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. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.
Development.
The full theory of the finite-dimensional representations of the Lie algebra of the Lorentz group is deduced using the general framework of the representation theory of semisimple Lie algebras. The finite-dimensional representations of the connected component formula_1 of the full Lorentz group are obtained by employing the Lie correspondence and the matrix exponential. The full finite-dimensional representation theory of the universal covering group (and also the spin group, a double cover) formula_2 of formula_1 is obtained, and explicitly given in terms of action on a function space in representations of formula_2 and formula_5. The representatives of time reversal and space inversion are given in space inversion and time reversal, completing the finite-dimensional theory for the full Lorentz group. The general properties of the ("m", "n") representations are outlined. Action on function spaces is considered, with the action on spherical harmonics and the Riemann P-functions appearing as examples. The infinite-dimensional case of irreducible unitary representations are realized for the formula_2 principal series and the complementary series. Finally, the Plancherel formula for formula_2 is given, and representations of 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, one conclusion of which is, roughly, "a classification of all unitary representations of the inhomogeneous Lorentz group amounts to a classification of all possible relativistic wave equations". The classification of the irreducible infinite-dimensional representations of the Lorentz group was established by Paul Dirac's doctoral student in theoretical physics, Harish-Chandra, later turned mathematician, in 1947. The corresponding classification for formula_8 was published independently by Bargmann and Israel Gelfand together with Mark Naimark in the same year.
Applications.
Many of the representations, both finite-dimensional and infinite-dimensional, are important in theoretical physics. Representations appear in the description of fields in classical field theory, most importantly the electromagnetic field, and of particles in relativistic quantum mechanics, as well as of both particles and quantum fields in quantum field theory and of various objects in string theory and beyond. The representation theory also provides the theoretical ground for the concept of spin. The theory enters into general relativity in the sense that in small enough regions of spacetime, physics is that of special relativity.
The finite-dimensional irreducible non-unitary representations together with the irreducible infinite-dimensional unitary representations of the "inhomogeneous" Lorentz group, the Poincare group, are the representations that have direct physical relevance.
Infinite-dimensional unitary representations of the Lorentz group appear by "restriction" of the irreducible infinite-dimensional unitary representations of the Poincaré group acting on the Hilbert spaces of relativistic quantum mechanics and quantum field theory. But these are also of mathematical interest and of "potential" direct physical relevance in other roles than that of a mere restriction. There were speculative theories, (tensors and spinors have infinite counterparts in the "expansors" of Dirac and the "expinors" of Harish-Chandra) consistent with relativity and quantum mechanics, but they have found no proven physical application. Modern speculative theories potentially have similar ingredients per below.
Classical field theory.
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. While second quantization and the Lagrangian formalism associated with it is not a fundamental aspect of QFT, it is the case that so far all quantum field theories can be approached this way, including the standard model. 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.
Relativistic quantum mechanics.
For the present purposes the following definition is made: A "relativistic wave function" is a set of functions on spacetime which transforms under an arbitrary proper Lorentz transformation as
where is an -dimensional matrix representative of belonging to some direct sum of the representations to be introduced below.
The most useful relativistic quantum mechanics "one-particle" theories (there are no fully consistent such theories) are the Klein–Gordon equation and the Dirac equation in their original setting. They are relativistically invariant and their solutions transform under the Lorentz group as Lorentz scalars () and bispinors respectively (). The electromagnetic field is a relativistic wave function according to this definition, transforming under .
The infinite-dimensional representations may be used in the analysis of scattering.
Quantum field theory.
In quantum field theory, the demand for relativistic invariance enters, among other ways in that the S-matrix necessarily must be Poincaré invariant. This has the implication that there is one or more infinite-dimensional representation of the Lorentz group acting on Fock space. 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.
The transformations of field operators illustrate the complementary role played by the finite-dimensional representations of the Lorentz group and the infinite-dimensional unitary representations of the Poincare group, witnessing the deep unity between mathematics and physics. For illustration, consider the definition an -component field operator: A relativistic field operator is a set of operator valued functions on spacetime which transforms under proper Poincaré transformations according to
Here is the unitary operator representing on the Hilbert space on which is defined and is an -dimensional representation of the Lorentz group. The transformation rule is the second Wightman axiom of quantum field theory.
