Representation theory of the Lorentz group

Hendrik Antoon Lorentz (right) after whom the Lorentz group is named and Albert Einstein whose special theory of relativity is the main source of application. Photo taken by Paul Ehrenfest 1921.

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; 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, 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.

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 Template:Math of the full Lorentz group Template:Math 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) Template:Math of Template:Math is obtained, and explicitly given in terms of action on a function space in representations of SL(2, C) and sl(2, C). 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-function appearing as examples. The infinite-dimensional case of irreducible unitary representations is classified and realized for the principal series and the complementary series. Finally, the Plancherel formula for Template:Math is given.

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 programme, 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 non-technical introduction contains some prerequisite material for readers not familiar with representation theory. The Lie algebra basis and other adopted conventions are given in conventions and Lie algebra bases.

Non-technical introduction to representation theory

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 finite-dimensional and infinite-dimensional representations of the Lorentz group. The reader familiar with these concepts should skip by. Template:Hidden begin

Symmetry groups

A 90° clockwise rotation of the square around its center. This is one of the symmetries of a square.

The mathematical notion of a group and the notion of symmetry in both mathematics and physics are intimately related. A group has the simple property that if one element of a group is multiplied by another, the result is another element of the group. The same can, mutatis mutandis, be said of symmetries. Apply one symmetry operation (physically or by changing coordinate system), and then another one. The result is that of applying a single symmetry operation. Else they don't qualify as a symmetry operations the present context. Group theory is thus the mathematical language in which symmetries of nature are expressed. These may relate to very concrete symmetries of physical objects, like the symmetries of a square. One then speaks of the symmetry group associated with the object.

In the case of a square, the symmetry group, called the [[w:Dihedral group|dihedral group Template:Math]], is finite. For instance, only some rotations, and some reflections in the plane, will make the transformed square look exactly like it did before the symmetry operation. Other objects possess higher symmetry. The sphere is the extreme example. It possesses full rotational symmetry and reflectional symmetry. Rotate or reflect a ball with any kind of rotation or reflection about any plane through the origin, and it will look exactly the same as before the symmetry operation.

A central fact is that the symmetry groups can be represented by matrices. In the case of Template:Math for the square, the matrix representation is composed of eight Template:Math matrices. In the case of the symmetries of a sphere, the matrix group is the orthogonal group of three dimensions. These are Template:Math matrices.

Symmetry of space and time

Main articles: Symmetry (physics)#Spacetime symmetries, Spacetime symmetries, and Lorentz group

The sphere, a maximally symmetric object.

Less obvious is that space itself possesses symmetry. It too looks the same, no matter how one rotates it, so it has rotational symmetry. In fact, in this case, it is more practical to use passive rotations, meaning the observer rotates himself and does not attempt to physically rotate the universe. 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. (Envisage the coordinate system being fixed in the rotated observer. Then actively rotate the observer only.) In this way, every point in space obtains new coordinates, just 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. 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 best though of as just having the observer set his clock at minus what it shows and then have the clock's hands move counterclockwise. Physical time progresses forward as always.

Lorentz transformations

Main article: Lorentz transformation

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 four-dimensional 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 four-vector representing an event in Lorentz frame transforms as

$x^{\prime }=\left[\begin{array}{c}x{\text{'}}^0\\ x{\text{'}}^1\\ x{\text{'}}^2\\ x{\text{'}}^3\end{array}\right]=\left[\begin{array}{cccc}{\lambda }_{00}& {\lambda }_{01}& {\lambda }_{02}& {\lambda }_{03}\\ {\lambda }_{10}& {\lambda }_{11}& {\lambda }_{12}& {\lambda }_{13}\\ {\lambda }_{20}& {\lambda }_{21}& {\lambda }_{22}& {\lambda }_{23}\\ {\lambda }_{30}& {\lambda }_{31}& {\lambda }_{32}& {\lambda }_{33}\end{array}\right]\left[\begin{array}{c}{x}^0\\ x^1\\ x^2\\ x^3\end{array}\right],$

or on short form

$x^{\prime }=\mathrm{\Lambda }x.$

Multiplication table and representations

Main articles: Cayley table and Representation theory

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 new set of elements, finite-dimensional or infinite-dimensional matrices, giving the same multiplication table after mapping the old elements to the new elements in a one-to-one fashion. 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).

Ordinary Lorentz transformations matrices do not suffice

Lorentz boost of an electric charge, the charge is at rest in one frame or the other.

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 at this point that representation theory enters the picture. The electromagnetic field is usually envisaged by assignment to each point in space a three-dimensional vector representing the electric field and another three-dimensional vector representing the magnetic field. When space is rotated, the 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, and in a way showing that the two vectors certainly 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.

Finite-dimensional representations by matrices

The problem of representation theory of the Lorentz group is, in the finite-dimensional case, to find new sets of matrices, not necessarily Template:Math in size that satisfies the same multiplication table as the matrices in the original Lorentz group. Returning to the example of the electromagnetic field, what is needed here are Template:Math-matrices that can be applied to a Template:Math-dimensional column vector containing the all together six components of the electromagnetic field. Thus one is looking for Template:Math-matrices such that

$F^{\prime }=\left[\begin{array}{c}E{\text{'}}^1\\ E{\text{'}}^2\\ E{\text{'}}^3\\ B{\text{'}}^1\\ B{\text{'}}^2\\ B{\text{'}}^3\end{array}\right]=\left[\begin{array}{cccccc}\mathrm{\Pi }(\mathrm{\Lambda }{)}_{00}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{01}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{02}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{03}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{04}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{05}\\ \mathrm{\Pi }(\mathrm{\Lambda }{)}_{10}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{11}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{12}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{13}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{14}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{15}\\ \mathrm{\Pi }(\mathrm{\Lambda }{)}_{20}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{21}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{22}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{23}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{24}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{25}\\ \mathrm{\Pi }(\mathrm{\Lambda }{)}_{30}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{31}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{32}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{33}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{34}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{35}\\ \mathrm{\Pi }(\mathrm{\Lambda }{)}_{40}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{41}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{42}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{43}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{44}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{45}\\ \mathrm{\Pi }(\mathrm{\Lambda }{)}_{50}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{51}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{52}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{53}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{54}& \mathrm{\Pi }(\mathrm{\Lambda }{)}_{55}\end{array}\right]\left[\begin{array}{c}{E}^1\\ E^2\\ E^3\\ B^1\\ B^2\\ B^3\end{array}\right],$

in short

$F^{\prime }=\mathrm{\Pi }(\mathrm{\Lambda })F,$

correctly expresses the transformation of the electromagnetic field under the Lorentz transformation Template:Math. The same reasoning can be applied to Dirac's bispinors. While these have Template:Math components, the original Template:Math-matrices in the Lorentz group will not do the job properly, not even when restricted to mere rotations. Another Template:Math-representation is needed.

The sections dedicated to finite-dimensional representations are dedicated to exposing all such representations by finite-dimensional matrices that respect the multiplication table.

Infinite-dimensional representations by action on vector spaces of functions

Infinite-dimensional representations are usually realized as acting on sets of real or complex-valued functions on a set Template:Mvar endowed with a group action. A set being endowed with a group action Template:Mvar means, in essence, that if Template:Math and Template:Math that Template:Math with Template:Math. Now if Template:Math denotes the set of all complex-valued functions on Template:Mvar, which is a vector space, a representation Template:Mvar of Template:Mvar can be defined by

$(\mathrm{\Pi }(g))f(x)=f(A(g^{-1})x),f\in {ℂ}^X,g\in G,x\in X.$

The point to make is that again one has

$(\mathrm{\Pi }(g))f\in {ℂ}^X.$

and one has a representation of Template:Mvar. This representation of Template:Mvar is finite-dimensional if and only if Template:Mvar is a finite set. This method is very general, and one typically explores vector spaces of more specialized functions on sets close at hand. To illustrate this procedure, consider a group Template:Mvar of Template:Math-dimensional matrices as a subset of Euclidean space Template:Math, and let the space of functions be polynomials, perhaps of some maximum degree Template:Mvar, or even homogeneous polynomials of degree Template:Mvar, all defined on Template:Math. Then restrict those functions to Template:Math. Now observe that the set Template:Math automatically comes equipped with group actions, namely

$L_{g}h=gh,R_{g}h=hg,C_{g}h=gh{g}^{-1},g,h\in G.$

Here Template:Math denotes left action (by Template:Mvar), Template:Math denotes right action (by Template:Mvar), and Template:Math denotes conjugation (by Template:Mvar). 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 Template:Mvar on Template:Math.

The goal in the infinite-dimensional 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.

Infinite-dimensional representations viewed as infinite-dimensional matrices

In order to relate this to the finite-dimensional case, one may chose a basis for the vector space of functions and simply then examine what happens to the basis functions under a given transformation. Take image of the first basis function under a transformation, expressed as a linear combination the basis functions. Explicitly, if Template:Math is a basis, compute

$\begin{array}{cc}\mathrm{\Pi }(\mathrm{\Lambda })f_1& ={\lambda }_{11}{f}_1+{\lambda }_{21}{f}_2+\cdots ,\\ \mathrm{\Pi }(\mathrm{\Lambda })f_2& ={\lambda }_{12}{f}_1+{\lambda }_{22}{f}_2+\cdots ,\\ & ⋮\end{array}$

The coefficients of the basis functions in this expression is then the first column in a representative matrix. Proceed. In general, the resulting matrix is countably infinite in dimension:

$\mathrm{\Pi }(\mathrm{\Lambda })=\left[\begin{array}{ccc}{\lambda }_{11}& {\lambda }_{12}& \cdots \\ {\lambda }_{21}& {\lambda }_{22}& \cdots \\ ⋮& ⋮& \ddots \end{array}\right]$

Again, it is required that the set of infinite matrices obtained this way stand in one-to-one correspondence with the original Template:Math-matrices and that the multiplication table is the right one - the one of the Template:Math-matrices. It should be emphasized that in the infinite-dimensional case, one is rarely concerned with these 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 algebra

Main article: Lie algebra

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 can near the identity 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 one-to-one 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 finite-dimensional vector space, in the case of the Lorentz Lie algebra the dimension is Template:Math, 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.

