“Representation theory of the Lorentz group”的意思、由来-开放百科全书

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;^[1] physics itself must be made out of them. Indeed, special relativity together with quantum mechanics are the two physical theories that are most thoroughly established,^[2] 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

of the full Lorentz group {{math|O(3; 1)}} are obtained by employing the Lie correspondence and the matrix exponential. The full finite-dimensional representation theory of the universal covering group (and also the spin group, a double cover)

is obtained, and explicitly given in terms of action on a function space in representations of

and

. The representatives of time reversal and space inversion are given in space inversion and time reversal, completing the finite-dimensional theory for the full Lorentz group. The general properties of the (m, n) representations are outlined. Action on function spaces is considered, with the action on spherical harmonics and the Riemann P-functions appearing as examples. The infinite-dimensional case of irreducible unitary representations are realized for the

principal series and the complementary series. Finally, the Plancherel formula for

is given, and representations of {{math|SO(3, 1)}} are classified and realized for Lie algebras.

The development of the representation theory has historically followed the development of the more general theory of representation theory of semisimple groups, largely due to Élie Cartan and Hermann Weyl, but the Lorentz group has also received special attention due to its importance in physics. Notable contributors are physicist E. P. Wigner and mathematician Valentine Bargmann with their Bargmann–Wigner program,^[3] 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.^[4] 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,^[5] in 1947. The corresponding classification for

was published independently by Bargmann and Israel Gelfand together with Mark Naimark in the same year.

The non-technical introduction contains some prerequisite material for readers not familiar with representation theory. Standard results used here from general finite-dimensional representation theory are outlined in technical introduction to finite-dimensional representation theory. The Lie algebra basis and other adopted conventions are given in conventions and Lie algebra bases. There is a substantially detailed article on the present subject to be found at Wikiversity.

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.

Symmetry of space and time

Space itself possesses symmetry. It looks the same no matter how one rotates it, and the resulting rotational symmetry is referred to as isotropy of space. In the present case it is common to use passive rotations, meaning that the observer^[6] rotates themself. Mathematically, the active operation of a rotation is performed by multiplying position vectors by a rotation matrix. A passive rotation is accomplished by rotating only the basis vectors of the coordinate system. (The coordinate system can be thought of as being fixed in the rotated observer. The observer is physically rotated.) In this way, every point in space obtains new coordinates as if it was somehow physically rigidly rotated.

The Lorentz group contains all rotation matrices, extended to four dimensions with zeros in the first row and the first column except for the upper left element which is one, as elements.

There are, in addition, matrices that effect Lorentz boosts. These can be thought of, in the passive view, as (instantly!) giving the coordinate system (and with it the observer) a velocity in a chosen direction.

Finally, two special transformations are used to invert the coordinate system in space, space inversion, and in time, time reversal. In the first case, the space coordinate axes are reversed. The latter is reversal of the time direction. This is thought of, in the passive view, as having the observer set their clock at minus what it shows and then have the clock's hands move counterclockwise. Physical time progresses forward.

Mathematically the Lorentz group is defined as the set of transformations preserving the bilinear form

where the left-hand side is the Minkowski inner product of two events in spacetime, and the right-hand side is the spacetime interval, see classical group for mathematical detail.

Lorentz transformations

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

Multiplication table and representations

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.^[7] The same holds true in the case of an infinite group like the rotation group SO(3) or the Lorentz group. The multiplication table is just harder to visualize in the case of a group of uncountable size (same size as the set of reals). One way to do this is to wellorder the elements of the group with an ordinal number {{mvar|ρ}} being the order type. The "infinite Cayley table" is then indexed by two ordinals {{math|0 ≤ α, β ≤ ρ}} written in Cantor normal form.

Ordinary Lorentz transformations matrices do not suffice

The objects to be transformed may be something else than ordinary physical objects extending in three spatial dimensions (and time, unless the frame is the rest frame). It is for these objects that representation theory is needed for the mathematical description of transformations induced by ordinary Lorentz transformations of spacetime. For instance, the electromagnetic field is often (naively) envisaged by the assignment to each point in spacetime of a three-dimensional vector representing the electric field and another three-dimensional vector representing the magnetic field.

