rotation group SO(3) or the Lorentz group. The multiplication table is just harder to visualize in the case of a group of uncountable size (same size as the set of reals). One way to do this is to wellorder the elements of the group with an ordinal number ρ being the order type. The "infinite Cayley table" is then indexed by two ordinals 0 ≤ α, β ≤ ρ written in Cantor normal form.
Ordinary Lorentz transformations matrices do not sufficeEdit
The objects to be transformed may be something else than ordinary physical objects extending in three spatial dimensions (and time, unless the frame is the rest frame). It is for these objects that representation theory is needed for the mathematical description of transformations induced by ordinary Lorentz transformations of spacetime. For instance, the electromagnetic field is often (naively) envisaged by the assignment to each point in spacetime of a threedimensional vector representing the electric field and another threedimensional vector representing the magnetic field.
When space is rotated, the classically expected thing happens. The electric field and the magnetic field vectors at a designated point rotate with preserved length and angle between them.
Under Lorentz boosts they behave differently, showing that the two vectors aren't separate physical objects. The electric and magnetic components mix. See the illustration on the right. The electromagnetic field tensor displays the manifestly covariant mathematical structure of the electromagnetic field. It has six independent components at an event.^{[nb 6]}
Finitedimensional representations by matricesEdit
The problem of representation theory of the Lorentz group is, in the finitedimensional case, to find new sets of matrices, not necessarily 4 × 4 in size that satisfies the same multiplication table as the matrices in the original Lorentz group. Returning to the example of the electromagnetic field, 6 × 6 matrices are needed that can be applied to a sixdimensional column vector containing the all together six components of the electromagnetic field. Thus 6 × 6matrices are sought for, such that
or in short
correctly expresses the transformation of the electromagnetic field under the Lorentz transformation Λ.^{[nb 7]} The same reasoning can be applied to Dirac's bispinors. While these have 4 components, the original 4 × 4matrices in the Lorentz group are inappropriate, even when restricted to rotations. Another 4 × 4representation is needed.
The sections dedicated to finitedimensional representations are dedicated to exposing all such representations by finitedimensional matrices that respect the multiplication table.
Infinitedimensional representations by action on vector spaces of functionsEdit
Infinitedimensional representations are usually realized as acting on sets of real or complexvalued functions on a set X endowed with a group action. A set being endowed with a group action A means, in essence, that if x ∈ X and g ∈ G that A(g)x = y with y ∈ X. If denotes the set of all complexvalued functions on X, which is a vector space, a representation Π of G can be defined by^{[3]}
The point to make is that again
and there is representation of G. This representation of G is finitedimensional if and only if X is a finite set. This method is very general, and typically vector spaces of more specialized functions on sets close at hand are explored. Illustrating this procedure, a group G of ndimensional matrices as a subset of Euclidean space are considered, and the space of functions polynomials, perhaps of some maximum degree d, or even homogeneous polynomials of degree d, all defined on Those functions are then restricted to The set X = G automatically comes equipped with group actions, namely
Here L_{g} denotes left action (by g), R_{g} denotes right action (by g), and C_{g} denotes conjugation (by g). With this sort of action, the vectors being acted on are functions. The resulting representations are (when the functions are unrestricted), in the first and second cases respectively, the left regular representation and the right regular representation of G on ^{[3]}
The goal in the infinitedimensional case of the representation theory is to classify all different possible representations, and to exhibit them in terms of vector spaces of functions and the action of the standard representation on the arguments of the functions.
Infinitedimensional representations viewed as infinitedimensional matricesEdit
In order to relate representations on infinitedimensional spaces to the finitedimensional cases, an ordered basis is chosen for the vector space of functions and the action on basis functions under a given transformation is examined. Explicitly, if f_{1}, f_{2}, ... is a basis, compute
The coefficients of the basis functions in the expression for each transformed basis function is a column in the representative matrix. In general, the resulting matrix is countably infinite in dimension:^{[nb 8]}
Again, it is required that the set of infinite matrices obtained this way stand in onetoone correspondence with the original 4 × 4matrices and that the multiplication table is the right one – the one of the 4 × 4matrices.^{[nb 9]} It should be emphasized that in the infinitedimensional case, the concern is rarely the complete matrices. They are exposed here only to highlight the common thread. But individual matrix elements are frequently computed, especially for the Lie algebra (below).
Lie algebraEdit
The Lorentz group is a Lie group and has as such a Lie algebra, The Lie algebra is a vector space of matrices that can be said to model the group near the identity. It is endowed with a multiplication operation, the Lie bracket. With it, the product in the group near the identity can be expressed in Lie algebraic terms (but not in a particularly simple way). The link between the (matrix) Lie algebra and the (matrix) Lie group is the matrix exponential. It is onetoone near the identity in the group.
Due to this it often suffices to find representations of the Lie algebra. Lie algebras are much simpler objects than Lie groups to work with. Due to the fact that the Lie algebra is a finitedimensional vector space, in the case of the Lorentz Lie algebra the dimension is 6, one need only find a finite number of representative matrices of the Lie algebra, one for each element of a basis of the Lie algebra as a vector space. The rest follow from extension by linearity, and the representation of the group is obtained by exponentiation.
One possible choice of basis for the Lie algebra is, in the standard representation, given in conventions and Lie algebra bases.
ApplicationsEdit
Many of the representations, both finitedimensional and infinitedimensional, 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.^{[4]}
The finitedimensional irreducible nonunitary representations together with the irreducible infinitedimensional unitary representations of the inhomogeneous Lorentz group, the Poincare group, are the representations that have direct physical relevance.^{[5]}^{[6]}
Infinitedimensional unitary representations of the Lorentz group appear by restriction of the irreducible infinitedimensional 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.^{[7]} There were speculative theories,^{[8]}^{[9]} (tensors and spinors have infinite counterparts in the expansors of Dirac and the expinors of HarishChandra) consistent with relativity and quantum mechanics, but they have found no proven physical application. Modern speculative theories potentially have similar ingredients per below.
MathematicsEdit
From the point of view that the goal of mathematics is to classify and characterize, the representation theory of the Lorentz group is since 1947 a finished chapter. But in association with the Bargmann–Wigner programme, there are (as of 2006) yet unresolved purely mathematical problems, linked to the infinitedimensional unitary representations.^{[citation needed]}
The irreducible infinitedimensional 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 infinitedimensional unitary representations of the (generalized) Poincaré group are the socalled tachyonic representations. Tachyons appear in the spectrum of bosonic strings and are associated with instability of the vacuum.^{[10]}^{[11]} 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.^{[12]}
One open problem (as of 2006) is the completion of the Bargmann–Wigner programme for the isometry group SO(D − 2, 1) of the de Sitter spacetime dS_{D−2}. Ideally, the physical components of wave functions would be realized on the hyperboloid dS_{D−2} of radius μ > 0 embedded in and the corresponding O(D−2, 1) covariant wave equations of the infinitedimensional unitary representation to be known.^{[11]}
It is common in mathematics to regard the Lorentz group to be, foremost, the Möbius group to which it is isomorphic. The group may be represented in terms of a set of functions defined on the Riemann sphere. These are the Riemann Pfunctions, which are expressible as hypergeometric functions.^{[citation needed]}
Classical field theoryEdit
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.^{[13]} While second quantization and the Lagrangian formalism associated with it is not a fundamental aspect of QFT,^{[14]} it is the case that so far all quantum field theories can be approached this way, including the standard model.^{[15]} 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.^{[13]}
Relativistic quantum mechanicsEdit
For the present purposes the following definition is made:^{[16]} A relativistic wave function is a set of n functions ψ^{α} on spacetime which transforms under an arbitrary proper Lorentz transformation Λ as
where D[Λ] is an ndimensional matrix representative of Λ belonging to some direct sum of the (m, n) representations to be introduced below.
The most useful relativistic quantum mechanics oneparticle theories (there are no fully consistent such theories) are the Klein–Gordon equation^{[17]} and the Dirac equation^{[18]} in their original setting. They are relativistically invariant and their solutions transform under the Lorentz group as Lorentz scalars ((m, n) = (0, 0)) and bispinors respectively ((0, 1/2) ⊕ (1/2, 0)). The electromagnetic field is a relativistic wave function according to this definition, transforming under (1, 0) ⊕ (0, 1).^{[19]}
The infinitedimensional representations may be used in the analysis of scattering.^{[20]}
Quantum field theoryEdit
In QFT, the demand for relativistic invariance enters, among other ways in that the Smatrix necessarily must be Poincaré invariant.^{[21]} This has the implication that there is one or more infinitedimensional representation of the Lorentz group acting on Fock space.^{[nb 10]} 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.^{[22]}
The transformations of field operators illustrate the complementary role played by the finitedimensional representations of the Lorentz group and the infinitedimensional unitary representations of the Poincare group, witnessing the deep unity between mathematics and physics.^{[23]} For illustration, consider the definition an ncomponent field operator:^{[24]} A relativistic field operator is a set of n operator valued functions on spacetime which transforms under proper Poincaré transformations (Λ, a) according to^{[25]}^{[26]}
Here U[Λ, a] is the unitary operator representing (Λ, a) on the Hilbert space on which Ψ is defined and D is an ndimensional 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 m and spin s (or helicity), it is deduced that^{[27]}^{[nb 11]}