By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass and spin (or helicity), it is deduced that
where are interpreted as creation and annihilation operators respectively. The creation operator transforms according to
and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin of the particle. The connection between the two are the "wave functions", also called "coefficient functions"
that carry "both" the indices operated on by Lorentz transformations and the indices operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection. To exhibit the connection, subject both sides of equation to a Lorentz transformation resulting in for e.g. ,
where is the non-unitary Lorentz group representative of and is a unitary representative of the so-called "Wigner rotation" associated to and that derives from the representation of the Poincaré group, and 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 representation under which it is supposed to transform, 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.
Speculative theories.
In theories in which spacetime can have more than dimensions, the generalized Lorentz groups of the appropriate dimension take the place of .
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. This results in a relativistically invariant theory in any spacetime dimension. 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. The corresponding result for superstring theory is again deduced demanding Lorentz invariance, but now with supersymmetry. In these theories the Poincaré algebra is replaced by a supersymmetry algebra which is a -graded Lie algebra extending the Poincaré algebra. The structure of such an algebra is to a large degree fixed by the demands of Lorentz invariance. In particular, the fermionic operators (grade ) belong to a or representation space of the (ordinary) Lorentz Lie algebra. The only possible dimension of spacetime in such theories is 10.
Finite-dimensional representations.
Representation theory of groups in general, and Lie groups in particular, is a very rich subject. The Lorentz group has some properties that makes it "agreeable" and others that make it "not very agreeable" within the context of representation theory; the group is simple and thus semisimple, but is not connected, and none of its components are simply connected. Furthermore, the Lorentz group is not compact.
For finite-dimensional representations, the presence of semisimplicity means that the Lorentz group can be dealt with the same way as other semisimple groups using a well-developed theory. In addition, all representations are built from the irreducible ones, since the Lie algebra possesses the "complete reducibility property". But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, cannot be dealt with in all the aspects as in the simple framework that applies to simply connected, compact groups. Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional unitary representations exist. Lack of simple connectedness gives rise to spin representations of the group. The non-connectedness means that, for representations of the full Lorentz group, time reversal and space inversion has to be dealt with separately.
History.
The development of the finite-dimensional representation theory of the Lorentz group mostly follows that of the subject in general. Lie theory originated with Sophus Lie in 1873. By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing. In 1913 the theorem of highest weight for representations of simple Lie algebras, the path that will be followed here, was completed by Élie Cartan. Richard Brauer was 1935–38 largely responsible for the development of the Weyl-Brauer matrices describing how spin representations of the Lorentz Lie algebra can be embedded in Clifford algebras. The Lorentz group has also historically received special attention in representation theory, see History of infinite-dimensional unitary representations below, due to its exceptional importance in physics. Mathematicians Hermann Weyl and Harish-Chandra and physicists Eugene Wigner and Valentine Bargmann made substantial contributions both to general representation theory and in particular to the Lorentz group. Physicist Paul Dirac was perhaps the first to manifestly knit everything together in a practical application of major lasting importance with the Dirac equation in 1928.
The Lie algebra.
The irreducible complex linear representations of the complexification, formula_14 of the Lie algebra formula_15 of the Lorentz group are to be found. A convenient basis for formula_15 is given by the three generators of rotations and the three generators 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
The components of and separately satisfy the commutation relations of the Lie algebra formula_18 and, moreover, they commute with each other,
where are indices which each take values , and is the three-dimensional Levi-Civita symbol. Let formula_20 and formula_21 denote the complex linear span of and respectively.
One has the isomorphisms
where formula_22 is the complexification of formula_23
The utility of these isomorphisms comes from the fact that all irreducible representations of formula_18, and hence all irreducible complex linear representations of formula_25 are known. The irreducible complex linear representation of formula_22 is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of formula_27
The unitarian trick.
The Lie algebra formula_28 is the Lie algebra of formula_29 It contains the compact subgroup with Lie algebra formula_30 The latter is a compact real form of formula_31 Thus from the first statement of the unitarian trick, representations of are in one-to-one correspondence with holomorphic representations of formula_29
By compactness, Peter–Weyl theorem applies to , and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of are precisely the tensor products of irreducible unitary representations of .