When space is rotated, the classically expected thing happens. The electric field and the magnetic field vectors at a designated point rotate with preserved length and angle between them.

Under Lorentz boosts they behave differently, showing that the two vectors aren't separate physical objects. The electric and magnetic components mix. See the illustration on the right. The electromagnetic field tensor displays the manifestly covariant mathematical structure of the electromagnetic field. It has six independent components at an event.^[8]

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 {{math|4 × 4}} in size that satisfies the same multiplication table as the matrices in the original Lorentz group. Returning to the example of the electromagnetic field, {{math|6 × 6}} matrices are needed that can be applied to a six-dimensional column vector containing the all together six components of the electromagnetic field. Thus {{math|6 × 6}}-matrices are sought for, such that

correctly expresses the transformation of the electromagnetic field under the Lorentz transformation {{math|Λ}}.^[9] The same reasoning can be applied to Dirac's bispinors. While these have {{math|4}} components, the original {{math|4 × 4}}-matrices in the Lorentz group are inappropriate, even when restricted to rotations. Another {{math|4 × 4}}-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 {{mvar|X}} endowed with a group action. A set being endowed with a group action {{mvar|A}} means, in essence, that if {{math|x ∈ X}} and {{math|g ∈ G}} that {{math|A(g)x {{=}} y}} with {{math|y ∈ X}}. If

denotes the set of all complex-valued functions on {{mvar|X}}, which is a vector space, a representation {{mvar|Π}} of {{mvar|G}} can be defined by^[10]

and there is representation of {{mvar|G}}. This representation of {{mvar|G}} is finite-dimensional if and only if {{mvar|X}} is a finite set. This method is very general, and typically vector spaces of more specialized functions on sets close at hand are explored. Illustrating this procedure, a group {{mvar|G}} of {{math|n}}-dimensional matrices as a subset of Euclidean space

are considered, and the space of functions polynomials, perhaps of some maximum degree {{mvar|d}}, or even homogeneous polynomials of degree {{mvar|d}}, all defined on

Those functions are then restricted to

The set {{math|X {{=}} G}} automatically comes equipped with group actions, namely

Here {{math|L_g}} denotes left action (by {{mvar|g}}), {{math|R_g}} denotes right action (by {{mvar|g}}), and {{math|C_g}} denotes conjugation (by {{mvar|g}}). With this sort of action, the vectors being acted on are functions. The resulting representations are (when the functions are unrestricted), in the first and second cases respectively, the left regular representation and the right regular representation of {{mvar|G}} on

^[10]

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 representations on infinite-dimensional spaces to the finite-dimensional cases, an ordered basis is chosen for the vector space of functions and the action on basis functions under a given transformation is examined. Explicitly, if {{math|f₁, f₂, ...}} is a basis, compute

The coefficients of the basis functions in the expression for each transformed basis function is a column in the representative matrix. In general, the resulting matrix is countably infinite in dimension:^[11]

Again, it is required that the set of infinite matrices obtained this way stand in one-to-one correspondence with the original {{math|4 × 4}}-matrices and that the multiplication table is the right one – the one of the {{math|4 × 4}}-matrices.^[12] It should be emphasized that in the infinite-dimensional case, the concern is rarely the complete matrices. They are exposed here only to highlight the common thread. But individual matrix elements are frequently computed, especially for the Lie algebra (below).

Lie 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 {{math|6}}, one need only find a finite number of representative matrices of the Lie algebra, one for each element of a basis of the Lie algebra as a vector space. The rest follow from extension by linearity, and the representation of the group is obtained by exponentiation.

One possible choice of basis for the Lie algebra is, in the standard representation, given in conventions and Lie algebra bases.