( X1 )
where a^{†}, a are interpreted as creation and annihilation operators respectively. The creation operator a^{†} transforms according to^{[27]}^{[28]}
and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finitedimensional nonunitary representation of the Lorentz group, while the creation operator transforms under the infinitedimensional unitary representation of the Poincare group characterized by the mass and spin (m, s) of the particle. The connection between the two are the wave functions, also called coefficient functions
that carry both the indices (x, α) operated on by Lorentz transformations and the indices (p, σ) operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection.^{[29]} To exhibit the connection, subject both sides of equation (X1) to a Lorentz transformation resulting in for e.g. u,
where D is the nonunitary Lorentz group representative of Λ and D^{(s)} is a unitary representative of the socalled Wigner rotation R associated to Λ and p that derives from the representation of the Poincaré group, and 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 (m, n) representation under which it is supposed to transform,^{[nb 12]} 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.^{[nb 13]}
Speculative theoriesEdit
In theories in which spacetime can have more than D = 4 dimensions, the generalized Lorentz groups O(D − 1; 1) of the appropriate dimension take the place of O(3; 1).^{[nb 14]}
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.^{[30]} This results in a relativistically invariant theory in any spacetime dimension.^{[31]} 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.^{[32]} 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 Z_{2}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 1) belong to a (0, 1/2) or (1/2, 0) representation space of the (ordinary) Lorentz Lie algebra.^{[33]} The only possible dimension of spacetime in such theories is 10.^{[34]}
Finitedimensional representationsEdit
Representation theory of groups in general, and Lie groups in particular, is a very rich subject. The full Lorentz group is no exception. 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. Perhaps most importantly, the Lorentz group is not compact.^{[35]}
For finitedimensional representations, the presence of semisimplicity means that the Lorentz group can be dealt with the same way as other semisimple groups using a welldeveloped theory. In addition, all representations are built from the irreducible ones, since the Lie algebra possesses the complete reducibility property.^{[nb 15]}^{[36]} But, the noncompactness 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. Noncompactness implies, for a connected simple Lie group, that no nontrivial finitedimensional unitary representations exist.^{[37]} Lack of simple connectedness gives rise to spin representations of the group.^{[38]} The nonconnectedness means that, for representations of the full Lorentz group, time reversal and space inversion has to be dealt with separately.^{[39]}^{[40]}
HistoryEdit
The development of the finitedimensional representation theory of the Lorentz group mostly follows that of the subject in general. Lie theory originated with Sophus Lie in 1873.^{[41]}^{[42]} By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing.^{[43]}^{[44]} 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.^{[45]}^{[46]} Richard Brauer was 1935–38 largely responsible for the development of the WeylBrauer matrices describing how spin representations of the Lorentz Lie algebra can be embedded in Clifford algebras.^{[47]}^{[48]} The Lorentz group has also historically received special attention in representation theory, see History of infinitedimensional unitary representations below, due to its exceptional importance in physics. Mathematicians Hermann Weyl^{[45]}^{[49]}^{[41]}^{[50]}^{[51]} and HarishChandra^{[52]}^{[53]} and physicists Eugene Wigner^{[54]}^{[55]} and Valentine Bargmann^{[56]}^{[57]}^{[58]} made substantial contributions both to general representation theory and in particular to the Lorentz group.^{[59]} 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.^{[60]}^{[61]}^{[nb 16]}
Technical introduction to finitedimensional representation theoryEdit
 For an overview of the strategy employed for the Lie algebra, overview of classification of Lie algebra representations is referred to.
 Techniques specific to noncompact groups can be found in unitarian trick and Lie algebra representations and Weyl's Unitarian trick.
 The passage from representations of a Lie algebra to the Lie group is covered in projective representations and matrix Lie group representations from Lie algebra representations.
The same material can be found, in more detailed form (in appropriate sequence and inline), in the Wikiversity version.
The Lie algebraEdit
According to the strategy, the irreducible complex linear representations of the complexification, of the Lie algebra of the Lorentz group are to be found. A convenient basis for is given by the three generators J_{i} of rotations and the three generators K_{i} of boosts. They are explicitly given in conventions and Lie algebra bases.
The Lie algebra is complexified, and the basis is changed to the components of its two ideals^{[62]}
The components of A = (A_{1}, A_{2}, A_{3}) and B = (B_{1}, B_{2}, B_{3}) separately satisfy the commutation relations of the Lie algebra and, moreover, they commute with each other,^{[63]}
where i, j, k are indices which each take values 1, 2, 3, and ε_{ijk} is the threedimensional LeviCivita symbol. Let and denote the complex linear span of A and B respectively.
One has the isomorphisms^{[64]}^{[nb 17]}