By appeal to simple connectedness, the second statement of the unitarian trick is applied. The objects in the following list are in one-to-one correspondence:
Tensor products of representations appear at the Lie algebra level as either of
where is the identity operator. Here, the latter interpretation, which follows from , is intended. The highest weight representations of formula_22 are indexed by for . (The highest weights are actually , but the notation here is adapted to that of formula_37) The tensor products of two such complex linear factors then form the irreducible complex linear representations of formula_31
Finally, the formula_39-linear representations of the real forms of the far left, formula_15, and the far right, formula_25 in are obtained from the formula_42-linear representations of formula_28 characterized in the previous paragraph.
The ("μ", "ν")-representations of sl(2, C).
The complex linear representations of the complexification of formula_44 obtained via isomorphisms in , stand in one-to-one correspondence with the real linear representations of formula_27 The set of all "real linear" irreducible representations of formula_22 are thus indexed by a pair . The complex linear ones, corresponding precisely to the complexification of the real linear formula_18 representations, are of the form , while the conjugate linear ones are the . All others are real linear only. The linearity properties follow from the canonical injection, the far right in , of formula_22 into its complexification. Representations on the form or are given by "real" matrices (the latter are not irreducible). Explicitly, the real linear -representations of formula_22 are
where formula_51 are the complex linear irreducible representations of formula_22 and formula_53 their complex conjugate representations. (The labeling is usually in the mathematics literature , but half-integers are chosen here to conform with the labeling for the formula_54 Lie algebra.) Here the tensor product is interpreted in the former sense of . These representations are concretely realized below.
The ("m", "n")-representations of so(3; 1).
Via the displayed isomorphisms in and knowledge of the complex linear irreducible representations of formula_28 upon solving for and , all irreducible representations of formula_56 and, by restriction, those of formula_15 are obtained. The representations of formula_15 obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible. Since formula_15 is semisimple, all its representations can be built up as direct sums of the irreducible ones.
Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers and , conventionally written as one of
where is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given by
where is the -dimensional unit matrix and
are the -dimensional irreducible representations of formula_62 also termed spin matrices or "angular momentum matrices". These are explicitly given as
where denotes the Kronecker delta. In components, with , , the representations are given by
Off-diagonal direct sums.
Since for any irreducible representation for which it is essential to operate over the field of complex numbers, the direct sum of representations and have particular relevance to physics, since it permits to use linear operators over real numbers.
The group.
The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence. The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras. The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted formula_65
If formula_66 for some vector space is a representation, a representation of the connected component of is defined by
This definition applies whether the resulting representation is projective or not.
Surjectiveness of exponential map for SO(3, 1).
From a practical point of view, it is important whether the first formula in can be used for all elements of the "group". It holds for all formula_67, however, in the general case, e.g. for formula_2, not all are in the image of .
But formula_69 "is" surjective. One way to show this is to make use of the isomorphism formula_70 the latter being the Möbius group. It is a quotient of formula_71 (see the linked article). The quotient map is denoted with formula_72 The map formula_73 is onto. Apply with being the differential of at the identity. Then
Since the left hand side is surjective (both and are), the right hand side is surjective and hence formula_75 is surjective. Finally, recycle the argument once more, but now with the known isomorphism between and formula_76 to find that is onto for the connected component of the Lorentz group.
Fundamental group.
The Lorentz group is "doubly connected", i. e. is a group with two equivalence classes of loops as its elements.
To exhibit the fundamental group of , the topology of its covering group formula_2 is considered. By the polar decomposition theorem, any matrix formula_78 may be "uniquely" expressed as
where is unitary with determinant one, hence in , and is Hermitian with trace zero. The trace and determinant conditions imply:
The manifestly continuous one-to-one map is a homeomorphism with continuous inverse given by (the locus of is identified with formula_81)
explicitly exhibiting that formula_2 is simply connected. But formula_84 where formula_85 is the center of formula_2. Identifying and amounts to identifying with , which in turn amounts to identifying antipodal points on formula_87 Thus topologically,
where last factor is not simply connected: Geometrically, it is seen (for visualization purposes, formula_89 may be replaced by formula_90) that a path from to in formula_91 "is" a loop in formula_92 since and are antipodal points, and that it is not contractible to a point. But a path from to , thence to again, a loop in formula_89 and a "double loop" (considering , where formula_94 is the covering map) in formula_92 that "is" contractible to a point (continuously move away from "upstairs" in formula_89 and shrink the path there to the point ). Thus is a group with two equivalence classes of loops as its elements, or put more simply, is "doubly connected".