The metric signature to be used below is Template:Math and the metric is given by Template:Math. 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 standard representation, given by

$\begin{array}{cc}{J}_1& =J^{23}=-J^{32}=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right),J_2=J^{31}=-J^{13}=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right),J_3=J^{12}=-J^{21}=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right),\\ K_1& =J^{01}=J^{10}=i\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),K_2=J^{02}=J^{20}=i\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),K_3=J^{03}=J^{30}=i\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right).\end{array}$

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.

Mathematics

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 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 Poincare 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 (as of 2006) is the completion of the Bargmann–Wigner programme for the isometry group Template:Math of the de Sitter spacetime Template:Math. Ideally, one would like to see the physical components of wave functions realized on the hyperboloid Template:Math of radius Template:Math embedded in Template:Math and the corresponding Template:Math covariant wave equations of the infinite-dimensional unitary representation to be known.

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 P-functions, which are expressible as hypergeometric functions.

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, one begins with 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 purpose one may make the following definition: A relativistic wave function is a set of Template:Mvar functions Template:Math on spacetime which transforms under an arbitrary proper Lorentz transformation Template:Math as

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

where Template:Math is an Template:Math-dimensional matrix representative of Template:Math belonging to some direct sum of the Template:Math 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 (Template:Math) and bispinors respectively (Template:Math). The electromagnetic field is a relativistic wave function according to this definition, transforming under Template:Math.

Quantum field theory

In QFT, 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 one may deduce a realization of the generators of the Lorentz group.

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 of some Template:Math-component field operator: Given a matrix representation as above, a relativistic field operator is a set of Template:Mvar operator valued functions on spacetime which transforms under proper Lorentz transformations Template:Math according to

${\mathrm{\Psi }}^{\alpha }(x)\to \mathrm{\Psi }{\text{'}}^{\alpha }(x^{\prime })=U[\mathrm{\Lambda }]{\mathrm{\Psi }}^{\alpha }(x)U[{\mathrm{\Lambda }}^{-1}]=𝒟{[{\mathrm{\Lambda }}^{-1}{]}^{\alpha }}_{\beta }{\mathrm{\Psi }}^{\beta }(\mathrm{\Lambda }x)$

By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass Template:Mvar and spin Template:Mvar (or helicity), one finds

${\mathrm{\Psi }}^{\alpha }(x)=\sum _{\sigma }\int dp[a(𝐩,\sigma )u^{\alpha }(𝐩,\sigma )e^{ip\cdot x}+a^{†}(𝐩,\sigma )v^{\alpha }(𝐩,\sigma )e^{-ip\cdot x}],$

where Template:Math are interpreted as creation and annihilation operators respectively. The creation operator Template:Math transforms according to

$a^{†}(𝐩,\sigma )\to a{\text{'}}^{†}({𝐩}^{\prime },\sigma )=U[\mathrm{\Lambda }]a^{†}(𝐩,\sigma )U[{\mathrm{\Lambda }}^{-1}]=a^{†}(\mathrm{\Lambda }𝐩,\rho )D^{(s)}{[R(\mathrm{\Lambda },𝐩{)}^{-1}{]}^{\rho }}_{\sigma },$

and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin Template:Math of the particle. The connection between the two is the wave function, also called cofficient function

$v^{\alpha }(𝕡,\sigma )e^{-ip\cdot x}$

that carries both the indices Template:Math operated on by Lorentz transformations and the indices Template:Math operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection. 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 Template:Math 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 Template:Math dimensions, the generalized Lorentz groups Template:Math of the appropriate dimension take the place of Template:Math.

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 [[w:graded Lie algebra|Template:Math-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 Template:Math) belong to a (0, ½) or (½, 0) 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 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.

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, one has to deal with time reversal and space inversion 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.

Strategy

Classification of the finite-dimensional irreducible representations generally consists of two steps. The first step is to hypothesize the existence of representations. One assumes heuristically that all representations that a priori could exist, do exist. One investigates the properties of these hypothetical representations, primarily using the Lie algebra. The goal of this study is twofold. First, some of these hypothetical representations may not exist. The goal in this situation is to show that their existence would imply a falsehood such as Template:Math. If this can be done, then the initial hypothesis that the representation existed must be false, and one can therefore exclude these hypothetical representations from later studies. Second, one can better understand the representations that do exist. These representations must have enough structure to manifest the symmetries of the group action, but describing this structure may not be easy. Before a classification has been completed, it is unclear which representations fall into the first class and which fall into the second.

If this first step of the classification is successful, it results in a tentative classification of the possible representations. This is often a short list. Each list entry is a single representation or a family of related representations, and ideally, the entry gives requirements so specific that they can be met by at most a single representation. The second step consists of explicit construction of the representations on this list. If successful, it justifies the existence hypotheses made in the first step. The results of investigations performed in the first step provide hints about how to construct the representations, i.e. construction of a vector space Template:Mvar and a specified Lie algebra action on Template:Mvar, since most of the properties they must have are then known.

For finite-dimensional irreducible representations of finite-dimensional semisimple Lie algebras the general result is Cartan's theorem of highest weight. It provides a classification of the irreducible representations in terms of the weights of the Lie algebra.

For some semisimple Lie algebras, especially non-compact ones, it is easier to proceed indirectly via Weyl's unitarian trick instead of applying Cartan's theorem directly. In the present case of Template:Math one sets up a chain of isomorphisms between Lie algebras and other correspondences preserving irreducible representations, so that the representations may be obtained from representations of Template:Math. See equation Template:EquationNoteand references around it. It is essential here that Template:Math is compact, since then the irreducible representations of Template:Math are simply tensor products of irreducible representations of Template:Math, that can all be obtained from the irreducible representations of Template:Math.

Then the classification part. Cartan's theorem is applied to Template:Math (together with knowledge of its highest weights) and one obtains a classification of the representations of Template:Math via Template:EquationNote. An explicit construction of the representations of Template:Math is then given (which is not much more difficult to obtain than the more basic Template:Math representations), thus completing the task with the Template:Math representations of Template:Math as the final result.

Representative matrices may be obtained by choice of basis in the representation space. An explicit formula for matrix elements is presented and some common representations are listed.

The Lie correspondence is subsequently employed for obtaining group representations of the connected component of the Lorentz group, Template:Math. This is effected by taking the matrix exponential of the matrices of the Lie algebra representation, a topic which is investigated in some depth. A subtlety arises due to the (in physics parlance) doubly connected nature of Template:Math. This results in the projective representations or two-value representations that are actually spin representations of the covering group Template:Math.

The Lie correspondence gives results only for the connected component of the groups, and thus the components of the full Lorentz that contain the operations of time reversal and space inversion are treated separately, mostly from physical considerations, by defining representatives for the space inversion and time reversal matrices.

The Lie algebra

Wilhelm Killing, Independent discoverer of Lie algebras. The simple Lie algebras were first classified by him in 1888.

According to the general representation theory of Lie groups, one first looks for the representations of the complexification, Template:Math of the Lie algebra Template:Math of the Lorentz group. A convenient basis for Template:Math is given by the three generators Template:Math of rotations and the three generators Template:Math of boosts. They are explicitly given in conventions and Lie algebra bases.

Now complexify the Lie algebra, and then change basis to the components of

$𝐀=\frac{𝐉+i𝐊}{2},𝐁=\frac{𝐉-i𝐊}{2}.$

One may verify that the components of Template:Math and Template:Math separately satisfy the commutation relations of the Lie algebra su(2) and moreover that they commute with each other,

$[A_i,A_j]=i{\epsilon }_{ijk}{A}_k,[B_i,B_j]=i{\epsilon }_{ijk}{B}_k,[A_i,B_j]=0,$

where Template:Math are indices which each take values Template:Math, and Template:Math is the three-dimensional Levi-Civita symbol. Let Template:Math and Template:Math denote the complex linear span of Template:Math and Template:Math respectively.

One has the isomorphisms Template:NumBlk

where sl(2, C) is the complexification of su(2) ≈ Template:Math ≈ Template:Math.

The utility of these isomorphisms comes from the fact that all irreducible representations of su(2) are known. Every irreducible representation of su(2) is isomorphic to one of the highest weight representations. Moreover, there is a one-to-one correspondence between linear representations of su(2) and complex linear representations of sl(2, C).

The unitarian trick

Hermann Weyl, inventor the unitarian trick. There are several concepts and formulas in representation theory named after Weyl, e.g. the Weyl group and the Weyl character formula.
Photo courtesy of ETH-Bibliothek Zürich, Bildarchiv

In Template:EquationNote, all isomorphisms are Template:Math-linear (the last is just a defining equality). The most important part of the manipulations below is that the Template:Math-linear (irreducible) representations of a (real or complex) Lie algebra are in one-to-one correspondence with Template:Math-linear (irreducible) representation of its complexification. With this in mind, it is seen that the Template:Math-linear representations of the real forms of the far left, Template:Math, and the far right, Template:Math, in Template:EquationNotecan be obtained from the Template:Math-linear representations of Template:Math.

The manipulations to obtain representations of a non-compact algebra (here so(3; 1)), and subsequently the non-compact group itself, from qualitative knowledge about unitary representations of a compact group (here Template:Math) is a variant of Weyl's so-called unitarian trick. The trick specialized to Template:Math can be summarized concisely.