( A1 )
where is the complexification of
The utility of these isomorphisms comes from the fact that all irreducible representations of , and hence (see strategy) all irreducible complex linear representations of are known. According to the final conclusion in strategy, the irreducible complex linear representation of is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of
The unitarian trickEdit
The Lie algebra is the Lie algebra of It contains the compact subgroup SU(2) × SU(2) with Lie algebra The latter is a compact real form of Thus from the first statement of the unitarian trick, representations of SU(2) × SU(2) are in onetoone correspondence with holomorphic representations of
By compactness, Peter–Weyl theorem applies to SU(2) × SU(2),^{[65]} and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of SU(2) × SU(2) are precisely the tensor products of irreducible unitary representations of SU(2).^{[66]}
By appeal to simple connectedness, the second statement of the unitarian trick is applied. The objects in the following list are in onetoone correspondence:
 Holomorphic representations of
 Smooth representations of SU(2) × SU(2)
 Real linear representations of
 Complex linear representations of
Tensor products of representations appear at the Lie algebra level as either of^{[nb 18]}

( A0 )
where Id is the identity operator. Here, the latter interpretation, which follows from (G6), is intended. The highest weight representations of are indexed by μ for μ = 0, 1/2, 1, .... (The highest weights are actually 2μ = 0, 1, 2, ..., but the notation here is adapted to that of ) The tensor products of two such complex linear factors then form the irreducible complex linear representations of
Finally, the linear representations of the real forms of the far left, , and the far right, ^{[nb 19]} in (A1) are obtained from the linear representations of characterized in the previous paragraph.
The representationsEdit
The real linear representations for and follow here assuming the complex linear representations of are known. Explicit realizations and group representations are given later.
The (μ, ν)representations of sl(2, C)Edit
The complex linear representations of the complexification of obtained via isomorphisms in (A1), stand in onetoone correspondence with the real linear representations of ^{[67]} The set of all, at least real linear, irreducible representations of are thus indexed by a pair (μ, ν). The complex linear ones, corresponding precisely to the complexification of the real linear representations, are of the form (μ, 0), while the conjugate linear ones are the (0, ν).^{[67]} All others are real linear only. The linearity properties follow from the canonical injection, the far right in (A1), of into its complexification. Representations on the form (ν, ν) or (μ, ν) ⊕ (ν, μ) are given by real matrices (the latter are not irreducible). Explicitly, the real linear (μ, ν)representations of are
where are the complex linear irreducible representations of and their complex conjugate representations. (The labeling is usually in the mathematics literature 0, 1, 2, …, but halfintegers are chosen here to conform with the labeling for the Lie algebra.) Here the tensor product is interpreted in the former sense of (A0). These representations are concretely realized below.
The (m, n)representations of so(3; 1)Edit
Via the displayed isomorphisms in (A1) and knowledge of the complex linear irreducible representations of upon solving for J and K, all irreducible representations of and, by restriction, those of are obtained. The representations of 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.^{[64]} Since is semisimple,^{[64]} 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 halfintegers m = μ and n = ν, conventionally written as one of
where V is a finitedimensional vector space. These are, up to a similarity transformation, uniquely given by^{[nb 20]}

( A2 )
where 1_{n} is the ndimensional unit matrix and
are the (2n + 1)dimensional irreducible representations of also termed spin matrices or angular momentum matrices. These are explicitly given as^{[68]}
where δ denotes the Kronecker delta. In components, with −m ≤ a, a′ ≤ m, −n ≤ b, b′ ≤ n, the representations are given by^{[69]}
Common representationsEdit
m = 0  1/2  1  3/2  