Projective representations.
Since has two elements, some representations of the Lie algebra will yield projective representations. Once it is known whether a representation is projective, formula 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 in ) is used to represent the group element in the standard representation.
For the Lorentz group, the -representation is projective when is a half-integer. See the section spinors.
For a projective representation of , it holds that
since any loop in 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 is a double-valued function. It is not possible to consistently chose a sign to obtain a continuous representation of all of , but this is possible "locally" around any point.
The covering group SL(2, C).
Consider formula_5 as a "real" Lie algebra with basis
where the sigmas are the Pauli matrices. From the relations
is obtained
which are exactly on the form of the -dimensional version of the commutation relations for formula_15 (see conventions and Lie algebra bases below). Thus, the map , , extended by linearity is an isomorphism. Since formula_2 is simply connected, it is the universal covering group of .
A geometric view.
Let be a path from to , denote its homotopy class by and let be the set of all such homotopy classes. Define the set
and endow it with the multiplication operation
where formula_101 is the path multiplication of formula_102 and formula_103:
With this multiplication, becomes a group isomorphic to formula_105 the universal covering group of . Since each has two elements, by the above construction, there is a 2:1 covering map . According to covering group theory, the Lie algebras formula_106 and formula_107 of are all isomorphic. The covering map is simply given by .
An algebraic view.
For an algebraic view of the universal covering group, let formula_2 act on the set of all Hermitian matrices formula_109 by the operation
The action on formula_109 is linear. An element of formula_109 may be written in the form
The map is a group homomorphism into formula_112 Thus formula_113 is a 4-dimensional representation of formula_2. Its kernel must in particular take the identity matrix to itself, and therefore . Thus for in the kernel so, by Schur's lemma, is a multiple of the identity, which must be since . The space formula_109 is mapped to Minkowski space , via
The action of on formula_109 preserves determinants. The induced representation of formula_2 on formula_118 via the above isomorphism, given by
preserves the Lorentz inner product since
This means that belongs to the full Lorentz group . By the main theorem of connectedness, since formula_2 is connected, its image under in is connected, and hence is contained in .
It can be shown that the Lie map of formula_121 is a Lie algebra isomorphism: formula_122 The map is also onto.
Thus formula_2, since it is simply connected, is the universal covering group of , isomorphic to the group of above.
Non-surjectiveness of exponential mapping for SL(2, C).
The exponential mapping formula_124 is not onto. The matrix
is in formula_105 but there is no formula_126 such that .
In general, if is an element of a connected Lie group with Lie algebra formula_127 then, by ,
 
The matrix can be written
Realization of representations of and and their Lie algebras.
The complex linear representations of formula_5 and formula_2 are more straightforward to obtain than the formula_130 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 formula_5 are exactly the -representations. They can be exponentiated too. The -representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here).
The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of and there is no factor of in the exponential mapping compared to the physics convention used elsewhere. Let the basis of formula_5 be
This choice of basis, and the notation, is standard in the mathematical literature.
Complex linear representations.
The irreducible holomorphic -dimensional representations formula_133 can be realized on the space of homogeneous polynomial of degree in 2 variables formula_134 the elements of which are
The action of formula_2 is given by
The associated formula_5-action is, using and the definition above, for the basis elements of formula_138
With a choice of basis for formula_139, these representations become matrix Lie algebras.
Real linear representations.
The -representations are realized on a space of polynomials formula_140 in formula_141 homogeneous of degree in formula_142 and homogeneous of degree in formula_143 The representations are given by
By employing again it is found that
In particular for the basis elements,
 + \overline{z_2}\frac{\partial}{\partial \overline{z_2}} \\
\phi_{\mu,\nu}(X) &= -z_2\frac{\partial}{\partial z_1} - \overline{z_2}\frac{\partial}{\partial \overline{z_1}} \\
Properties of the ("m", "n") representations.