Let Template:Math be a finite-dimensional complex vector space. The following statements are equivalent, in the sense that if one of them holds, then there is a uniquely determined (modulo choice of basis for Template:Math) corresponding representation (either via given Lie algebra isomorphisms, or via complexification of Lie algebras per above, or via restriction to real forms, or via the exponential mapping (to be introduced), or, finally, via a standard mechanism (also to be introduced) for obtaining Lie algebra representations given group representations) of the appropriate type for the other groups and Lie algebras:

• There is a representation of Template:Math on Template:Math.
• There is a representation of Template:Math on Template:Math.
• There is a holomorphic representation of Template:Math on Template:Math.
• There is a representation of Template:Math on Template:Math.
• There is a representation of Template:Math on Template:Math.
• There is a complex linear representation of Template:Math on Template:Math.

If one representation is irreducible, then all of them are. In this list, direct products (groups) or direct sums (Lie algebras) may be introduced (if done consistently). The essence of the trick is that the starting point in the above list is immaterial. Both qualitative knowledge (like existence theorems for one item on the list) and concrete realizations for one item on the list will translate and propagate, respectively, to the others.

Now, the representations of Template:Math, which is the Lie algebra of Template:Math, are supposed to be irreducible. This means that they must be tensor products of complex linear representations of Template:Math, as can be seen by restriction to the subgroup Template:Math, a compact group to which the Peter–Weyl theorem applies. The irreducible unitary representations of Template:Math are precisely the tensor products of irreducible unitary representations of Template:Math. These stand in one-to-one correspondence with the holomorphic representations of Template:Math and these, in turn, are in one-to-one correspondence with the complex linear representations of Template:Math because Template:Math is simply connected.

For Template:Math, there exists the highest weight representations (obtainable, via the trick, from the corresponding Template:Math-representations), here indexed by Template:Math for Template:Math. The tensor products of two complex linear factors then form the irreducible complex linear representations of Template:Math. For reference, if Template:Math and Template:Math are representations of a Lie algebra Template:Math, then their tensor product Template:Math is given by either of

where Template:Math is the identity operator. Here, the latter interpretation is intended. The not necessarily complex linear representations of Template:Math come using another variant of the unitarian trick as is shown in the last Lie algebra isomorphism in Template:EquationNote.

The representations

The representations for all Lie algebras and groups involved in the unitarian trick can now be obtained. The real linear representations for Template:Math and Template:Math follow here assuming the complex linear representations of Template:Math are known. Explicit realizations and group representations are given later.

sl(2, C)

The complex linear representations of the complexification of Template:Math, Template:Math, obtained via isomorphisms in Template:EquationNote, stand in one-to-one correspondence with the real linear representations of Template:Math. The set of all, at least real linear, irreducible representations of Template:Math are thus indexed by a pair Template:Math. The complex linear ones, corresponding precisely to the complexification of the real linear Template:Math representations, are of the form Template:Math, while the conjugate linear ones are the Template:Math. All others are real linear only. The linearity properties follow from the canonical injection, the far right in Template:EquationNote, of Template:Math into its complexification. Representations on the form Template:Math or Template:Math are given by real matrices (the latter is not irreducible). Explicitly, the real linear Template:Math-representations of Template:Math are

${\varphi }_{\mu ,\nu }(X)={\varphi }_{\mu }\otimes {\bar{\varphi }}_{\nu }(X)={\varphi }_{\mu }(X)\otimes {\mathrm{I}\mathrm{d}}_{\nu +1}+{\mathrm{I}\mathrm{d}}_{\mu +1}\otimes \overline{{\varphi }_{\nu }(X)},X\in 𝔰𝔩(2,ℂ)$

where Template:Math are the complex linear irreducible representations of Template:Math and Template:Math their complex conjugate representations. Here the tensor product is interpreted in the former sense of Template:EquationNote. These representations are concretely realized below.

so(3; 1)

Via the displayed isomorphisms in Template:EquationNoteand knowledge of the complex linear irreducible representations of Template:Math, upon solving for Template:Math and Template:Math, all irreducible representations of Template:Math_C, and, by restriction, those of Template:Math are known. It's worth noting that the representations of Template:Math 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 Template:Math 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 Template:Math and Template:Math, conventionally written as one of

$(m,n)\equiv D^{(m,n)}\equiv {\pi }_{m,n}.$

The notation Template:Math is usually reserved for the group representations. Let Template:Math, where Template:Mvar is a vector space, denote the irreducible representations of Template:Math according to this classification. These are, up to a similarity transformation, uniquely given by

where the Template:Math are the Template:Math-dimensional irreducible spin Template:Mvar [[Representation theory of SU(2)|representations of Template:Math]] ≈ Template:Math and Template:Math is the Template:Mvar-dimensional unit matrix.

Explicit formula for matrix elements

Let Template:Math, where Template:Mvar is a vector space, denote the irreducible representations of Template:Math according to the Template:Math classification. In components, with Template:Math, Template:Math, the representations are given by

$\begin{array}{cc}({\pi }_{m,n}(J_i){)}_{a^{\prime }{b}^{\prime },ab}& ={\delta }_{b^{\prime }b}(J_i^{(m)}{)}_{a^{\prime }a}+{\delta }_{a^{\prime }a}(J_i^{(n)}{)}_{b^{\prime }b},\\ ({\pi }_{m,n}(K_i){)}_{a^{\prime }{b}^{\prime },ab}& =i({\delta }_{a^{\prime }a}(J_i^{(n)}{)}_{b^{\prime }b}-{\delta }_{b^{\prime }b}(J_i^{(m)}{)}_{a^{\prime }a}),\end{array}$

where Template:Math is the Kronecker delta and the Template:Math are the Template:Math-dimensional irreducible representations of Template:Math, also termed spin matrices or angular momentum matrices. These are explicitly given as

$\begin{array}{cc}{\left(J_1^{(j)}\right)}_{a^{\prime }a}& =\frac{1}{2}(\sqrt{(j-a)(j+a+1)}{\delta }_{a^{\prime },a+1}+\sqrt{(j+a)(j-a+1)}{\delta }_{a^{\prime },a-1}),\\ {\left(J_2^{(j)}\right)}_{a^{\prime }a}& =\frac{1}{2i}(\sqrt{(j-a)(j+a+1)}{\delta }_{a^{\prime },a+1}-\sqrt{(j+a)(j-a+1)}{\delta }_{a^{\prime },a-1}),\\ {\left(J_3^{(j)}\right)}_{a^{\prime }a}& =a{\delta }_{a^{\prime }a}.\end{array}$

Common representations

	Template:Mvar=0	½	1
Template:Mvar=0	scalar	Weyl spinor bispinor	self-dual 2-form 2-form field
½	Weyl spinor (right-handed)	4-vector	Rarita–Schwinger field
1	anti-self-dual 2-form		traceless symmetric tensor
Purple: Template:Math complex irreducible representations   Black: Template:Math Bold: Template:Math

Since for any irreducible representation for which Template:Math it is essential to operate over the field of complex numbers, the direct sum of representations Template:Math and Template:Math has a particular relevance to physics, since it permits to use linear operators over real numbers.

• (0, 0) is the Lorentz scalar representation. This representation is carried by relativistic scalar field theories.
• (½, 0) is the left-handed Weyl spinor and (0, ½) is the right-handed Weyl spinor representation. Fermionic supersymmetry generators transform under one of these representations.
• (½, 0) ⊕ (0, ½) is the bispinor representation. (See also Dirac spinor and Weyl spinors and bispinors below.)
• (½, ½) is the four-vector representation. The four-momentum of a particle (either massless or massive) transforms under this representation.
• (1, 0) is the self-dual 2-form field representation and (0, 1) is the anti-self-dual 2-form field representation.
• (1, 0) ⊕ (0, 1) is the adjoint representation and the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation.
• (1, ½) ⊕ (½, 1) is the Rarita–Schwinger field representation.
• (1, 1) is the spin 2 representation of a traceless symmetric tensor field. A physical example is the traceless part of the energy-momentum tensor Template:Math.
• (Template:Sfrac, 0) ⊕ (0, Template:Sfrac) would be the symmetry of the hypothesized gravitino. It can be obtained from the (1, ½) ⊕ (½, 1)-representation.

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 Template:Math. It is one-to-one in a neighborhood of the identity.

The Lie correspondence

Sophus Lie, the originator of Lie theory. The theory of manifolds was not discovered in Lie's time, so he worked locally with subsets of Template:Math. The structure would today be called a local group.

The Lie correspondence and some results based on it needed here and below are stated for reference. If Template:Mvar denotes a Lie group and Template:Math a Lie algebra, let Template:Math denote the group generated by Template:Math, the image of the Lie algebra under the exponential mapping, and let Template:Math denote the Lie algebra of Template:Math. The Lie correspondence reads in modern language as follows:

• There is a one-to-one correspondence between connected and simply connected Lie groups Template:Math and Lie algebras Template:Math under which Template:Math corresponds to Template:Math and Template:Math to Template:Math. Equivalently, Template:Math and Template:Math. Template:EquationRef

A linear Lie group is one that has at least one faithful finite-dimensional representation. The following are some corollaries that will be used in the sequel:

• A connected linear Lie group Template:Math is abelian if and only if Template:Math is abelian. Template:EquationRef
• A connected subgroup Template:Math with Lie algebra Template:Math of a connected linear Lie group Template:Math is normal if and only if Template:Math is an ideal. Template:EquationRef
• If Template:Math are linear Lie groups with Lie algebras Template:Math and Template:Math is a group homomorphism, then Template:Math, its pushforward at the identity, is a Lie algebra homomorphism and Template:Math for every Template:Math. Template:EquationRef

Lie algebra representations from group representations

Using the above theorem it is always possible to pass from a representation of a Lie group Template:Math to a representation of its Lie algebra Template:Math. If Template:Math is a group representation for some vector space Template:Math, then its pushforward (differential) at the identity, or Lie map, Template:Math is a Lie algebra representation. It is explicitly computed using

This, of course, holds for the Lorentz group in particular, but not all Lie algebra representations arise this way because their corresponding group representations may not exist as proper representations, i.e. they are projective, see below.