n = 0  Scalar (1)  Lefthanded Weyl spinor (2) 
Selfdual 2form (3) 
(4) 
1/2  Righthanded Weyl spinor (2) 
4vector (4)  (6)  (8) 
1  Antiselfdual 2form (3) 
(6)  Traceless symmetric tensor (9) 
(12) 
3/2  (4)  (8)  (12)  (16) 
 The (0, 0) representation is the onedimensional trivial representation and is carried by relativistic scalar field theories.
 Fermionic supersymmetry generators transform under one of the (0, 1/2) or (1/2, 0) representations.^{[33]}
 The fourmomentum of a particle (either massless or massive) transforms under the (1/2, 1/2) representation.
 A physical example of a traceless symmetric tensor field is the traceless^{[nb 21]} part of the energymomentum tensor T^{μν}.^{[70]}^{[nb 22]}
Offdiagonal direct sums
Since for any irreducible representation for which m ≠ n it is essential to operate over the field of complex numbers, the direct sum of representations (m, n) and (n, m) have particular relevance to physics, since it permits to use linear operators over real numbers.
 (1/2, 0) ⊕ (0, 1/2) is the bispinor representation. See also Dirac spinor and Weyl spinors and bispinors below.
 (1, 1/2) ⊕ (1/2, 1) is the Rarita–Schwinger field representation.
 (3/2, 0) ⊕ (0, 3/2) would be the symmetry of the hypothesized gravitino.^{[nb 23]} It can be obtained from the (1, 1/2) ⊕ (1/2, 1) representation.^{[71]}
 (1, 0) ⊕ (0, 1) is the representation of a parityinvariant 2form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation.
The groupEdit
The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence.^{[72]} The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras.^{[73]} The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted The general theory is summarized in technical introduction to finitedimensional representation theory.
If for some vector space V is a representation, a representation Π of the connected component of G is defined by

(G2)
This definition applies whether the resulting representation is projective or not.
Surjectiveness of exponential map for SO(3, 1)Edit
From a practical point of view, it is important whether the first formula in (G2) can be used for all elements of the group. It holds for all , however, in the general case, e.g. for , not all g ∈ G are in the image of exp.
But is surjective. One way to show this is to make use of the isomorphism the latter being the Möbius group. It is a quotient of (see the linked article). The quotient map is denoted with The map is onto.^{[74]} Apply (Lie) with π being the differential of p at the identity. Then
Since the left hand side is surjective (both exp and p are), the right hand side is surjective and hence is surjective.^{[75]} Finally, recycle the argument once more, but now with the known isomorphism between SO(3; 1)^{+} and to find that exp is onto for the connected component of the Lorentz group.
Fundamental groupEdit
The Lorentz group is doubly connected, i. e. π_{1}(SO(3; 1)) is a group with two equivalence classes of loops as its elements.
To exhibit the fundamental group of SO(3; 1)^{+}, the topology of its covering group is considered. By the polar decomposition theorem, any matrix may be uniquely expressed as^{[76]}
where u is unitary with determinant one, hence in SU(2), and h is Hermitian with trace zero. The trace and determinant conditions imply:^{[77]}
The manifestly continuous onetoone map is a homeomorphism with continuous inverse given by (the locus of u is identified with )
explicitly exhibiting that is simply connected. But where is the center of . Identifying λ and −λ amounts to identifying u with −u, which in turn amounts to identifying antipodal points on Thus topologically,^{[77]}
where last factor is not simply connected: Geometrically, it is seen (for visualization purposes, may be replaced by ) that a path from u to −u in is a loop in since u and −u are antipodal points, and that it is not contractible to a point. But a path from u to −u, thence to u again, a loop in and a double loop (considering p(ue^{h}) = p(−ue^{h}), where is the covering map) in that is contractible to a point (continuously move away from −u "upstairs" in and shrink the path there to the point u).^{[77]} Thus π_{1}(SO(3; 1)) is a group with two equivalence classes of loops as its elements, or put more simply, SO(3; 1) is doubly connected.
Projective representationsEdit
Since π_{1}(SO(3; 1)^{+}) has two elements, some representations of the Lie algebra will yield projective representations.^{[78]}^{[nb 24]} Once it is known whether a representation is projective, formula (G2) applies to all group elements and all representations, including the projective ones — with the understanding that the representative of a group element will depend on which element in the Lie algebra (the X in (G2)) is used to represent the group element in the standard representation.
For the Lorentz group, the (m, n)representation is projective when m + n is a halfinteger. See the section spinors.
For a projective representation Π of SO(3; 1)^{+}, it holds that^{[77]}

(G5)
since any loop in SO(3; 1)^{+} traversed twice, due to the double connectedness, is contractible to a point, so that its homotopy class is that of a constant map. It follows that Π is a doublevalued function. It is not possible to consistently chose a sign to obtain a continuous representation of all of SO(3; 1)^{+}, but this is possible locally around any point.^{[37]}
The covering group SL(2, C)Edit
Consider as a real Lie algebra with basis
where the sigmas are the Pauli matrices. From the relations

(J1)
is obtained

(J2)
which are exactly on the form of the 3dimensional version of the commutation relations for (see conventions and Lie algebra bases below). Thus, the map J_{i} ↔ j_{i}, K_{i} ↔ k_{i}, extended by linearity is an isomorphism. Since is simply connected, it is the universal covering group of SO(3; 1)^{+}.
A geometric viewEdit
Let p_{g}(t), 0 ≤ t ≤ 1 be a path from 1 ∈ SO(3; 1)^{+} to g ∈ SO(3; 1)^{+}, denote its homotopy class by [p_{g}] and let π_{g} be the set of all such homotopy classes. Define the set

(C1)
and endow it with the multiplication operation

(C2)
where is the path multiplication of and :
With this multiplication, G becomes a group isomorphic to ^{[79]} the universal covering group of SO(3; 1)^{+}. Since each π_{g} has two elements, by the above construction, there is a 2:1 covering map p : G → SO(3; 1)^{+}. According to covering group theory, the Lie algebras and of G are all isomorphic. The covering map p : G → SO(3; 1)^{+} is simply given by p(g, [p_{g}]) = g.
An algebraic viewEdit
For an algebraic view of the universal covering group, let act on the set of all Hermitian 2×2 matrices by the operation^{[77]}

(C3)
The action on is linear. An element of may be written in the form

(C4)
The map P is a group homomorphism into Thus is a 4dimensional representation of . Its kernel must in particular take the identity matrix to itself, A^{†}IA = A^{†}A = I and therefore A^{†} = A^{−1}. Thus AX = XA for A in the kernel so, by Schur's lemma,^{[nb 25]} A is a multiple of the identity, which must be ±I since det A = 1.^{[80]} The space is mapped to Minkowski space M^{4}, via