The representations, defined above via (as restrictions to the real form formula_144) of tensor products of irreducible complex linear representations and of formula_138 are irreducible, and they are the only irreducible representations.
Dimension.
The representations are -dimensional. This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of formula_2 and formula_22. For a Lie general algebra formula_107 the Weyl dimension formula,
applies, where is the set of positive roots, is the highest weight, and is half the sum of the positive roots. The inner product formula_150 is that of the Lie algebra formula_127 invariant under the action of the Weyl group on formula_152 the Cartan subalgebra. The roots (really elements of formula_153 are via this inner product identified with elements of formula_154 For formula_138 the formula reduces to , "where the present notation must be taken into account". The highest weight is . By taking tensor products, the result follows.
Faithfulness.
If a representation of a Lie group is not faithful, then is a nontrivial normal subgroup. There are three relevant cases.
In the case of , the first case is excluded since is semi-simple. The second case (and the first case) is excluded because is simple. For the third case, is isomorphic to the quotient formula_156 But formula_85 is the center of formula_158 It follows that the center of is trivial, and this excludes the third case. The conclusion is that every representation and every projective representation for finite-dimensional vector spaces are faithful.
By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial non-discrete normal subgroups replaced by (one-dimensional) nontrivial ideals in the Lie algebra, and the center of replaced by the center of formula_159The center of any semisimple Lie algebra is trivial and formula_15 is semi-simple and simple, and hence has no non-trivial ideals.
A related fact is that if the corresponding representation of formula_2 is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding formula_2 representation is not faithful, but is .
Non-unitarity.
The Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary. This is due to the non-compactness of the Lorentz group. In fact, a connected simple non-compact Lie group cannot have "any" nontrivial unitary finite-dimensional representations. There is a topological proof of this. Let , where is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group . Then where is the compact subgroup of consisting of unitary transformations of . The kernel of is a normal subgroup of . Since is simple, is either all of , in which case is trivial, or is trivial, in which case is faithful. In the latter case is a diffeomorphism onto its image, and is a Lie group. This would mean that is an embedded non-compact Lie subgroup of the compact group . This is impossible with the subspace topology on since all "embedded" Lie subgroups of a Lie group are closed If were closed, it would be compact, and then would be compact, contrary to assumption.
In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of and used in the construction are Hermitian. This means that is Hermitian, but is anti-Hermitian. The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.
Restriction to SO(3).
The representation is, however, unitary when restricted to the rotation subgroup , but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an representation have -invariant subspaces of highest weight (spin) , where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) is -dimensional. So for example, the (, ) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.
Since the angular momentum operator is given by , the highest spin in quantum mechanics of the rotation sub-representation will be and the "usual" rules of addition of angular momenta and the formalism of 3-j symbols, 6-j symbols, etc. applies.
Spinors.
It is the -invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the representation has spin if is half-integral. The simplest are and , the Weyl-spinors of dimension . Then, for example, and are a spin representations of dimensions and respectively. According to the above paragraph, there are subspaces with spin both and in the last two cases, so these representations cannot likely represent a "single" physical particle which must be well-behaved under . It cannot be ruled out in general, however, that representations with multiple subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out "unphysical components", leaving only a single spin.
Construction of pure spin representations for any (under ) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints.
Dual representations.
The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation:
Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If is an element of the Weyl group of a semisimple Lie algebra, then . In the case of formula_138 the Weyl group is . It follows that each is isomorphic to its dual formula_166 The root system of formula_167 is shown in the figure to the right. The Weyl group is generated by formula_168 where formula_169 is reflection in the plane orthogonal to as ranges over all roots. Inspection shows that so . Using the fact that if are Lie algebra representations and , then , the conclusion for is
Complex conjugate representations.