Group representations from Lie algebra representations

Here Template:Mvar is a finite-dimensional vector space, Template:Math is the set of all invertible linear transformations on Template:Mvar and Template:Math is its Lie algebra. The maps Template:Mvar and Template:Math are Lie algebra and group representations respectively, and Template:Math is the exponential mapping. The diagram commutes only up to a sign if Template:Math is projective.

Given a Template:Math representation, one may try to construct a representation of Template:Math, the identity component of the Lorentz group, by using the exponential mapping. Since Template:Math is a matrix Lie group, the exponential mapping is simply the matrix exponential. If Template:Mvar is an element of so(3; 1) in the standard representation, then

is a Lorentz transformation by general properties of Lie algebras. Motivated by this and the Lie correspondence theorem stated above, let Template:Math for some vector space Template:Mvar be a representation and tentatively define a representation Template:Math of Template:Math by first setting

The subscript Template:Math indicates a small open set containing the identity. Its precise meaning is defined below. There are at least two potential problems with this definition. The first is that it is not obvious that this yields a group homomorphism, or even a well defined map at all (local existence). The second problem is that for a given Template:Math there may not be exactly one Template:Math such that Template:Math (local uniqueness). The soundness of the tentative definition Template:EquationNoteis shown in several steps below:

1. Template:Math is a local homomorphism.
2. Template:Math defined along a path using properties of Template:Math is a global homomorphism.
3. The exponential mapping Template:Math is surjective.
4. Template:Math defined along a path coincides with Template:Math with Template:Math.

Local existence and uniqueness

A theorem based on the inverse function theorem states that the map Template:Math is one-to-one for Template:Math small enough Template:EquationRef. This makes the map well-defined. The qualitative form of the Baker–Campbell–Hausdorff formula then guarantees that it is a group homomorphism, still for Template:Math small enough Template:EquationRef. Let Template:Math denote image under the exponential mapping of the open set in Template:Math where conditions Template:EquationNoteand Template:EquationNoteboth hold. Let Template:Math, then

This shows that the map Template:Math is a well-defined group homomorphism on Template:Math.

Global existence and uniqueness

Technically, formula Template:EquationNoteis used to define Template:Math near the identity. For other elements Template:Math one chooses a path from the identity to Template:Mvar and defines Template:Math along that path by partitioning it finely enough so that formula Template:EquationNotecan be used again on the resulting factors in the partition. In detail, one sets

where the Template:Mvar are on the path and the factors on the far right are uniquely defined by Template:EquationNoteprovided that all Template:Math and, for all conceivable pairs Template:Math of points on the path between Template:Math and Template:Math, Template:Math as well. For each Template:Math take, by the inverse function theorem, the unique Template:Mvar such that Template:Math = Template:Math and obtain Template:NumBlk

By compactness of the path there is an Template:Mvar large enough so that Template:Math is well defined, possibly depending on the partition and/or the path, whether Template:Math is close to the identity or not.

Partition independence

It turns out that the result is always independent of the partitioning of the path. To demonstrate the independence of a chosen path, one employs the Baker–Campbell–Hausdorff formula. It shows that Template:Math is a group homomorphism for elements in Template:Math.

To see this, first fix a partitioning used in Template:EquationNote. Then insert a new point Template:Math somewhere on the path, say

$g=\cdots (g_{i+1}{h}^{-1})(h{g}_i^{-1})\cdots ,\cdots {\mathrm{\Pi }}_U(g_{i+1}{h}^{-1}){\mathrm{\Pi }}_U(h{g}_i^{-1})\cdots .$

But

$\cdots {\mathrm{\Pi }}_U(g_{i+1}{h}^{-1}){\mathrm{\Pi }}_U(h{g}_i^{-1})\cdots =\cdots {\mathrm{\Pi }}_U(g_{i+1}{h}^{-1}h{g}_i^{-1})\cdots =\cdots {\mathrm{\Pi }}_U(g_{i+1}{g}_i^{-1})\cdots$

as a consequence of the Baker–Campbell–Hausdorff formula and the conditions on the original partitioning. Thus, adding a point on the path has no effect on the definition of Template:Math.

Then, for any two given partitions of a given path, they have common refinement, their union. This refinement can be reached from any of the two partitionings by, one-by-one, adding points from the other partition. No individual addition changes the definition of Template:Math, hence, since there are finitely many points in each partition, the value of Template:Math must have been the same for the two partitionings to begin with.

Path independence

For simply connected groups, the construction will be independent of the path as well, yielding a well defined representation. In that case formula Template:EquationNotecan unambiguously be used directly. Simply connected spaces have the property that any two paths can be continuously deformed into each other. Any such deformation is called a homotopy and is usually chosen as a continuous function Template:Math from the unit square Template:Math into the group. For Template:Math the image is one of the paths, for Template:Math the other, for intermediate Template:Math, an intermediate path results, but endpoints are kept fixed.

One deforms the path, a little bit at a time, using the previous result, the independence of partitioning. Each consecutive deformation is so small that two consecutive deformed paths can be partitioned using the same partition points. Thus two consecutive deformed paths yield the same value for Template:Math. But any two pairs of consecutive deformations need not have the same choice partition points, so the actual path laid out in the group as one progresses through the deformation does indeed change.

Using compactness arguments, in a finite number of steps, the original (Template:Math) path is deformed into the other (Template:Math) without affecting the value of Template:Math.

Global homomorphism

The map Template:Math is, by the Baker-Campbell-Hausdorff formula, a local homomorphism. To show that Template:Math is a global homomorphism, consider two elements Template:Math. Lay out paths Template:Math from the identity to them and define a path Template:Math going along Template:Math for Template:Math and along Template:Math for Template:Math. This is a path from the identity to Template:Math. Select adequate partitionings for Template:Math. This corresponds to a choice of "times" Template:Math and Template:Math. Divide the first set with 2 and divide the second set with 2 and add ½ and so obtain a new (adequate) set of "times" to be used for Template:Math. Direct computation shows that, with these partitionings (and hence all partitionings), Template:Math.

Surjectiveness of exponential mapping

From a practical point of view, it is important that formula Template:EquationNotecan be used for all elements of the group. The Lie correspondence theorem above guarantees that Template:EquationNoteholds for all Template:Math, but provides no guarantee that all Template:Math are in the image of Template:Math. For general Lie groups, this is not the case, especially not for non-compact groups, as for example for Template:Math, the universal covering group of Template:Math. It will be treated in this respect below.

But Template:Math is surjective. One way to see this is to make use of the isomorphism Template:Math, the latter being the Möbius group. It is a quotient of Template:Math (see the linked article). Let Template:Math denote the quotient map. Now Template:Math is onto. Apply the Lie correspondence theorem with Template:Math being the differential at the identity of Template:Math. Then for all Template:Math Template:Math. Since the left hand side is surjective (both Template:Math and Template:Math are), the right hand side is surjective and hence Template:Math is surjective. Finally, recycle the argument once more, but now with the known isomorphism between Template:Math and Template:Math to find that Template:Math is onto for the connected component of the Lorentz group.

Consistency

From the way Template:Math has been defined for elements far from the identity, it not immediately clear that formula Template:EquationNoteholds for all elements of Template:Math, i.e. that one can take Template:Math in Template:EquationNote. But, in summary,

• Template:Math is a uniquely constructed homomorphism.
• Using Template:EquationNotewith Template:Math as defined here, then one ends up with the Template:Math one started with since Template:Math was defined that way near the identity, and Template:EquationNotedepends only on an arbitrarily small neighborhood of the identity.
• Template:Math is surjective.

Hence Template:EquationNoteholds everywhere. One finally unconditionally writes

Fundamental group

The above construction relies on simple connectedness. The result needs modifications for non-simply connected groups per below. To exhibit the fundamental group of Template:Math, one may consider first the topology of its [[#The covering group|covering group Template:Math]]. By the polar decomposition theorem, any matrix Template:Math may be uniquely expressed as

$\lambda =u{e}^h,\det u=1,\mathrm{tr}v=0,$

where Template:Mvar is unitary with determinant one, hence in Template:Math, and Template:Mvar is Hermitian with trace zero. The trace and determinant conditions imply

$h=\left(\begin{array}{cc}c& a-ib\\ a+ibc& -c\end{array}\right),u=\left(\begin{array}{cc}d+ie& f+ig\\ -f+ig& d-ie\end{array}\right),d^2+e^2+f^2+g^2=1,$

with Template:Math unconstrained and Template:Math constrained to the 3-sphere Template:Math. It follows that the manifestly continuous one-to-one map Template:Math is a homeomorphism (hence preserves the fundamental group). Since Template:Math is simply connected for all Template:Mvar and Template:Math is simply connected for Template:Math and since simple connectedness is preserved under cartesian products, it follows that Template:Math is simply connected. Now, Template:Math, where Template:Math is the center of Template:Math. Identifying Template:Math and Template:Math amounts to identifying Template:Math with Template:Math, which in turn amounts to identifying antipodal points on Template:Math. Thus topologically,

$SO(3;1)\approx {ℝ}^3\times S^3/Z_2,$

where last factor is not simply connected: Geometrically, it is easy to see (for visualization purposes, replace Template:Math by Template:Math) that a path from Template:Math to Template:Math in Template:Math is a loop in Template:Math since Template:Math and Template:Math are antipodal points, and that it is not contractible to a point. But a path from Template:Math to Template:Math, thence to Template:Math again, a loop in Template:Math and a double loop (considering Template:Math, where Template:Mvar is the covering map Template:Math) in Template:Math that is contractible to a point (continuously move away from Template:Math "upstairs" in Template:Math and shrink the path there to the point Template:Math). Thus Template:Math is a two-element group with two equivalence classes of loops as its elements – or put more simply, Template:Math is doubly connected.