(C5)
The action of P(A) on preserves determinants. The induced representation p of on via the above isomorphism, given by

(C6)
preserves the Lorentz inner product since
This means that p(A) belongs to the full Lorentz group SO(3; 1). By the main theorem of connectedness, since is connected, its image under p in SO(3; 1) is connected, and hence is contained in SO(3; 1)^{+}.
It can be shown that the Lie map of is a Lie algebra isomorphism: ^{[nb 26]} The map P is also onto.^{[nb 27]}
Thus , since it is simply connected, is the universal covering group of SO(3; 1)^{+}, isomorphic to the group G of above.
Nonsurjectiveness of exponential mapping for SL(2, C)Edit
The exponential mapping is not onto.^{[81]} The matrix

(S6)
is in but there is no such that q = exp(Q).^{[nb 28]}
In general, if g is an element of a connected Lie group G with Lie algebra then, by (Lie),

(S7)
The matrix q can be written

(S8)
Realization of representations of SL(2, C) and sl(2, C) and their Lie algebrasEdit
The complex linear representations of and are more straightforward to obtain than the 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 are exactly the (μ, ν)representations. They can be exponentiated too. The (μ, 0)representations are complex linear and are (isomorphic to) the highest weightrepresentations. These are usually indexed with only one integer (but halfintegers are used here).
The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of i and there is no factor of i in the exponential mapping compared to the physics convention used elsewhere. Let the basis of be^{[82]}

(S1)
This choice of basis, and the notation, is standard in the mathematical literature.
Complex linear representationsEdit
The irreducible holomorphic (n + 1)dimensional representations can be realized on the space of homogeneous polynomial of degree n in 2 variables ^{[83]}^{[84]} the elements of which are
The action of is given by^{[85]}^{[86]}

(S2)
The associated action is, using (G6) and the definition above, for the basis elements of ^{[87]}

(S5)
With a choice of basis for , these representations become matrix Lie algebras.
Real linear representationsEdit
The (μ, ν)representations are realized on a space of polynomials in homogeneous of degree μ in and homogeneous of degree ν in ^{[84]} The representations are given by^{[88]}

(S6)
By employing (G6) again it is found that

(S7)
In particular for the basis elements,:

(S8)
Properties of the (m, n) representationsEdit
The (m, n) representations, defined above via (A1) (as restrictions to the real form ) of tensor products of irreducible complex linear representations π_{m = μ} and π_{n = ν} of are irreducible, and they are the only irreducible representations.^{[65]}
 Irreducibility follows from the unitarian trick^{[89]} and that a representation Π of SU(2) × SU(2) is irreducible if and only if Π = Π_{μ} ⊗ Π_{ν},^{[nb 29]} where Π_{μ}, Π_{ν} are irreducible representations of SU(2).
 Uniqueness follows from that the Π_{m} are the only irreducible representations of SU(2), which is one of the conclusions of the theorem of the highest weight.^{[90]}
DimensionEdit
The (m, n) representations are (2m + 1)(2n + 1)dimensional.^{[91]} This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of and . For a Lie general algebra the Weyl dimension formula,^{[92]}
applies, where R^{+} is the set of positive roots, ρ is the highest weight, and δ is half the sum of the positive roots. The inner product is that of the Lie algebra invariant under the action of the Weyl group on the Cartan subalgebra. The roots (really elements of are via this inner product identified with elements of For the formula reduces to dim π_{μ} = 2μ + 1 = 2m + 1, where the present notation must be taken into account. The highest weight is 2μ.^{[93]} By taking tensor products, the result follows.
FaithfulnessEdit
If a representation Π of a Lie group G is not faithful, then N = ker Π is a nontrivial normal subgroup.^{[94]} There are three relevant cases.
 N is nondiscrete and abelian.
 N is nondiscrete and nonabelian.
 N is discrete. In this case N ⊂ Z, where Z is the center of G.^{[nb 30]}
In the case of SO(3; 1)^{+}, the first case is excluded since SO(3; 1)^{+} is semisimple.^{[nb 31]} The second case (and the first case) is excluded because SO(3; 1)^{+} is simple.^{[nb 32]} For the third case, SO(3; 1)^{+} is isomorphic to the quotient But is the center of It follows that the center of SO(3; 1)^{+} is trivial, and this excludes the third case. The conclusion is that every representation Π : SO(3; 1)^{+} → GL(V) and every projective representation Π : SO(3; 1)^{+} → PGL(W) for V, W finitedimensional vector spaces are faithful.
By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial nondiscrete normal subgroups replaced by (onedimensional) nontrivial ideals in the Lie algebra,^{[95]} and the center of SO(3; 1)^{+} replaced by the center of The center of any semisimple Lie algebra is trivial^{[96]} and is semisimple and simple, and hence has no nontrivial ideals.
A related fact is that if the corresponding representation of is faithful, then the representation is projective. Conversely, if the representation is nonprojective, then the corresponding representation is not faithful, but is 2:1.
NonunitarityEdit
The (m, n) Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary.^{[nb 33]} This is due to the noncompactness of the Lorentz group. In fact, a connected simple noncompact Lie group cannot have any nontrivial unitary finitedimensional representations.^{[37]} There is a topological proof of this.^{[97]} Let u : G → GL(V), where V is finitedimensional, be a continuous unitary representation of the noncompact connected simple Lie group G. Then u(G) ⊂ U(V) ⊂ GL(V) where U(V) is the compact subgroup of GL(V) consisting of unitary transformations of V. The kernel of u is a normal subgroup of G. Since G is simple, ker u is either all of G, in which case u is trivial, or ker u is trivial, in which case u is faithful. In the latter case u is a diffeomorphism onto its image,^{[98]} u(G) ≅ G and u(G) is a Lie group. This would mean that u(G) is an embedded noncompact Lie subgroup of the compact group U(V). This is impossible with the subspace topology on u(G) ⊂ U(V) since all embedded Lie subgroups of a Lie group are closed^{[99]} If u(G) were closed, it would be compact,^{[nb 34]} and then G would be compact,^{[nb 35]} contrary to assumption.^{[nb 36]}
In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of A and B used in the construction are Hermitian. This means that J is Hermitian, but K is antiHermitian.^{[100]} The nonunitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentzinvariant positive definite norm.^{[101]}
Restriction to SO(3)Edit
The (m, n) representation is, however, unitary when restricted to the rotation subgroup SO(3), but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an (m, n) representation have SO(3)invariant subspaces of highest weight (spin) m + n, m + n − 1, … ,  m − n,^{[102]} where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) j is (2j + 1)dimensional. So for example, the (1/2, 1/2) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.
Since the angular momentum operator is given by J = A + B, the highest spin in quantum mechanics of the rotation subrepresentation will be (m + n)ℏ and the "usual" rules of addition of angular momenta and the formalism of 3j symbols, 6j symbols, etc. applies.^{[103]}
SpinorsEdit
It is the SO(3)invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the (m, n) representation has spin if m + n is halfintegral. The simplest are ( 1/2, 0) and (0, 1/2), the Weylspinors of dimension 2. Then, for example, (0, 3/2) and (1, 1/2) are a spin representations of dimensions 23/2 + 1 = 4 and (2 + 1)(21/2 + 1) = 6 respectively. Note that, according to the above paragraph, there are subspaces with spin both 3/2 and 1/2 in the last two cases, so these representations cannot likely represent a single physical particle which must be wellbehaved under SO(3). It cannot be ruled out in general, however, that representations with multiple SO(3) subrepresentations with different spin can represent physical particles with welldefined spin. It may be that there is a suitable relativistic wave equation that projects out unphysical components, leaving only a single spin.^{[104]}
Construction of pure spin n/2 representations for any n (under SO(3)) from the irreducible representations involves taking tensor products of the Diracrepresentation with a nonspin representation, extraction of a suitable subspace, and finally imposing differential constraints.^{[105]}
Dual representationsEdit
The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation:
 The set of weights of the dual representation of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation.^{[106]}
 Two irreducible representations are isomorphic if and only if they have the same highest weight.^{[nb 37]}
 For each semisimple Lie algebra there exists a unique element w_{0} of the Weyl group such that if μ is a dominant integral weight, then w_{0} ⋅ (−μ) is again a dominant integral weight.^{[107]}
 If is an irreducible representation with highest weight μ_{0}, then has highest weight w_{0} ⋅ (−μ).^{[107]}
Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If −I is an element of the Weyl group of a semisimple Lie algebra, then w_{0} = −I. In the case of the Weyl group is W = {I, −I}.^{[108]} It follows that each π_{μ}, μ = 0, 1, … is isomorphic to its dual The root system of is shown in the figure to the right.^{[nb 38]} The Weyl group is generated by where is reflection in the plane orthogonal to γ as γ ranges over all roots.^{[nb 39]} Inspection shows that w_{α} ⋅ w_{β} = −I so −I ∈ W. Using the fact that if π, σ are Lie algebra representations and π ≅ σ, then Π ≅ Σ,^{[109]} the conclusion for SO(3; 1)^{+} is
Complex conjugate representationsEdit
If π is a representation of a Lie algebra, then is a representation, where the bar denotes entrywise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication.^{[110]} In general, every irreducible representation π of can be written uniquely as π = π^{+} + π^{−}, where^{[111]}
with holomorphic (complex linear) and antiholomorphic (conjugate linear). For since is holomorphic, is antiholomorphic. Direct examination of the explicit expressions for and in equation (S8) below shows that they are holomorphic and antiholomorphic respectively. Closer examination of the expression (S8) also allows for identification of and for as
Using the above identities (interpreted as pointwise addition of functions), for SO(3; 1)^{+} yields
where the statement for the group representations follow from exp(X) = exp(X). It follows that the irreducible representations (m, n) have real matrix representatives if and only if m = n. Reducible representations on the form (m, n) ⊕ (n, m) have real matrices too.
The adjoint representation, the Clifford algebra, and the Dirac spinor representationEdit
In general representation theory, if (π, V) is a representation of a Lie algebra then there is an associated representation of on End(V), also denoted π, given by

(I1)
Likewise, a representation (Π, V) of a group G yields a representation Π on End(V) of G, still denoted Π, given by^{[112]}

(I2)
If π and Π are the standard representations on and if the action is restricted to 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 or ) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular.
Applying this to the Lorentz group, if (Π, V) is a projective representation, then direct calculation using (G5) shows that the induced representation on End(V) is a proper representation, i.e. a representation without phase factors.
In quantum mechanics this means that if (π, H) or (Π, H) is a representation acting on some Hilbert space H, then the corresponding induced representation acts on the set of linear operators on H. As an example, the induced representation of the projective spin (1/2, 0) ⊕ (0, 1/2) representation on End(H) is the nonprojective 4vector (1/2, 1/2) representation.^{[113]}
For simplicity, consider only the "discrete part" of End(H), that is, given a basis for H, the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4vector representation of above on this simplified End(H) has an invariant 4dimensional subspace that is spanned by the four gamma matrices.^{[114]} (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, whose complexification is generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the (0, 0), a pseudoscalar irrep, also the (0, 0), but with parity inversion eigenvalue −1, see the next section below, the already mentioned vector irrep, (1/2, 1/2), a pseudovector irrep, (1/2, 1/2) with parity inversion eigenvalue +1 (not −1), and a tensor irrep, (1, 0) ⊕ (0, 1).^{[115]} The dimensions add up to 1 + 1 + 4 + 4 + 6 = 16. In other words,

(I3)
where, as is customary, a representation is confused with its representation space.
The (1/2, 0) ⊕ (0, 1/2) spin representationEdit
The sixdimensional representation space of the tensor (1, 0) ⊕ (0, 1)representation inside has two roles. The^{[116]}

(I4)
where are the gamma matrices, the sigmas, only 6 of which are nonzero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,^{[117]}