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.^[13]

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.^[14]^[15]

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.^[16] There were speculative theories,^[17]^[19] (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.{{citation needed|date=September 2017}}

The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the little group of the Poincaré group of spacelike vectors in higher spacetime dimension. The corresponding infinite-dimensional unitary representations of the (generalized) Poincaré group are the so-called tachyonic representations. Tachyons appear in the spectrum of bosonic strings and are associated with instability of the vacuum.^[18]^[19] 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.^[20]

One open problem (as of 2006) is the completion of the Bargmann–Wigner programme for the isometry group {{math|SO(D – 2, 1)}} of the de Sitter spacetime {{math|dS_{D – 2}}}. Ideally, the physical components of wave functions would be realized on the hyperboloid {{math|dS_{D – 2}}} of radius {{math|μ > 0}} embedded in

and the corresponding {{math|O(D − 2, 1)}} covariant wave equations of the infinite-dimensional unitary representation to be known.^[19]

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.{{citation needed|date=September 2017}}

Classical field theory

While the electromagnetic field together with the gravitational field are the only classical fields providing accurate descriptions of nature, other types of classical fields are important too. In the approach to quantum field theory (QFT) referred to as second quantization, the starting point is one or more classical fields, where e.g. the wave functions solving the Dirac equation are considered as classical fields prior to (second) quantization.^[21] While second quantization and the Lagrangian formalism associated with it is not a fundamental aspect of QFT,^[22] it is the case that so far all quantum field theories can be approached this way, including the standard model.^[23] 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.^[21]

Relativistic quantum mechanics

For the present purposes the following definition is made:^[24] A relativistic wave function is a set of {{mvar|n}} functions {{math|ψ^α}} on spacetime which transforms under an arbitrary proper Lorentz transformation {{math|Λ}} as

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

The most useful relativistic quantum mechanics one-particle theories (there are no fully consistent such theories) are the Klein–Gordon equation^[25] and the Dirac equation^[26] in their original setting. They are relativistically invariant and their solutions transform under the Lorentz group as Lorentz scalars ({{math|(m, n) {{=}} (0, 0)}}) and bispinors respectively ({{math|(0, {{sfrac|1|2}}) ⊕ ({{sfrac|1|2}}, 0)}}). The electromagnetic field is a relativistic wave function according to this definition, transforming under {{math|(1, 0) ⊕ (0, 1)}}.^[27]

The infinite-dimensional representations may be used in the analysis of scattering.^[28]

Quantum field theory

In QFT, the demand for relativistic invariance enters, among other ways in that the S-matrix necessarily must be Poincaré invariant.^[29] This has the implication that there is one or more infinite-dimensional representation of the Lorentz group acting on Fock space.^[30] One way to guarantee the existence of such representations is the existence of a Lagrangian description (with modest requirements imposed, see the reference) of the system using the canonical formalism, from which a realization of the generators of the Lorentz group may be deduced.^[31]

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.^[32] For illustration, consider the definition an {{mvar|n}}-component field operator:^[33] A relativistic field operator is a set of {{mvar|n}} operator valued functions on spacetime which transforms under proper Poincaré transformations {{math|(Λ, a)}} according to^[34]^[35]

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

By considerations of differential constraints that the field operator must be subjected to in order to describe a single particle with definite mass {{mvar|m}} and spin {{mvar|s}} (or helicity), it is deduced that^[36]^[37]

where {{math|a^†, a}} are interpreted as creation and annihilation operators respectively. The creation operator {{math|a^†}} transforms according to^[36]^[38]

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

that carry both the indices {{math|(x, α)}} operated on by Lorentz transformations and the indices {{math|(p, σ)}} operated on by Poincaré transformations. This may be called the Lorentz–Poincaré connection.^[39] To exhibit the connection, subject both sides of equation {{EquationNote|X1|(X1)}} to a Lorentz transformation resulting in for e.g. {{mvar|u}},

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

All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the {{math|(m, n)}} representation under which it is supposed to transform,^[40] 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.^[41]

Speculative theories

In theories in which spacetime can have more than {{math|D {{=}} 4}} dimensions, the generalized Lorentz groups {{math|O(D − 1; 1)}} of the appropriate dimension take the place of {{math|O(3; 1)}}.^[42]

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.^[43] This results in a relativistically invariant theory in any spacetime dimension.^[44] 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.^[45] 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 {{math|Z₂}}-graded Lie algebra extending the Poincaré algebra. The structure of such an algebra is to a large degree fixed by the demands of Lorentz invariance. In particular, the fermionic operators (grade {{math|1}}) belong to a {{math|(0, {{sfrac|1|2}})}} or {{math|({{sfrac|1|2}}, 0)}} representation space of the (ordinary) Lorentz Lie algebra.^[46] The only possible dimension of spacetime in such theories is 10.^[47]