Projective representations

For a group that is connected but not simply connected, such as Template:Math, the result may depend on the homotopy class of the chosen path. The result, when using Template:EquationNote, will then depend on which Template:Mvar in the Lie algebra is used to obtain the representative matrix for Template:Mvar.

Since Template:Math per above has two elements, not all representations of the Lie algebra will yield representations of the group, but some will instead yield projective representations. Once these conclusions have been reached, and once one knows whether a representation is projective, there is no need to be concerned about paths and partitions. Formula Template:EquationNoteapplies to all group elements and all representations, including the projective ones.

For the Lorentz group, the Template:Math-representation is projective when Template:Math is a half-integer. See the section spinors.

For a projective representation Template:Math of Template:Math, it holds that

since any loop in Template:Math 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 Template:Math is a double-valued function. One cannot consistently chose a sign to obtain a continuous representation of all of Template:Math, but this is possible locally around any point.

The covering group

Consider Template:Math as a real Lie algebra with basis

$(\frac{1}{\sqrt{2}}{\sigma }_1,\frac{1}{\sqrt{2}}{\sigma }_2,\frac{1}{\sqrt{2}}{\sigma }_3,\frac{i}{\sqrt{2}}{\sigma }_1,\frac{i}{\sqrt{2}}{\sigma }_2,\frac{i}{\sqrt{2}}{\sigma }_3)\equiv (j_1,j_2,j_3,k_1,k_2,k_3),$

where the sigmas are the Pauli matrices. From the relations

one obtains

which are exactly on the form of the Template:Math-dimensional version of the commutation relations for Template:Math (see conventions and Lie algebra bases below). Thus, one may map Template:Math, Template:Math, and extend by linearity to obtain an isomorphism. Since Template:Math is simply connected, it is the universal covering group of Template:Math.

A geometric view

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

Let Template:Math denote the set of path homotopy classes Template:Math of paths Template:Math, from Template:Math to Template:Math and define the set Template:NumBlk and endow it with the multiplication operation Template:NumBlk The dot on the far right denotes path multiplication.

With this multiplication, Template:Mvar is a group and Template:Math, the universal covering group of Template:Math. By the above construction, there is, since each Template:Math has two elements, a 2:1 covering map Template:Math and an isomorphism Template:Math. According to covering group theory, the Lie algebras Template:Math, Template:Math and Template:Math of Template:Math are all isomorphic. The covering map Template:Math is simply given by Template:Math.

An algebraic view

For an algebraic view of the universal covering group, let Template:Math act on the set of all Hermitian Template:Gaps matrices Template:Math by the operation Template:NumBlk

Since Template:Math is Hermitian, Template:Math is again Hermitian because Template:Math, and also Template:Math, so the action is linear as well. An element of Template:Math may generally be written in the form Template:NumBlk for Template:Mvar real, showing that Template:Math is a 4-dimensional real vector space. Moreover, Template:Math meaning that Template:Math is a group homomorphism into Template:Math. Thus Template:Math is a 4-dimensional representation of Template:Math. Its kernel must in particular take the identity matrix to itself, Template:Math. Thus Template:Math for Template:Mvar in the kernel so, by Schur's lemma, Template:Mvar is a multiple of the identity, which must be Template:Math since Template:Math. Now map Template:Math to spacetime Template:Math endowed with the Lorentz metric, Minkowski space, via Template:NumBlk The action of Template:Math on Template:Math preserves determinants since Template:Math. The induced representation Template:Math of Template:Math on Template:Math, via the above isomorphism, given by Template:NumBlk will preserve the Lorentz inner product since

Template:Math.

This means that Template:Math belongs to the full Lorentz group Template:Math. By the main theorem of connectedness, since Template:Math is connected, its image under Template:Math in Template:Math is connected as well, and hence is contained in Template:Math.

It can be shown that the Lie map of Template:Math, Template:Math is a Lie algebra isomorphism (its kernel is Template:Math and must therefore be an isomorphism for dimensional reasons). The map Template:Math is also onto.

Thus Template:Math, since it is simply connected, is the universal covering group of Template:Math, isomorphic to the group Template:Math of above.

Representations of Template:Math and Template:Math

The complex linear representations of Template:Math and Template:Math are more straightforward to obtain than the Template:Math representations. If Template:Math is a representation of Template:Math with highest weight Template:Math, then the complexification of Template:Math is a complex linear representation of Template:Math. All complex linear representation of Template:Math are of this form. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are obtained by exponentiation. By simple connectedness of Template:Math, this always yields a representation of the group as opposed to in the Template:Math case. The real linear representations of Template:Math are exactly the Template:Math-representations presented earlier. They can be exponentiated too. The Template:Math-representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer.

It is also possible to obtain representations of Template:Math directly. This will be done below. Then, using the unitarian trick, going the other way, one finds Template:Math-,Template:Math-,Template:Math-,Template:Math-, and Template:Math-representations as well as Template:Math-representations (via Template:EquationNote) and, possibly projective, Template:Math-representations (via projection from Template:Math, see below, or exponentiation).

The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of Template:Math and there is no factor of Template:Math in the exponential mapping compared to the physics convention used elsewhere. Let the basis of Template:Math be Template:NumBlk This choice of basis, and the notation, is standard in the mathematical literature.

Concrete realization

The irreducible holomorphic Template:Math-dimensional representations of Template:Math, Template:Math, can be realized on a set of functions Template:Math where each Template:Math is a homogeneous polynomial of degree Template:Math in 2 variables. The elements of Template:Math appears as Template:Math. The action of Template:Math is given by

The associated Template:Math-action is, using Template:EquationNoteand the definition above, given by

Defining Template:Math and using the chain rule one finds

The basis elements of Template:Math are then represented by

on the space Template:Math (all Template:Math). By employing the unitarian trick one obtains representations for Template:Math, Template:Math, Template:Math, and Template:Math, all are obtained by restriction of either Template:EquationNoteor Template:EquationNote. They are formally identical to Template:EquationNoteor Template:EquationNote. With a choice of basis for Template:Math, all these representations become matrix groups or matrix Lie algebras.

The Template:Math-representations are realized on a space of polynomials Template:Math in Template:Math, homogeneous of degree Template:Math in Template:Math and homogeneous of degree Template:Math in Template:Math. The representations are given by

By carrying out the same steps as above, one finds

from which the expressions

for the basis elements follow.

Non-surjectiveness of exponential mapping

This diagram shows the web of maps discussed in the text. Here Template:Math is a finite-dimensional vector space carrying representations of Template:Math, Template:Math, Template:Math, Template:Math, Template:Math is the exponential mapping, Template:Math is the covering map from Template:Math onto Template:Math and Template:Math is the Lie algebra isomorphism induced by it. The maps Template:Math and the two Template:Math are representations. the picture is only partially true when Template:Math is projective.

Unlike in the case Template:Math, the exponential mapping Template:Math is not onto. The conjugacy classes of Template:Math are represented by the matrices

but there is no element Template:Math in Template:Math such that Template:Math.

In general, if Template:Math is an element of a connected Lie group Template:Math with Lie algebra Template:Math, then

This follows from the compactness of a path from the identity to Template:Math and the one-to-one nature of Template:Math near the identity. In the case of the matrix Template:Math, one may write

The kernel of the covering map Template:Math of above is Template:Math, a normal subgroup of Template:Math. The composition Template:Math is onto. If a matrix Template:Math is not in the image of Template:Math, then there is a matrix Template:Math equivalent to it with respect to Template:Math, meaning Template:Math, that is in the image of Template:Math. The condition for equivalence is Template:Math. In the case of the matrix Template:Math, one may solve for Template:Math in the equation Template:Math. One finds

As a corollary, since the covering map Template:Math is a homomorphism,the mapping version of the Lie correspondence Template:EquationNotecan be used to provide a proof of the surjectiveness of Template:Math for Template:Math. Let Template:Math denote the isomorphism between Template:Math and Template:Math. Refer to the commutative diagram. One has Template:Math for all Template:Math. Since Template:Math is onto, Template:Math is onto, and hence Template:Math is onto as well.

Template:Math-representations from Template:Math-representations

By the first isomorphism theorem, a representation Template:Math of Template:Math descends to a representation Template:Math of Template:Math if and only if Template:Math. Refer to the commutative diagram. If this condition holds, then both elements in the fiber Template:Math will be mapped by Template:Math to the same representative, and the expression Template:Math makes sense. One may thus define Template:Math. In particular, if Template:Math is faithful, i.e. having kernel = Template:Math, then there is no corresponding proper representation of Template:Math, but there is a projective one as was shown in a previous section, corresponding to the two possible choices of representative in each fiber Template:Math.

Lie algebra representations of Template:Math are obtained from Template:Math-representations simply by composition with Template:Math.

Template:Math-representations from Template:Math-representations

Template:Math-representations can be obtained from non-projective Template:Math-representations by composition with the projection map Template:Math. These are always representations since they are compositions of group homomorphisms. Such a representation is never faithful because Template:Math. If the Template:Math-representation is projective, then the resulting Template:Math-representation would be projective as well. Instead, the isomorphism Template:Math can be employed, composed with Template:Math. This is always a non-projective representation.

Properties of the (m, n) representations

The Template:Math representations are irreducible, and they are the only irreducible representations.

• Irreducibility follows from the unitarian trick and that a representation Template:Math of Template:Math is irreducible if and only if Template:Math, where Template:Math are irreducible representations of Template:Math.
• Uniqueness follows from that the Template:Math are the only irreducible representations of Template:Math, which is one of the conclusions of the theorem of the highest weight.