If is a representation of a Lie algebra, then formula_171 is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication. In general, every irreducible representation of formula_172 can be written uniquely as , where
with formula_174 holomorphic (complex linear) and formula_175 anti-holomorphic (conjugate linear). For formula_138 since formula_177 is holomorphic, formula_178 is anti-holomorphic. Direct examination of the explicit expressions for formula_179 and formula_180 in equation below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression also allows for identification of formula_174 and formula_175 for formula_183 as
Using the above identities (interpreted as pointwise addition of functions), for yields
where the statement for the group representations follow from = . It follows that the irreducible representations have real matrix representatives if and only if . Reducible representations on the form have real matrices too.
The adjoint representation, the Clifford algebra, and the Dirac spinor representation.
In general representation theory, if is a representation of a Lie algebra formula_127 then there is an associated representation of formula_127 on , also denoted , given by
Likewise, a representation of a group yields a representation on of , still denoted , given by
If and are the standard representations on formula_188 and if the action is restricted to formula_189 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 formula_190 or formula_191) 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 is a projective representation, then direct calculation using shows that the induced representation on is a proper representation, i.e. a representation without phase factors.
In quantum mechanics this means that if or is a representation acting on some Hilbert space , then the corresponding induced representation acts on the set of linear operators on . As an example, the induced representation of the projective spin representation on is the non-projective 4-vector (, ) representation.
For simplicity, consider only the "discrete part" of , that is, given a basis for , the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4-vector representation of above on this simplified has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, formula_192 whose complexification is formula_193 generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the , a pseudoscalar irrep, also the , but with parity inversion eigenvalue , see the next section below, the already mentioned vector irrep, , a pseudovector irrep, with parity inversion eigenvalue +1 (not −1), and a tensor irrep, . The dimensions add up to . In other words,
where, as is customary, a representation is confused with its representation space.
The spin representation.
The six-dimensional representation space of the tensor -representation inside formula_194 has two roles. The
where formula_195 are the gamma matrices, the sigmas, only of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,
and hence constitute a representation (in addition to spanning a representation space) sitting inside formula_192 the spin representation. For details, see bispinor and Dirac algebra.
The conclusion is that every element of the complexified formula_194 in (i.e. every complex matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on formula_198 making it a space of bispinors.
Reducible representations.
There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained by taking direct sums, tensor products, and quotients of the irreducible representations. Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e.g. formula_199 and the Poincaré group. These representations are in general not irreducible.
The Lorentz group and its Lie algebra have the "complete reducibility property". This means that every representation reduces to a direct sum of irreducible representations. The reducible representations will therefore not be discussed.
Space inversion and time reversal.
The (possibly projective) representation is irreducible as a representation , the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If it can be extended to a representation of all of , the full Lorentz group, including space parity inversion and time reversal. The representations can be extended likewise.
Space parity inversion.
For space parity inversion, the adjoint action of on formula_15 is considered, where is the standard representative of space parity inversion, , given by
It is these properties of and under that motivate the terms "vector" for and pseudovector or "axial vector" for . In a similar way, if is any representation of formula_15 and is its associated group representation, then acts on the representation of by the adjoint action, for formula_202 . If is to be included in , then consistency with requires that
holds, where and are defined as in the first section. This can hold only if and have the same dimensions, i.e. only if . When then can be extended to an irreducible representation of , the orthochronous Lorentz group. The parity reversal representative does not come automatically with the general construction of the representations. It must be specified separately. The matrix (or a multiple of modulus −1 times it) may be used in the representation.
If parity is included with a minus sign (the matrix ) in the representation, it is called a pseudoscalar representation.
Time reversal.
Time reversal , acts similarly on formula_15 by
By explicitly including a representative for , as well as one for , a representation of the full Lorentz group 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 , in addition to the and generate the group. These are interpreted as generators of translations. The time-component is the Hamiltonian . The operator satisfies the relation
in analogy to the relations above with formula_15 replaced by the full Poincaré algebra. By just cancelling the 's, the result would imply that for every state with positive energy in a Hilbert space of quantum states with time-reversal invariance, there would be a state with negative energy . Such states do not exist. The operator is therefore chosen antilinear and antiunitary, so that it anticommutes with , resulting in , and its action on Hilbert space likewise becomes antilinear and antiunitary. It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix. This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, 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, , 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 and , charge conjugation symmetry , has nothing directly to do with Lorentz invariance.
Action on function spaces.