(I5)
and hence constitute a representation (in addition to spanning a representation space) sitting inside the (1/2, 0) ⊕ (0, 1/2) spin representation. For details, see bispinor and Dirac algebra.
The conclusion is that every element of the complexified in End(H) (i.e. every complex 4×4 matrix) has well defined Lorentz transformation properties. In addition, it has a spinrepresentation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on making it a space of bispinors.
Reducible representationsEdit
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. and the Poincaré group. These representations are in general not irreducible.
The Lorentz group and its Lie algebra have the complete reducibility property. This means that every representation reduces to a direct sum of irreducible representations. The reducible representations will therefore not be discussed.
Space inversion and time reversalEdit
The (possibly projective) (m, n) representation is irreducible as a representation SO(3; 1)^{+}, the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If m = n it can be extended to a representation of all of O(3; 1), the full Lorentz group, including space parity inversion and time reversal. The representations (m, n) ⊕ (n, m) can be extended likewise.^{[118]}
Space parity inversionEdit
For space parity inversion, the adjoint action Ad_{P} of P ∈ SO(3; 1) on is considered, where P is the standard representative of space parity inversion, P = diag(1, −1, −1, −1), given by

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

(F2)
holds, where A and B are defined as in the first section. This can hold only if A_{i} and B_{i} have the same dimensions, i.e. only if m = n. When m ≠ n then (m, n) ⊕ (n, m) can be extended to an irreducible representation of SO(3; 1)^{+}, the orthochronous Lorentz group. The parity reversal representative Π(P) does not come automatically with the general construction of the (m, n) representations. It must be specified separately. The matrix β = i γ^{0} (or a multiple of modulus −1 times it) may be used in the (1/2, 0) ⊕ (0, 1/2)^{[119]} representation.
If parity is included with a minus sign (the 1×1 matrix [−1]) in the (0,0) representation, it is called a pseudoscalar representation.
Time reversalEdit
Time reversal T = diag(−1, 1, 1, 1), acts similarly on by^{[120]}

(F3)
By explicitly including a representative for T, as well as one for P, a representation of the full Lorentz group O(3; 1) is obtained. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the P^{μ}, in addition to the J^{i} and K^{i} generate the group. These are interpreted as generators of translations. The timecomponent P^{0} is the Hamiltonian H. The operator T satisfies the relation^{[121]}

(F4)
in analogy to the relations above with replaced by the full Poincaré algebra. By just cancelling the i's, the result THT^{−1} = −H would imply that for every state Ψ with positive energy E in a Hilbert space of quantum states with timereversal invariance, there would be a state Π(T^{−1})Ψ with negative energy −E. Such states do not exist. The operator Π(T) is therefore chosen antilinear and antiunitary, so that it anticommutes with i, resulting in THT^{−1} = H, and its action on Hilbert space likewise becomes antilinear and antiunitary.^{[122]} It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix.^{[123]} This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, Π is not a representation.
When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, (1/2, 0) ⊕ (0, 1/2), is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.
The third discrete symmetry entering in the CPT theorem along with P and T, charge conjugation symmetry C, has nothing directly to do with Lorentz invariance.^{[124]}
Action on function spacesEdit
If V is a vector space of functions of a finite number of variables n, then the action on a scalar function given by

(H1)
produces another function Πf ∈ V. Here Π_{x} is an ndimensional representation, and Π is a possibly infinitedimensional representation. A special case of this construction is when V is a space of functions defined on the a linear group G itself, viewed as a ndimensional manifold embedded in (with m the dimension of the matrices).^{[125]} 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 finitedimensional representations.^{[65]} The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex noncompact groups, e.g.
The following exemplifies action of the Lorentz group and the rotation subgroup on some function spaces.
Euclidean rotationsEdit
The subgroup SO(3) of threedimensional Euclidean rotations has an infinitedimensional representation on the Hilbert space
where are the spherical harmonics. An arbitrary square integrable function f one the unit sphere can be expressed as^{[126]}

(H2)
where the f_{lm} are generalized Fourier coefficients.
The Lorentz group action restricts to that of SO(3) and is expressed as

(H4)
where the D^{l} are obtained from the representatives of odd dimension of the generators of rotation.
The Möbius groupEdit
The identity component of the Lorentz group is isomorphic to the Möbius group M. This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection, the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere.
In the plane, a Möbius transformation characterized by the complex numbers a, b, c, d acts on the plane according to^{[127]}

.
(M1)
and can be represented by complex matrices

(M2)
since multiplication by a nonzero complex scalar does not change f. These are elements of and are unique up to a sign (since ±Π_{f} give the same f), hence
The Riemann PfunctionsEdit
The Riemann Pfunctions, 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 Pfunctions are expressed as^{[128]}

(T1)
where the a, b, c, α, β, γ, α′, β′, γ′ are complex constants. The Pfunction on the right hand side can be expressed using standard hypergeometric functions. The connection is^{[129]}

(T2)
The set of constants 0, ∞, 1 in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation.^{[130]} Its exponents, i. e. solutions of the indicial equation, for expansion around the singular point 0 are 0 and 1 − c ,corresponding to the two linearly independent solutions,^{[nb 40]} and for expansion around the singular point 1 they are 0 and c − a − b.^{[131]} Similarly, the exponents for ∞ are a and b for the two solutions.^{[132]}
One has thus

(T3)
where the condition (sometimes called Riemann's identity)^{[133]}
on the exponents of the solutions of Riemann's differential equation has been used to define γ′.
The first set of constants on the left hand side in (T1), a, b, c denotes the regular singular points of Riemann's differential equation. The second set, α, β, γ, are the corresponding exponents at a, b, c for one of the two linearly independent solutions, and, accordingly, α′, β′, γ′ are exponents at a, b, c for the second solution.
Define an action of the Lorentz group on the set of all Riemann Pfunctions by first setting

(T4)
where A, B, C, D are the entries in

(T5)
for Λ = p(λ) ∈ SO(3; 1)^{+} a Lorentz transformation.
Define

(T6)
where P is a Riemann Pfunction. The resulting function is again a Riemann Pfunction. 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