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.^[48]

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.^[49]^[50] 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.^[51] Lack of simple connectedness gives rise to spin representations of the group.^[52] The non-connectedness means that, for representations of the full Lorentz group, time reversal and space inversion has to be dealt with separately.^[53]^[54]

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.^[55]^[56] By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing.^[57]^[58] 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.^[59]^[60] 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.^[61]^[62] 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^[59]^[63]^[55]^[64]^[65] and Harish-Chandra^[66]^[67] and physicists Eugene Wigner^[68]^[69] and Valentine Bargmann^[70]^[71]^[72] made substantial contributions both to general representation theory and in particular to the Lorentz group.^[73] 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.^[74]^[75]^[76]

Technical introduction to finite-dimensional representation theory

The same material can be found, in more detailed form (in appropriate sequence and inline), in the Wikiversity version.

The Lie algebra

According to the strategy, the irreducible complex linear representations of the complexification,

of the Lie algebra

of the Lorentz group are to be found. A convenient basis for

is given by the three generators {{math|J_i}} of rotations and the three generators {{math|K_i}} of boosts. They are explicitly given in conventions and Lie algebra bases.

The Lie algebra is complexified, and the basis is changed to the components of its two ideals^[77]

The components of {{math|A {{=}} (A₁, A₂, A₃)}} and {{math|B {{=}} (B₁, B₂, B₃)}} separately satisfy the commutation relations of the Lie algebra

and, moreover, they commute with each other,^[78]

where {{math|i, j, k}} are indices which each take values {{math|1, 2, 3}}, and {{math|ε_ijk}} is the three-dimensional Levi-Civita symbol. Let

and

denote the complex linear span of {{math|A}} and {{math|B}} respectively.

The utility of these isomorphisms comes from the fact that all irreducible representations of

, and hence (see strategy) all irreducible complex linear representations of

are known. According to the final conclusion in strategy, the irreducible complex linear representation of

is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of

The unitarian trick

The Lie algebra

is the Lie algebra of

It contains the compact subgroup {{math|SU(2) × SU(2)}} with Lie algebra

The latter is a compact real form of

Thus from the first statement of the unitarian trick, representations of {{math|SU(2) × SU(2)}} are in one-to-one correspondence with holomorphic representations of

By compactness, Peter–Weyl theorem applies to {{math|SU(2) × SU(2)}},^[81] and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of {{math|SU(2) × SU(2)}} are precisely the tensor products of irreducible unitary representations of {{math|SU(2)}}.^[82]

By appeal to simple connectedness, the second statement of the unitarian trick is applied. The objects in the following list are in one-to-one correspondence:

Tensor products of representations appear at the Lie algebra level as either of^[83]

where {{math|Id}} is the identity operator. Here, the latter interpretation, which follows from {{EquationNote|G6|(G6)}}, is intended. The highest weight representations of

are indexed by {{mvar|μ}} for {{math|1=μ = 0, 1/2, 1, ...}}. (The highest weights are actually {{math|2μ {{=}} 0, 1, 2, ...}}, but the notation here is adapted to that of

) The tensor products of two such complex linear factors then form the irreducible complex linear representations of

Finally, the

-linear representations of the real forms of the far left,

, and the far right,

^[84] in {{EquationNote|A1|(A1)}} are obtained from the

-linear representations of

characterized in the previous paragraph.

The representations

The real linear representations for

and

follow here assuming the complex linear representations of

are known. Explicit realizations and group representations are given later.