If is a vector space of functions of a finite number of variables , then the action on a scalar function formula_205 given by
produces another function . Here is an -dimensional representation, and is a possibly infinite-dimensional representation. A special case of this construction is when is a space of functions defined on the a linear group itself, viewed as a -dimensional manifold embedded in formula_206 (with the dimension of the matrices). This is the setting in which the Peter–Weyl theorem and the Borel–Weil theorem are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations. The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. formula_158
The following exemplifies action of the Lorentz group and the rotation subgroup on some function spaces.
Euclidean rotations.
The subgroup of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space
where formula_209 are the spherical harmonics. An arbitrary square integrable function one the unit sphere can be expressed as
where the are generalized Fourier coefficients.
The Lorentz group action restricts to that of and is expressed as
where the are obtained from the representatives of odd dimension of the generators of rotation.
The Möbius group.
The identity component of the Lorentz group is isomorphic to the Möbius group . 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 acts on the plane according to
and can be represented by complex matrices
since multiplication by a nonzero complex scalar does not change . These are elements of formula_2 and are unique up to a sign (since give the same ), hence formula_211
The Riemann P-functions.
The Riemann P-functions, solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as
where the are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is
The set of constants in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation. Its "exponents", i. e. solutions of the indicial equation, for expansion around the singular point are and ,corresponding to the two linearly independent solutions, and for expansion around the singular point they are and . Similarly, the exponents for are and for the two solutions.
One has thus
where the condition (sometimes called Riemann's identity)
on the exponents of the solutions of Riemann's differential equation has been used to define .
The first set of constants on the left hand side in , denotes the regular singular points of Riemann's differential equation. The second set, , are the corresponding exponents at for one of the two linearly independent solutions, and, accordingly, are exponents at for the second solution.
Define an action of the Lorentz group on the set of all Riemann P-functions by first setting
where are the entries in
for a Lorentz transformation.
Define
where is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Möbius transformation of the argument is that of shifting the poles to new locations, hence changing the critical points, but there is no change in the exponents of the differential equation the new function satisfies. The new function is expressed as
where
Infinite-dimensional unitary representations.
History.
The Lorentz group and its double cover formula_2 also have infinite dimensional unitary representations, studied independently by , and at the instigation of Paul Dirac. This trail of development begun with where he devised matrices and necessary for description of higher spin (compare Dirac matrices), elaborated upon by , see also , and proposed precursors of the Bargmann-Wigner equations. In he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors. These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinite-dimensional generalization of spinors in his 1947 paper.
The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by and , based on an analogue for formula_2 of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in and .
The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3-dimensional hyperbolic space sitting in Minkowski space is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on formula_215 This theory is discussed in , , and the posthumous text of .
Principal series for SL(2, C).
The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup  of formula_216 Since the one-dimensional representations of correspond to the representations of the diagonal matrices, with non-zero complex entries and , they thus have the form
for an integer, real and with . The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when is replaced by . By definition the representations are realized on sections of line bundles on formula_218 which is isomorphic to the Riemann sphere. When , these representations constitute the so-called spherical principal series.
The restriction of a principal series to the maximal compact subgroup of  can also be realized as an induced representation of  using the identification , where is the maximal torus in  consisting of diagonal matrices with . It is the representation induced from the 1-dimensional representation , and is independent of . By Frobenius reciprocity, on  they decompose as a direct sum of the irreducible representations of  with dimensions with a non-negative integer.
Using the identification between the Riemann sphere minus a point and formula_219 the principal series can be defined directly on formula_220 by the formula
Irreducibility can be checked in a variety of ways:
Complementary series for SL(2, C).
The for , the complementary series is defined on formula_220 for the inner product
with the action given by
The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of , each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of . Irreducibility can be proved by analyzing the action of formula_107 on the algebraic sum of these subspaces or directly without using the Lie algebra.
Plancherel theorem for SL(2, C).
The only irreducible unitary representations of formula_2 are the principal series, the complementary series and the trivial representation.
Since acts as on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided is taken to be even.
To decompose the left regular representation of  on formula_230 only the principal series are required. This immediately yields the decomposition on the subrepresentations formula_231 the left regular representation of the Lorentz group, and formula_232 the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with "k" even and the latter only those with .)