Dimension

The Template:Math representations are Template:Math-dimensional. It follows from the Weyl dimension formula. For a Lie algebra Template:Math it reads

$\dim {\pi }_{\mu }=\frac{{\mathrm{\Pi }}_{\alpha \in R^{+}}⟨\alpha ,\mu +\delta ⟩}{{\mathrm{\Pi }}_{\alpha \in R^{+}}⟨\alpha ,\delta ⟩},$

where Template:Math is the set of positive roots and Template:Math is half the sum of the positive roots. The inner product Template:Math is that of the Lie algebra Template:Math, invariant under the action of the Weyl group on Template:Math, the Cartan subalgebra. The roots (really elements of Template:Math are via this inner product identified with elements of Template:Math. For Template:Math, the formula reduces to Template:Math. By taking tensor products, the result follows.

A quicker approach is, of course, to simply count the dimensions in any concrete realization, such as the one given in [[#representations of SL(2, C) and sl(2, C)|representations of Template:Math and Template:Math]].

Faithfulness

If a representation Template:Math of a Lie group Template:Math is not faithful, then Template:Math is a nontrivial normal subgroup because Template:Math. There are three relevant cases.

1. Template:Math is non-discrete and abelian.
2. Template:Math is non-discrete and non-abelian.
3. Template:Math is discrete. In this case Template:Math, where Template:Math is the center of Template:Math.

In the case of Template:Math, the first case is excluded since Template:Math is semi-simple. The second case (and the first case) is excluded because Template:Math is simple. For the third case, Template:Math is isomorphic to the quotient Template:Math. But Template:Math is the center of Template:Math. It follows that the center of Template:Math is trivial, and this excludes the third case. The conclusion is that every representation Template:Math and every projective representation Template:Math for Template:Math 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 Template:Math replaced by the center of Template:Math. The center of any semisimple Lie algebra is trivial and Template:Math is semi-simple and simple, and hence has no non-trivial ideals.

A related fact is that if the corresponding representation of Template:Math is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding Template:Math representation is not faithful, but is Template:Math.

Non-unitarity

The Template:Math 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 Template:Math, where Template:Math is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group Template:Math. Then Template:Math where Template:Math is the compact subgroup of Template:Math consisting of unitary transformations of Template:Math. The kernel, Template:Math, of Template:Math is a normal subgroup of Template:Math. Since Template:Math is simple, Template:Math is either all of Template:Math, in which case Template:Math is trivial, or Template:Math is trivial, in which case Template:Math is faithful. In the latter case Template:Math is a diffeomorphism onto its image, Template:Math, and Template:Math is Lie group. This would mean that Template:Math is an embedded non-compact Lie subgroup of the compact group Template:Math. This is impossible with the subspace topology on Template:Math since all embedded Lie subgroups of a Lie group are closed If Template:Math were closed, it would be compact, and then Template:Math 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 Template:Math and Template:Math used in the construction are Hermitian. This means that Template:Math is Hermitian, but Template:Math 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 Template:Math representation is, however, unitary when restricted to the rotation subgroup Template:Math, but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an Template:Math representation have Template:Math-invariant subspaces of highest weight (spin) Template:Math, where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) Template:Math is Template:Math-dimensional. So for example, the (½, ½) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.

Since the angular momentum operator is given by Template:Math, the highest spin in quantum mechanics of the rotation sub-representation will be Template:Math 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 Template:Math-invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the Template:Math representation has spin if Template:Math is half-integral. The simplest are Template:Math and Template:Math, the Weyl-spinors of dimension Template:Math. Then, for example, Template:Math and Template:Math are a spin representations of dimensions Template:Math and Template:Math respectively. Note that, according to the above paragraph, there are subspaces with spin both Template:Math and Template:Math in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under Template:Math. It cannot be ruled out in general, however, that representations with multiple Template:Math 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 Template:Math representations for any Template:Math (under Template:Math) 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 root system Template:Math of Template:Math.

To see if the dual representation of an irreducible representation is isomorphic to the original representation one can consider the following theorems:

1. The set of weights of the dual representation of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation.
2. Two irreducible representations are isomorphic if and only if they have the same highest weight.
3. For each semisimple Lie algebra there exists a unique element Template:Math of the Weyl group such that if Template:Math is a dominant integral weight, then Template:Math is again a dominant integral weight.
4. If Template:Math is an irreducible representation with highest weight Template:Math, then Template:Math has highest weight Template:Math.

Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. One sees that if Template:Math is an element of the Weyl group of a semisimple Lie algebra, then Template:Math. In the case of Template:Math, the Weyl group is Template:Math. It follows that each Template:Math is isomorphic to its dual Template:Math. The root system of Template:Math is shown in the figure to the right. The Weyl group is generated by Template:Math where Template:Math is reflection in the plane orthogonal to Template:Math as Template:Math ranges over all roots. One sees that Template:Math so Template:Math. Then using the fact that if Template:Math are Lie algebra representations and Template:Math, then Template:Math. The conclusion for Template:Math is

${\pi }_{m,n}^{*}\cong {\pi }_{m,n},{\mathrm{\Pi }}_{m,n}^{*}\cong {\mathrm{\Pi }}_{m,n},2m,2n\in \mathbb{N}.$

Complex conjugate representations

If Template:Math is a representation of a Lie algebra, then Template:Math 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 Template:Math of Template:Math can be written uniquely as Template:Math, where

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

with Template:Math holomorphic (complex linear) and Template:Math anti-holomorphic (conjugate linear). For Template:Math, since Template:Math is holomorphic, Template:Math is anti-holomorphic. Direct examination of the explicit expressions for Template:Math and Template:Math in equation Template:EquationNotebelow shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression Template:EquationNotealso allows for identification of Template:Math and Template:Math for Template:Math as Template:Math and Template:Math.

Using the above identities (interpreted as pointwise addition of functions), for Template:Math yields

$\overline{{\pi }_{m,n}}=\overline{{\pi }_{m,n}^{+}+{\pi }_{m,n}^{-}}=\overline{{\pi }_m^{{\oplus }_{2n+1}}}+\stackrel{‾}{\stackrel{{\oplus }_{2m+1}}{\overline{{\pi }_n}}}={\pi }_n^{{\oplus }_{2m+1}}+\stackrel{{\oplus }_{2n+1}}{\overline{{\pi }_m}}={\pi }_{n,m}^{+}+{\pi }_{n,m}^{-}={\pi }_{n,m},\overline{{\mathrm{\Pi }}_{m,n}}={\mathrm{\Pi }}_{n,m},2m,2n\in \mathbb{N},$

where the statement for the group representations follow from Template:Math = Template:Math. It follows that the irreducible representations Template:Math have real matrix representatives if and only if Template:Math. Reducible representations on the form Template:Math have real matrices too.

Induced representations on the Clifford algebra and the Dirac spinor representation

Richard Brauer and wife Ilse 1970. Brauer generalized the spin representations of Lie algebras sitting inside Clifford algebras to spin higher than Template:Sfrac.Template:Paragraph breakPhoto courtesy of MFO.

In general representation theory, if Template:Math is a representation of a Lie algebra g, then there is an associated representation of g on Template:Math, also denoted Template:Mvar, given by

Likewise, a representation Template:Math of a group Template:Mvar yields a representation Template:Math on Template:Math of Template:Mvar, still denoted Template:Math, given by

Applying this to the Lorentz group, if Template:Math is a projective representation, then direct calculation using (G4) shows that the induced representation on Template:Math is, in fact, a proper representation, i.e. a representation without phase factors.

In quantum mechanics this means that if Template:Math or Template:Math is a representation acting on some Hilbert space Template:Math, then the corresponding induced representation acts on the set of linear operators on Template:Math. As an example, the induced representation of the projective spin Template:Math representation on Template:Math is the non-projective 4-vector ({½, ½) representation.

For simplicity, consider now only the "discrete part" of Template:Math, that is, given a basis for Template:Math, the set of constant matrices of various dimension, including possibly infinite dimensions. A general element of the full Template:Math is the sum of tensor products of a matrix from the simplified Template:Math and an operator from the left out part. The left out part consists of functions of spacetime, differential and integral operators and the like. See Dirac operator for an illustrative example. Also left out are operators corresponding to other degrees of freedom not related to spacetime, such as gauge degrees of freedom in gauge theories.

The induced 4-vector representation of above on this simplified Template:Math has an invariant 4-dimensional subspace that is spanned by the four gamma matrices. (Note the different metric convention in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, Template:Math, whose complexification is Template:Math, generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the Template:Math, a pseudoscalar irrep, also the Template:Math, but with parity inversion eigenvalue −1, see the next section below, the already mentioned vector irrep, Template:Math, a pseudovector irrep, Template:Math with parity inversion eigenvalue +1 (not −1), and a tensor irrep, Template:Math. The dimensions add up to Template:Math. In other words,

where, as is customary, a representation is confused with its representation space. This is, in fact, a reasonably convenient way to show that the algebra spanned by the gammas is 16-dimensional.

The Template:Math spin representation

The six-dimensional representation space of the tensor Template:Math-representation inside Template:Math has two roles. In particular, letting

where Template:Math are the gamma matrices, the Template:Math, only 6 of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,

and hence constitute a representation (in addition to being a representation space) sitting inside Template:Math, the Template:Math spin representation. For details, see bispinor and Dirac algebra.

The conclusion is that every element of the complexified Template:Math in Template:Math (i.e. every complex Template:Gaps 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 Template:Math, 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 in a standard manner 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. Template:Math. These representations are in general not irreducible, and are not discussed here. It is to be noted though that the Lorenz group and its Lie algebra have the complete reducibility property. This means that every representation reduces to a direct sum of irreducible representations.

Space inversion and time reversal

The (possibly projective) Template:Math representation is irreducible as a representation Template:Math, the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If Template:Math it can be extended to a representation of all of Template:Math, the full Lorentz group, including space parity inversion and time reversal. The representations Template:Math can be extended likewise.