The (μ, ν)-representations of sl(2, C)

The complex linear representations of the complexification of

obtained via isomorphisms in {{EquationNote|A1|(A1)}}, stand in one-to-one correspondence with the real linear representations of

^[85] The set of all, at least real linear, irreducible representations of

are thus indexed by a pair {{math|(μ, ν)}}. The complex linear ones, corresponding precisely to the complexification of the real linear

representations, are of the form {{math|(μ, 0)}}, while the conjugate linear ones are the {{math|(0, ν)}}.^[85] All others are real linear only. The linearity properties follow from the canonical injection, the far right in {{EquationNote|A1|(A1)}}, of

into its complexification. Representations on the form {{math|(ν, ν)}} or {{math|(μ, ν) ⊕ (ν, μ)}} are given by real matrices (the latter are not irreducible). Explicitly, the real linear {{math|(μ, ν)}}-representations of

are

where

are the complex linear irreducible representations of

and

their complex conjugate representations. (The labeling is usually in the mathematics literature {{math|0, 1, 2, …}}, but half-integers are chosen here to conform with the labeling for the

Lie algebra.) Here the tensor product is interpreted in the former sense of {{EquationNote|A0|(A0)}}. These representations are concretely realized below.

The (m, n)-representations of so(3; 1)

Via the displayed isomorphisms in {{EquationNote|A1|(A1)}} and knowledge of the complex linear irreducible representations of

upon solving for {{math|J}} and {{math|K}}, all irreducible representations of

and, by restriction, those of

are obtained. The representations of

obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible.^[79] Since

is semisimple,^[79] all its representations can be built up as direct sums of the irreducible ones.

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

where {{mvar|V}} is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given by^[86]

are the {{math|(2n + 1)}}-dimensional irreducible representations of

also termed spin matrices or angular momentum matrices. These are explicitly given as^[87]

where {{math|δ}} denotes the Kronecker delta. In components, with {{math|−m ≤ a, a′ ≤ m}}, {{math|−n ≤ b, b′ ≤ n}}, the representations are given by^[88]

Common representations

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

The group

The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence.^[94] The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras.^[95] The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted

The general theory is summarized in technical introduction to finite-dimensional representation theory.

for some vector space {{mvar|V}} is a representation, a representation {{math|Π}} of the connected component of {{mvar|G}} is defined by

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

Surjectiveness of exponential map for SO(3, 1)

From a practical point of view, it is important whether the first formula in {{EquationNote|G2|(G2)}} can be used for all elements of the group. It holds for all

, however, in the general case, e.g. for

, not all {{math|g ∈ G}} are in the image of {{math|exp}}.

But

is surjective. One way to show this is to make use of the isomorphism

the latter being the Möbius group. It is a quotient of

(see the linked article). The quotient map is denoted with

The map

is onto.^[96] Apply {{EquationNote|Lie|(Lie)}} with {{mvar|π}} being the differential of {{mvar|p}} at the identity. Then

Since the left hand side is surjective (both {{math|exp}} and {{mvar|p}} are), the right hand side is surjective and hence

is surjective.^[97] Finally, recycle the argument once more, but now with the known isomorphism between {{math|SO(3; 1)⁺}} and

to find that {{math|exp}} is onto for the connected component of the Lorentz group.

Fundamental group

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

To exhibit the fundamental group of {{math|SO(3; 1)⁺}}, the topology of its covering group

is considered. By the polar decomposition theorem, any matrix

may be uniquely expressed as^[98]

where {{mvar|u}} is unitary with determinant one, hence in {{math|SU(2)}}, and {{mvar|h}} is Hermitian with trace zero. The trace and determinant conditions imply:^[99]

The manifestly continuous one-to-one map is a homeomorphism with continuous inverse given by (the locus of {{mvar|u}} is identified with

)

explicitly exhibiting that

is simply connected. But

where

is the center of

. Identifying {{mvar|λ}} and {{math|−λ}} amounts to identifying {{mvar|u}} with {{math|−u}}, which in turn amounts to identifying antipodal points on

Thus topologically,^[99]

where last factor is not simply connected: Geometrically, it is seen (for visualization purposes,

may be replaced by

) that a path from {{mvar|u}} to {{math|−u}} in

is a loop in

since {{mvar|u}} and {{math|−u}} are antipodal points, and that it is not contractible to a point. But a path from {{mvar|u}} to {{math|−u}}, thence to {{mvar|u}} again, a loop in

and a double loop (considering {{math|p(ue^h) {{=}} p(−ue^h)}}, where

is the covering map) in

that is contractible to a point (continuously move away from {{math|−u}} "upstairs" in

and shrink the path there to the point {{mvar|u}}).^[99] Thus {{math|π₁(SO(3; 1))}} is a group with two equivalence classes of loops as its elements, or put more simply, {{math|SO(3; 1)}} is doubly connected.