The left and right regular representation and are defined on formula_230 by
Now if is an element of , the operator formula_235 defined by
is Hilbert–Schmidt. Define a Hilbert space  by
where
and formula_239 denotes the Hilbert space of Hilbert–Schmidt operators on formula_240 Then the map  defined on by
extends to a unitary of formula_230 onto .
The map  satisfies the intertwining property
If are in then by unitarity
Thus if formula_245 denotes the convolution of formula_246 and formula_247 and formula_248 then
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 formula_250 By a theorem of Jacques Dixmier and Paul Malliavin, every smooth compactly supported function on formula_251 is a finite sum of convolutions of similar functions, the inversion formula holds for such . It can be extended to much wider classes of functions satisfying mild differentiability conditions.
Classification of representations of SO(3, 1).
The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to "assume" they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinite-dimensional representation on a Hilbert space of is at hand. Since is a subgroup, is a representation of it as well. Each irreducible subrepresentation of is finite-dimensional, and the representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of if is unitary.
The steps are the following:
Step 1.
One suitable choice of basis and labeling is given by
If this were a "finite-dimensional" representation, then would correspond the lowest occurring eigenvalue of in the representation, equal to , and would correspond to the highest occurring eigenvalue, equal to . In the infinite-dimensional case, retains this meaning, but does not. For simplicity, it is assumed that a given occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown that the assumption is possible to avoid (with a slightly more complicated calculation) with the same results.
Step 2.
The next step is to compute the matrix elements of the operators and forming the basis of the Lie algebra of formula_37 The matrix elements of formula_254 and formula_255 (the "complexified" Lie algebra is understood) are known from the representation theory of the rotation group, and are given by
where the labels and have been dropped since they are the same for all basis vectors in the representation.
Due to the commutation relations
the triple is a vector operator and the Wigner–Eckart theorem applies for computation of matrix elements between the states represented by the chosen basis. The matrix elements of
where the superscript signifies that the defined quantities are the components of a spherical tensor operator of rank (which explains the factor as well) and the subscripts are referred to as in formulas below, are given by
Here the first factors on the right hand sides are Clebsch–Gordan coefficients for coupling with to get . The second factors are the "reduced matrix elements". They do not depend on or , but depend on and, of course, . For a complete list of non-vanishing equations, see .
Step 3.
The next step is to demand that the Lie algebra relations hold, i.e. that
This results in a set of equations for which the solutions are
where
Step 4.
The imposition of the requirement of unitarity of the corresponding representation of the "group" restricts the possible values for the arbitrary complex numbers and . Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning
This translates to
leading to
where is the angle of on polar form. For follows formula_266 and formula_267 is chosen by convention. There are two possible cases:
This shows that the representations of above are "all" infinite-dimensional irreducible unitary representations.
Explicit formulas.
Conventions and Lie algebra bases.
The metric of choice is given by , and the physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of basis for the Lie algebra is, in the 4-vector representation, given by:
The commutation relations of the Lie algebra formula_15 are:
In three-dimensional notation, these are
The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol above and in the sequel should be observed.
Weyl spinors and bispinors.
By taking, in turn, and and by setting
in the general expression , and by using the trivial relations and , it follows
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) and , whose elements and are called left- and right-handed Weyl spinors respectively. Given
their direct sum as representations is formed,
This is, up to a similarity transformation, the Dirac spinor representation of formula_37 It acts on the 4-component elements of , called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of formula_37 Expressions for the group representations are obtained by exponentiation.
Open problems.
The classification and characterization of the representation theory of the Lorentz group was completed in 1947. But in association with the Bargmann–Wigner programme, there are yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations.
The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the little group of the Poincaré group of spacelike vectors in higher spacetime dimension. The corresponding infinite-dimensional unitary representations of the (generalized) Poincaré group are the so-called "tachyonic representations". Tachyons appear in the spectrum of bosonic strings and are associated with instability of the vacuum. 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.
One open problem is the completion of the Bargmann–Wigner programme for the isometry group of the de Sitter spacetime . Ideally, the physical components of wave functions would be realized on the hyperboloid of radius embedded in formula_281 and the corresponding covariant wave equations of the infinite-dimensional unitary representation to be known.