(T6)
where

(T7)
Infinitedimensional unitary representationsEdit
HistoryEdit
The Lorentz group SO(3; 1)^{+} and its double cover also have infinite dimensional unitary representations, studied independently by Bargmann (1947), Gelfand & Naimark (1947) and HarishChandra (1947) at the instigation of Paul Dirac.^{[134]}^{[135]} This trail of development begun with Dirac (1936) where he devised matrices U and B necessary for description of higher spin (compare Dirac matrices), elaborated upon by Fierz (1939), see also Fierz & Pauli (1939), and proposed precursors of the BargmannWigner equations.^{[136]} In Dirac (1945) he proposed a concrete infinitedimensional representation space whose elements were called expansors as a generalization of tensors.^{[nb 41]} These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinitedimensional 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 HarishChandra (1951) and Gelfand & Graev (1953), based on an analogue for of the integration formula of Hermann Weyl for compact Lie groups.^{[137]} Elementary accounts of this approach can be found in Rühl (1970) and Knapp (2001).
The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3dimensional 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 This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008).
Principal series for SL(2, C)Edit
The principal series, or unitary principal series, are the unitary representations induced from the onedimensional representations of the lower triangular subgroup B of Since the onedimensional representations of B correspond to the representations of the diagonal matrices, with nonzero complex entries z and z^{−1}, they thus have the form
for k an integer, ν real and with z = re^{iθ}. The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when k is replaced by −k. By definition the representations are realized on L^{2} sections of line bundles on which is isomorphic to the Riemann sphere. When k = 0, these representations constitute the socalled spherical principal series.
The restriction of a principal series to the maximal compact subgroup K = SU(2) of G can also be realized as an induced representation of K using the identification G/B = K/T, where T = B ∩ K is the maximal torus in K consisting of diagonal matrices with  z  = 1. It is the representation induced from the 1dimensional representation z^{k}T, and is independent of ν. By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions k + 2m + 1 with m a nonnegative integer.
Using the identification between the Riemann sphere minus a point and the principal series can be defined directly on by the formula^{[138]}
Irreducibility can be checked in a variety of ways:
 The representation is already irreducible on B. This can be seen directly, but is also a special case of general results on irreducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition G = B ∪ BsB where s is the Weyl group element^{[139]}
 .
 The action of the Lie algebra of G can be computed on the algebraic direct sum of the irreducible subspaces of K can be computed explicitly and the it can be verified directly that the lowestdimensional subspace generates this direct sum as a module.^{[9]}^{[140]}
Complementary series for SL(2, C)Edit
The for 0 < t < 2, the complementary series is defined on for the inner product^{[141]}
with the action given by^{[142]}^{[143]}
The representations in the complementary series are irreducible and pairwise nonisomorphic. As a representation of K, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of K = SU(2). Irreducibility can be proved by analyzing the action of on the algebraic sum of these subspaces^{[9]}^{[140]} or directly without using the Lie algebra.^{[144]}^{[145]}
Plancherel theorem for SL(2, C)Edit
The only irreducible unitary representations of are the principal series, the complementary series and the trivial representation. Since −I acts as (−1)^{k} on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even.
To decompose the left regular representation of G on only the principal series are required. This immediately yields the decomposition on the subrepresentations the left regular representation of the Lorentz group, and the regular representation on 3dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with k = 0.)
The left and right regular representation λ and ρ are defined on by
Now if f is an element of C_{c}(G), the operator defined by
is Hilbert–Schmidt. Define a Hilbert space H by
where
and denotes the Hilbert space of Hilbert–Schmidt operators on ^{[nb 42]} Then the map U defined on C_{c}(G) by
extends to a unitary of onto H.
The map U satisfies the intertwining property
If f_{1}, f_{2} are in C_{c}(G) then by unitarity
Thus if denotes the convolution of and and then^{[146]}
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 By a theorem of Jacques Dixmier and Paul Malliavin, every smooth compactly supported function on is a finite sum of convolutions of similar functions, the inversion formula holds for such f. It can be extended to much wider classes of functions satisfying mild differentiability conditions.^{[65]}
Classification of representations of SO(3, 1)Edit
The strategy followed in the classification of the irreducible infinitedimensional representations is, in analogy to the finitedimensional case, to assume they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinitedimensional representation Π_{H} on a Hilbert space H of SO(3; 1)^{+} is at hand.^{[147]} Since SO(3) is a subgroup, Π_{H} is a representation of it as well. Each irreducible subrepresentation of SO(3) is finitedimensional, and the SO(3) representation is reducible into a direct sum of irreducible finitedimensional unitary representations of SO(3) if Π_{H} is unitary.^{[148]}
The steps are the following:^{[149]}
 Chose a suitable basis of common eigenvectors of J^{2} and J_{3}.
 Compute matrix elements of J_{1}, J_{2}, J_{3} and K_{1}, K_{2}, K_{3}.
 Enforce Lie algebra commutation relations.
 Require unitarity together with orthonormality of the basis.^{[nb 43]}
Step 1Edit
One suitable choice of basis and labeling is given by
If this were a finitedimensional representation, then j_{0} would correspond the lowest occurring eigenvalue j(j + 1) of J^{2} in the representation, equal to m − n, and j_{1} would correspond to the highest occurring eigenvalue, equal to m + n. In the infinitedimensional case, j_{0} ≥ 0 retains this meaning, but j_{1} does not.^{[70]} For simplicity, it is assumed that a given j occurs at most once in a given representation (this is the case for finitedimensional representations), and it can be shown^{[150]} that the assumption is possible to avoid (with a slightly more complicated calculation) with the same results.
Step 2Edit
The next step is to compute the matrix elements of the operators J_{1}, J_{2}, J_{3} and K_{1}, K_{2}, K_{3} forming the basis of the Lie algebra of The matrix elements of and (the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by^{[151]}^{[152]}
where the labels j_{0} and j_{1} have been dropped since they are the same for all basis vectors in the representation.
Due to the commutation relations
the triple (K_{i}, K_{i}, K_{i}) ≡ K is a vector operator^{[153]} and the Wigner–Eckart theorem^{[154]} applies for computation of matrix elements between the states represented by the chosen basis.^{[155]} The matrix elements of
where the superscript (1) signifies that the defined quantities are the components of a spherical tensor operator of rank k = 1 (which explains the factor √2 as well) and the subscripts 0, ±1 are referred to as q in formulas below, are given by^{[156]}