Projective representations

Since {{math|π₁(SO(3; 1)⁺)}} has two elements, some representations of the Lie algebra will yield projective representations.^[100]^[101] Once it is known whether a representation is projective, formula {{EquationNote|G2|(G2)}} applies to all group elements and all representations, including the projective ones — with the understanding that the representative of a group element will depend on which element in the Lie algebra (the {{mvar|X}} in {{EquationNote|G2|(G2)}}) is used to represent the group element in the standard representation.

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

For a projective representation {{math|Π}} of {{math|SO(3; 1)⁺}}, it holds that^[99]

since any loop in {{math|SO(3; 1)⁺}} traversed twice, due to the double connectedness, is contractible to a point, so that its homotopy class is that of a constant map. It follows that {{math|Π}} is a double-valued function. It is not possible to consistently chose a sign to obtain a continuous representation of all of {{math|SO(3; 1)⁺}}, but this is possible locally around any point.^[51]

The covering group SL(2, C)

which are exactly on the form of the {{math|3}}-dimensional version of the commutation relations for

(see conventions and Lie algebra bases below). Thus, the map {{math|J_i ↔ j_i}}, {{math|K_i ↔ k_i}}, extended by linearity is an isomorphism. Since

is simply connected, it is the universal covering group of {{math|SO(3; 1)⁺}}.

A geometric view

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

With this multiplication, {{mvar|G}} becomes a group isomorphic to

^[102] the universal covering group of {{math|SO(3; 1)⁺}}. Since each {{mvar|π_g}} has two elements, by the above construction, there is a 2:1 covering map {{math|p : G → SO(3; 1)⁺}}. According to covering group theory, the Lie algebras

and

of {{mvar|G}} are all isomorphic. The covering map {{math|p : G → SO(3; 1)⁺}} is simply given by {{math|p(g, [p_g]) {{=}} g}}.

An algebraic view

For an algebraic view of the universal covering group, let

act on the set of all Hermitian {{gaps|2|×|2}} matrices

by the operation^[99]

The map {{math|P}} is a group homomorphism into

Thus

is a 4-dimensional representation of

. Its kernel must in particular take the identity matrix to itself, {{math|A^†IA {{=}} A^†A {{=}} I}} and therefore {{math|A^† {{=}} A⁻¹}}. Thus {{math|AX {{=}} XA}} for {{mvar|A}} in the kernel so, by Schur's lemma,^[103] {{mvar|A}} is a multiple of the identity, which must be {{math|±I}} since {{math|det A {{=}} 1}}.^[104] The space

is mapped to Minkowski space {{math|M⁴}}, via

The action of {{math|P(A)}} on

preserves determinants. The induced representation {{math|p}} of

via the above isomorphism, given by

This means that {{math|p(A)}} belongs to the full Lorentz group {{math|SO(3; 1)}}. By the main theorem of connectedness, since

is connected, its image under {{math|p}} in {{math|SO(3; 1)}} is connected, and hence is contained in {{math|SO(3; 1)⁺}}.

It can be shown that the Lie map of

is a Lie algebra isomorphism:

^[105] The map {{math|P}} is also onto.^[106]

Thus

, since it is simply connected, is the universal covering group of {{math|SO(3; 1)⁺}}, isomorphic to the group {{mvar|G}} of above.

Non-surjectiveness of exponential mapping for SL(2, C)

In general, if {{mvar|g}} is an element of a connected Lie group {{mvar|G}} with Lie algebra

then, by {{EquationNote|Lie|(Lie)}},

Realization of representations of {{math|SL(2, C)}} and {{math|sl(2, C)}} and their Lie algebras

The complex linear representations of

and

are more straightforward to obtain than the

representations. They can be (and usually are) written down from scratch. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are related to the complex linear Lie algebra representations by exponentiation. The real linear representations of

are exactly the {{math|(μ, ν)}}-representations. They can be exponentiated too. The {{math|(μ, 0)}}-representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here).