Space parity inversion

For space parity inversion, one considers the adjoint action Template:Math of Template:Math on Template:Math, where Template:Math is the standard representative of space parity inversion, Template:Math, given by

It is these properties of Template:Math and Template:Math under Template:Mvar that motivate the terms vector for Template:Math and pseudovector or axial vector for Template:Math. In a similar way, if Template:Math is any representation of Template:Math and Template:Math is its associated group representation, then Template:Math acts on the representation of Template:Math by the adjoint action, Template:Math for Template:Math, Template:Math. If Template:Math is to be included in Template:Math, then consistency with Template:EquationNoterequires that

holds, where Template:Math and Template:Math are defined as in the first section. This can hold only if Template:Math and Template:Math have the same dimensions, i.e. only if Template:Math. When Template:Math then Template:Math can be extended to an irreducible representation of Template:Math, the orthocronous Lorentz group. The parity reversal representative Template:Math does not come automatically with the general construction of the Template:Math representations. It must be specified separately. The matrix Template:Math (or a multiple of modulus −1 times it) may be used in the Template:Math representation.

If parity is included with a minus sign (the Template:Math matrix Template:Math) in the Template:Math representation, it is called a pseudoscalar representation.

Time reversal

Time reversal Template:Math, acts similarly on Template:Math by

By explicitly including a representative for Template:Math, as well as one for Template:Math, one obtains a representation of the full Lorentz group Template:Math. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the Template:Math, in addition to the Template:Math and Template:Math generate the group. These are interpreted as generators of translations. The time-component Template:Math is the Hamiltonian Template:Math. The operator Template:Math satisfies the relation

in analogy to the relations above with Template:Math replaced by the full Poincaré algebra. By just cancelling the Template:Math's, the result Template:Math would imply that for every state Template:Math with positive energy Template:Math in a Hilbert space of quantum states with time-reversal invariance, there would be a state Template:Math with negative energy Template:Math. Such states do not exist. The operator Template:Math is therefore chosen antilinear and antiunitary, so that it anticommutes with Template:Math, resulting in Template:Math = Template:Math, 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 if one is very strict with terminology, Template:Math 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, Template:Nowrap, 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 Template:Math and Template:Math, charge conjugation symmetry Template:Math, has nothing directly to do with Lorentz invariance.

Action on function spaces

In the classification of the irreducible finite-dimensional representations of above it was never specified precisely how a representative of a group or Lie algebra element acts on vectors in the representation space. The action can be anything as long as it is linear. The point silently adopted was that after a choice of basis in the representation space, everything becomes matrices anyway.

If Template:Mvar is a vector space of functions of a finite number of variables Template:Mvar, then the action on a scalar function Template:Math given by Template:NumBlk produces another function Template:Math. Here Template:Math is an Template:Mvar-dimensional representation, and Template:Math is a possibly infinite-dimensional representation. A special case of this construction is when Template:Mvar is a space of functions defined on the group Template:Mvar itself, viewed as a Template:Mvar-dimensional manifold embedded in Template:Math. 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 completeness of the characters in this sense can thus be used to prove the existence of the highest weight representations. The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. Template:Math; in the present case, there is a one-to-one correspondence between representations of Template:Math and holomorphic representations of Template:Math. (A group representation is called holomorphic if its corresponding Lie algebra representation is complex linear.) This theorem too can be used to demonstrate the existence of the highest weight representations.

Euclidean rotations

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

The subgroup Template:Math of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space Template:Math}, where the Template:Math are spherical harmonics. Its elements are square integrable complex-valued functions on the sphere. The inner product on this space is given by Template:NumBlk

If Template:Mvar is an arbitrary square integrable function defined on the unit sphere Template:Math, then it can be expressed as Template:NumBlk where the expansion coefficients are given by

The Lorentz group action restricts to that of Template:Math and is expressed as

This action is unitary, meaning that Template:NumBlk The Template:Math can be obtained from the Template:Math of above using Clebsch–Gordan decomposition, but they are more easily directly expressed as an exponential of an odd-dimensional Template:Math-representation (the 3-dimensional one is exactly Template:Math). In this case the space Template:Math decomposes neatly into an infinite direct sum of irreducible odd finite-dimensional representations Template:Math according to

This is characteristic of infinite-dimensional unitary representations of Template:Math. If Template:Mvar is an infinite-dimensional unitary representation on a separable Hilbert space, then it decomposes as a direct sum of finite-dimensional unitary representations. Such a representation is thus never irreducible. All irreducible finite-dimensional representations Template:Math can be made unitary by an appropriate choice of inner product,

$⟨f,g{⟩}_U\equiv \underset{\mathrm{S}\mathrm{O}(3)}{\int }⟨\mathrm{\Pi }(R)f,\mathrm{\Pi }(R)g⟩dg=\frac{1}{8{\pi }^2}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\underset{0}{\overset{2\pi }{\int }}⟨\mathrm{\Pi }(R)f,\mathrm{\Pi }(R)g⟩\sin \theta d\phi d\theta d\psi ,f,g\in V,$

where the integral is the unique invariant integral over Template:Math normalized to Template:Math, here expressed using the Euler angles parametrization. The inner product inside the integral is any inner product on Template:Math.

The Möbius group

Main article: Möbius transformation, Lorentz group#Relation to the Möbius group

The identity component of the Lorentz group is isomorphic to the Möbius group Template:Math. 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 Template:Math acts on the plane according to

and can be represented by complex matrices

since multiplication by a nonzero complex scalar does not change Template:Mvar. These are elements of Template:Math and are unique up to a sign (since Template:Math give the same Template:Mvar), hence Template:Math.

The Riemann P-functions

Main article: Riemann's differential equation

The Riemann P-functions, solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as

where the Template:Math 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 Template:Math 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 Template:Math are Template:Math and Template:Math ,corresponding to the two linearly independent solutions, and for expansion around the singular point Template:Math they are Template:Math and Template:Math. Similarly, the exponents for Template:Math are Template:Mvar and Template:Mvar for the two solutions.

One has thus

where the condition (sometimes called Riemann's identity)

$\alpha +{\alpha }^{\prime }+\beta +{\beta }^{\prime }+\gamma +{\gamma }^{\prime }=1$

on the exponents of the solutions of Riemann's differential equation has been used to define Template:Math.

The first set of constants on the left hand side in Template:EquationNote, Template:Math denotes the regular singular points of Riemann's differential equation. The second set, Template:Math, are the corresponding exponents at Template:Math for one of the two linearly independent solutions, and, accordingly, Template:Math are exponents at Template:Math for the second solution.

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

where Template:Math are the entries in

for Template:Math a Lorentz transformation.

Define

where Template:Mvar is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Mobius 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 Template:Math and its double cover Template:Math also have infinite dimensional unitary representations, studied independently by Template:Harvtxt, Template:Harvtxt and Template:Harvtxt at the instigation of Paul Dirac. This trail of development begun with Template:Harvtxt where he devised matrices Template:Math and Template:Math necessary for description of higher spin (compare Dirac matrices), elaborated upon by Template:Harvtxt, see also Template:Harvtxt, and proposed precursors of the Bargmann-Wigner equations. In Template:Harvtxt 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 Template:Harvtxt and Template:Harvtxt, based on an analogue for Template:Math of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in Template:Harvtxt and Template:Harvtxt.

The theory of spherical functions for the Lorentz group, required for harmonic analysis on the 3-dimensional unit quasi-sphere in Minkowski space, or equivalently 3-dimensional hyperbolic space, is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on Template:Math. This theory is discussed in Template:Harvtxt, Template:Harvtxt, Template:Harvtxt and the posthumous text of Template:Harvtxt.

Principal series

The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup Template:Mvar of Template:Math. Since the one-dimensional representations of Template:Mvar correspond to the representations of the diagonal matrices, with non-zero complex entries Template:Mvar and Template:Math, they thus have the form

${\chi }_{\nu ,k}\left(\begin{array}{cc}z& 0\\ c& z^{-1}\end{array}\right)=r^{i\nu }{e}^{ik\theta },$

for Template:Mvar an integer, Template:Mvar real and with Template:Mvar. The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when Template:Mvar is replaced by Template:Math. By definition the representations are realized on L² sections of line bundles on Template:Math, which is isomorphic to the Riemann sphere. When Template:Math, these representations constitute the so-called spherical principal series.

The restriction of a principal series to the maximal compact subgroup Template:Math of Template:Mvar can also be realized as an induced representation of Template:Mvar using the identification Template:Math, where Template:Math is the maximal torus in Template:Mvar consisting of diagonal matrices with Template:Math. It is the representation induced from the 1-dimensional representation Template:Math, and is independent of Template:Mvar. By Frobenius reciprocity, on Template:Mvar they decompose as a direct sum of the irreducible representations of Template:Mvar with dimensions Template:Math with Template:Mvar a non-negative integer.

Using the identification between the Riemann sphere minus a point and Template:Math, the principal series can be defined directly on Template:Math by the formula

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

Irreducibility can be checked in a variety of ways:

• The representation is already irreducible on Template:Mvar. 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 Template:Math where Template:Mvar is the Weyl group element

$\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$.

• The action of the Lie algebra $𝔤$ of Template:Mvar can be computed on the algebraic direct sum of the irreducible subspaces of Template:Mvar can be computed explicitly and the it can be verified directly that the lowest-dimensional subspace generates this direct sum as a $𝔤$-module.

Complementary series

The for Template:Math, the complementary series is defined on [[w:L2 space|Template:Math functions]] Template:Mvar on Template:Math for the inner product

$(f,g)=\int \int \frac{f(z)\overline{g(w)}dzdw}{|z-w{|}^{2-t}}.$

with the action given by

${\pi }_t{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}^{-1}f(z)=|cz+d{|}^{-2-t}f\left(\frac{az+b}{cz+d}\right).$

The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of Template:Mvar, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of Template:Math. Irreducibility can be proved by analyzing the action of $𝔤$ on the algebraic sum of these subspaces or directly without using the Lie algebra.

Plancherel theorem

The only irreducible unitary representations of Template:Math are the principal series, the complementary series and the trivial representation. Since Template:Math acts as Template:Math on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided Template:Mvar is taken to be even.