The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of {{math|i}} and there is no factor of {{math|i}} in the exponential mapping compared to the physics convention used elsewhere. Let the basis of

be^[109]

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

Complex linear representations

The irreducible holomorphic {{math|(n + 1)}}-dimensional representations

can be realized on the space of homogeneous polynomial of degree {{math|n}} in 2 variables

^[110]^[111] the elements of which are

The associated

-action is, using {{EquationNote|G6|(G6)}} and the definition above, for the basis elements of

^[114]

Real linear representations

The {{math|(μ, ν)}}-representations are realized on a space of polynomials

homogeneous of degree {{mvar|μ}} in

and homogeneous of degree {{mvar|ν}} in

^[111] The representations are given by^[115]

Properties of the (m, n) representations

The {{math|(m, n)}} representations, defined above via {{EquationNote|A1|(A1)}} (as restrictions to the real form

) of tensor products of irreducible complex linear representations {{math|π_{m {{=}} μ}}} and {{math|π_{n {{=}} ν}}} of

are irreducible, and they are the only irreducible representations.^[116]

Dimension

The {{math|(m, n)}} representations are {{math|(2m + 1)(2n + 1)}}-dimensional.^[120] This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of

and

. For a Lie general algebra

the Weyl dimension formula,^[121]

applies, where {{math|R⁺}} is the set of positive roots, {{math|ρ}} is the highest weight, and {{math|δ}} is half the sum of the positive roots. The inner product

is that of the Lie algebra

invariant under the action of the Weyl group on

the Cartan subalgebra. The roots (really elements of

are via this inner product identified with elements of

For

the formula reduces to {{math|dim π_μ {{=}} 2μ + 1 {{=}} 2m + 1}}, where the present notation must be taken into account. The highest weight is {{math|2μ}}.^[122] By taking tensor products, the result follows.

Faithfulness

If a representation {{math|Π}} of a Lie group {{math|G}} is not faithful, then {{math|N {{=}} ker Π}} is a nontrivial normal subgroup.^[123] There are three relevant cases.

In the case of {{math|SO(3; 1)⁺}}, the first case is excluded since {{math|SO(3; 1)⁺}} is semi-simple.^[125] The second case (and the first case) is excluded because {{math|SO(3; 1)⁺}} is simple.^[126] For the third case, {{math|SO(3; 1)⁺}} is isomorphic to the quotient

But

is the center of

It follows that the center of {{math|SO(3; 1)⁺}} is trivial, and this excludes the third case. The conclusion is that every representation {{math|Π : SO(3; 1)⁺ → GL(V)}} and every projective representation {{math|Π : SO(3; 1)⁺ → PGL(W)}} for {{math|V, W}} finite-dimensional vector spaces are faithful.

By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial non-discrete normal subgroups replaced by (one-dimensional) nontrivial ideals in the Lie algebra,^[127] and the center of {{math|SO(3; 1)⁺}} replaced by the center of

The center of any semisimple Lie algebra is trivial^[128] and

is semi-simple and simple, and hence has no non-trivial ideals.

A related fact is that if the corresponding representation of

is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding

representation is not faithful, but is {{math|2:1}}.

Non-unitarity

The {{math|(m, n)}} Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary.^[129] 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.^[51] There is a topological proof of this.^[130] Let {{math|u : G → GL(V)}}, where {{math|V}} is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group {{mvar|G}}. Then {{math|u(G) ⊂ U(V) ⊂ GL(V)}} where {{math|U(V)}} is the compact subgroup of {{math|GL(V)}} consisting of unitary transformations of {{mvar|V}}. The kernel of {{math|u}} is a normal subgroup of {{mvar|G}}. Since {{mvar|G}} is simple, {{math|ker u}} is either all of {{mvar|G}}, in which case {{math|u}} is trivial, or {{math|ker u}} is trivial, in which case {{math|u}} is faithful. In the latter case {{math|u}} is a diffeomorphism onto its image,^[131] {{math|u(G) ≅ G}} and {{math|u(G)}} is a Lie group. This would mean that {{math|u(G)}} is an embedded non-compact Lie subgroup of the compact group {{math|U(V)}}. This is impossible with the subspace topology on {{math|u(G) ⊂ U(V)}} since all embedded Lie subgroups of a Lie group are closed^[132] If {{math|u(G)}} were closed, it would be compact,^[133] and then {{mvar|G}} would be compact,^[134] contrary to assumption.^[135]