To decompose the left regular representation of Template:Mvar on Template:Math, only the principal series are required. This immediately yields the decomposition on the subrepresentations Template:Math, the left regular representation of the Lorentz group, and Template:Math, the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with Template:Math.)

The left and right regular representation Template:Mvar and Template:Mvar are defined on Template:Math by

$\lambda (g)f(x)=f(g^{-1}x),\rho (g)f(x)=f(xg).$

Now if Template:Mvar is an element of Template:Math, the operator Template:Math defined by

${\pi }_{\nu ,k}(f)=\underset{G}{\int }f(g)\pi (g)dg$

is Hilbert–Schmidt. Define a Hilbert space Template:Mvar by

$H=\underset{k\ge 0}{⨁}HS(L^2(C))\otimes L^2(R,c_k({\nu }^2+k^2{)}^{1/2}d\nu ),$

where

$c_0=1/4{\pi }^{3/2},c_k=1/(2\pi {)}^{3/2}(k\ne 0)$

and Template:Math denotes the Hilbert space of Hilbert–Schmidt operators on Template:Math. Then the map Template:Mvar defined on Template:Math by

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

extends to a unitary of Template:Math onto Template:Mvar.

The map Template:Mvar satisfies the intertwining property

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

If Template:Math, Template:Math are in Template:Math then by unitarity

$(f_1,f_2)=\sum _{k\ge 0}{c}_k^2\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\mathrm{T}\mathrm{r}({\pi }_{\nu ,k}(f_1){\pi }_{\nu ,k}(f_2{)}^{*})({\nu }^2+k^2)d\nu .$

Thus if Template:Math denotes the convolution of Template:Math and Template:Math, and $f_2^{*}(g)=\overline{f_2(g^{-1})}$, then

$f(1)=\sum _{k\ge 0}{c}_k^2\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\mathrm{T}\mathrm{r}({\pi }_{\nu ,k}(f))({\nu }^2+k^2)d\nu .$

The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively. The Plancherel formula extends to all Template:Math in Template:Math. By a theorem of Jacques Dixmier and Paul Malliavin, every function Template:Mvar in $C_c^{\mathrm{\infty }}(G)$ is a finite sum of convolutions of similar functions, the inversion formula holds for such Template:Mvar. It can be extended to much wider classes of functions satisfying mild differentiability conditions.

Classification

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 Template:Math on a Hilbert space Template:Mvar of Template:Math is at hand. Since Template:Math is a subgroup, Template:Math is a representation of it as well. Each irreducible subrepresentation of Template:Math is finite-dimensional, and the Template:Math representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of Template:Math if Template:Math is unitary.

The steps are the following:

1. Chose a suitable basis of common eigenvectors of Template:Math and Template:Math.
2. Compute matrix elements of Template:Math and Template:Math.
3. Enforce Lie algebra commutation relations.
4. Require unitarity together with orthonormality of the basis.

Step 1

One may suitably choose a basis and label the basis vectors by

$|j_0{j}_1;jm⟩.$

If this was a finite-dimensional representation, then Template:Math would correspond the lowest occurring eigenvalue Template:Math of Template:Math in the representation, equal to Template:Math, and Template:Math would correspond to the highest occurring eigenvalue, equal to Template:Math. In the infinite-dimensional case, Template:Math retains this meaning, but Template:Math does not. One assumes for simplicity that a given Template:Mvar 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 Template:Math and Template:Math forming the basis of the Lie algebra of Template:Math. The matrix elements of

$J_{\pm }=J_1\pm i{J}_2,J_3$

(here one is operating in the comlpexified Lie algebra) are known from the representation theory of the rotation group, and are given by

$⟨jm|J_{+}|jm-1⟩=⟨jm-1|J_{-}|jm⟩=\sqrt{(j+m)(j-m+1)},⟨j,m|J_3|jm⟩=m,$

where the labels Template:Math and Template:Math have been dropped since they are the same for all basis vectors in the representation.

Due to the commutation relations

$[J_i,K_j]=i{ϵ}_{ijk}{K}_k,$

the triple Template:Math 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

$\begin{array}{cc}{K}_0^{(1)}& =K_3\\ K_{\pm 1}^{(1)}& =\mp \frac{1}{\sqrt{2}}(K_1\pm i{K}_2),\end{array}$

where the superscript Template:Math signifies that the defined quantities are the components of a spherical tensor operator of rank Template:Math (which explains the factor √2 as well) and the subscripts Template:Math are referred to as Template:Mvar in formulas below, are given by

$\begin{array}{cc}⟨j^{\prime }{m}^{\prime }|K_0^{(1)}|jm⟩& =⟨j^{\prime }{m}^{\prime }k=1q=0|jm⟩⟨j\Vert K^{(1)}\Vert j^{\prime }⟩\\ ⟨j^{\prime }{m}^{\prime }|K_{\pm 1}^{(1)}|jm⟩& =⟨j^{\prime }{m}^{\prime }k=1q=\pm 1|jm⟩⟨j\Vert K^{(1)}\Vert j^{\prime }⟩.\end{array}$

Here the first factors on the right hand sides are Clebsch–Gordan coefficients for coupling Template:Math with Template:Mvar to get Template:Mvar. The second factors are the reduced matrix elements. They do not depend on Template:Math or Template:Mvar, but depend on Template:Math and, of course, Template:Math. For a complete list of non-vanishing equations, see Template:Harvtxt.

Step 3

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

$[K_{\pm },K_3]=\pm J_{\pm },[K_{+},K_{-}]=-2J_3.$

This results in a set of equations for which the solutions are

$\begin{array}{c}⟨j\Vert K^{(1)}\Vert j⟩=i\frac{j_1{j}_0}{\sqrt{j(j+1)}},\\ ⟨j\Vert K^{(1)}\Vert j-1⟩=-B_j{\xi }_j\sqrt{j(2j-1)},\\ ⟨j-1\Vert K^{(1)}\Vert j⟩=B_j{\xi }_j^{-1}\sqrt{j(2j+1)},\\ \end{array}$

where

$B_j=\sqrt{\frac{(j^2-j_0^2)(j^2-j_1^2)}{j^2(4j^2-1)}},$

and

$\begin{array}{cc}{j}_0& =0,\frac{1}{2},1,\dots ,\\ j_1& \in ℂ,\\ {\xi }_j& \in ℂ.\\ \end{array}$

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 Template:Math and Template:Math. Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning

$\begin{array}{cc}{K}_{\pm }^{†}& =K_{\mp },\\ K_3^{†}& =K_3.\end{array}$

This translates to

$\begin{array}{cc}⟨j\Vert K^{(1)}\Vert j⟩& =\overline{⟨j\Vert K^{(1)}\Vert j⟩},\\ ⟨j\Vert K^{(1)}\Vert j-1⟩& =-\overline{⟨j-1\Vert K^{(1)}\Vert j⟩},\\ \end{array}$

leading to

$\begin{array}{cc}{j}_0(j_1+\overline{j_1})& =0,\\ |B_j|(|{\xi }_j{|}^2-e^{-2i{\beta }_j})& =0,\end{array}$

where Template:Math is the angle of Template:Math on polar form. For Template:Math one has $|{\zeta }_j{|}^2=1$, and Template:Math is chosen by convention. There are two possible cases. The first with Template:Math gives, with Template:Math, Template:Mvar real,

$\begin{array}{cc}⟨j\Vert K^{(1)}\Vert j⟩& =\frac{\nu j_0}{j(j+1)},\\ B_j& =\sqrt{\frac{(j^2-j_0^2)(j^2+{\nu }^2)}{4j^2-1}}.\end{array}$

This is principal series and the elements may be denoted Template:Math. For the other possibility, Template:Math, one has

$\begin{array}{cc}⟨j\Vert K^{(1)}\Vert j⟩& =0,\\ B_j& =\sqrt{\frac{(j^2-{\nu }^2)}{4j^2-1}}.\end{array}$

One needs to require that $B_j^2$ is real and positive for Template:Math (because Template:Math), leading to Template:Math. This is complementary series and its elements may be denoted Template:Math.

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

$\begin{array}{cc}{J}_1& =J^{23}=-J^{32}=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right),\\ J_2& =J^{31}=-J^{13}=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right),\\ J_3& =J^{12}=-J^{21}=i\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right),\\ K_1& =J^{01}=J^{10}=i\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\\ K_2& =J^{02}=J^{20}=i\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\\ K_3& =J^{03}=J^{30}=i\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right).\end{array}$

The commutation relations of the Lie algebra so(3; 1) are

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

In three-dimensional notation, these are

$[J_i,J_j]=i{ϵ}_{ijk}{J}_k,[J_i,K_j]=i{ϵ}_{ijk}{K}_k,[K_i,K_j]=-i{ϵ}_{ijk}{J}_k.$

The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol Template:Mvar above and in the sequel should be observed.

Weyl spinors and bispinors

Solutions to the Dirac equation transform under the Template:Math-representation. Dirac discovered the gamma matrices in his search for a relativistically invariant equation, then already known to mathematicians.

By taking, in turn, Template:Math, Template:Math and Template:Math, Template:Math and by setting

$J_i^{(\frac{1}{2})}=\frac{1}{2}{\sigma }_i$

in the general expression Template:EquationNote, and by using the trivial relations Template:Nowrap and Template:Math, one obtains

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) Template:Math and Template:Math, whose elements Template:Math and Template:Math are called left- and right-handed Weyl spinors respectively. Given Template:Math and Template:Math one may form their direct sum as representations,

This is, up to a similarity transformation, the Template:Math Dirac spinor representation of Template:Math. It acts on the 4-component elements Template:Math of Template:Math, 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 Template:Math. Expressions for the group representations are obtained by exponentiation.

See also

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

Freely available online references

• Template:Cite arXiv Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006).
• Template:Citation Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group.

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