In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of {{math|A}} and {{math|B}} used in the construction are Hermitian. This means that {{math|J}} is Hermitian, but {{math|K}} is anti-Hermitian.^[136] 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.^[137]

Restriction to SO(3)

The {{math|(m, n)}} representation is, however, unitary when restricted to the rotation subgroup {{math|SO(3)}}, but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an {{math|(m, n)}} representation have {{math|SO(3)}}-invariant subspaces of highest weight (spin) {{math|m + n, m + n − 1, … , {{abs| m − n}}}},^[138] where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) {{math|j}} is {{math|(2j + 1)}}-dimensional. So for example, the ({{sfrac|1|2}}, {{sfrac|1|2}}) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.

Since the angular momentum operator is given by {{math|J {{=}} A + B}}, the highest spin in quantum mechanics of the rotation sub-representation will be {{math|(m + n)ℏ}} and the "usual" rules of addition of angular momenta and the formalism of 3-j symbols, 6-j symbols, etc. applies.^[139]

Spinors

It is the {{math|SO(3)}}-invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the {{math|(m, n)}} representation has spin if {{math|m + n}} is half-integral. The simplest are {{math|( {{sfrac|1|2}}, 0)}} and {{math|(0, {{sfrac|1|2}})}}, the Weyl-spinors of dimension {{math|2}}. Then, for example, {{math|(0, {{sfrac|3|2}})}} and {{math|(1, {{sfrac|1|2}})}} are a spin representations of dimensions {{math|2{{sfrac|3|2}} + 1 {{=}} 4}} and {{math|(2 + 1)(2{{sfrac|1|2}} + 1) {{=}} 6}} respectively. Note that, according to the above paragraph, there are subspaces with spin both {{math|{{sfrac|3|2}}}} and {{math|{{sfrac|1|2}}}} in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under {{math|SO(3)}}. It cannot be ruled out in general, however, that representations with multiple {{math|SO(3)}} subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out unphysical components, leaving only a single spin.^[140]

Construction of pure spin {{math|{{sfrac|n|2}}}} representations for any {{math|n}} (under {{math|SO(3)}}) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints.^[141]

Dual representations

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

Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If {{math|−I}} is an element of the Weyl group of a semisimple Lie algebra, then {{math|w₀ {{=}} −I}}. In the case of

the Weyl group is {{math|W {{=}} {I, −I}}}.^[145] It follows that each {{math|π_μ, μ {{=}} 0, 1, …}} is isomorphic to its dual

The root system of

is shown in the figure to the right.^[146] The Weyl group is generated by

where

is reflection in the plane orthogonal to {{math|γ}} as {{math|γ}} ranges over all roots.^[147] Inspection shows that {{math|w_α ⋅ w_β {{=}} −I}} so {{math|−I ∈ W}}. Using the fact that if {{math|π, σ}} are Lie algebra representations and {{math|π ≅ σ}}, then {{math|Π ≅ Σ}},^[148] the conclusion for {{math|SO(3; 1)⁺}} is

Complex conjugate representations

If {{math|π}} is a representation of a Lie algebra, then

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.^[149] In general, every irreducible representation {{math|π}} of

can be written uniquely as {{math|π {{=}} π⁺ + π⁻}}, where^[150]

with

holomorphic (complex linear) and

anti-holomorphic (conjugate linear). For

since

is holomorphic,

is anti-holomorphic. Direct examination of the explicit expressions for

and

in equation {{EquationNote|S8|(S8)}} below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression {{EquationNote|S8|(S8)}} also allows for identification of

and

for

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

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

The adjoint representation, the Clifford algebra, and the Dirac spinor representation

In general representation theory, if {{math|(π, V)}} is a representation of a Lie algebra

then there is an associated representation of

on {{math|End(V)}}, also denoted {{mvar|π}}, given by

Likewise, a representation {{math|(Π, V)}} of a group {{mvar|G}} yields a representation {{math|Π}} on {{math|End(V)}} of {{mvar|G}}, still denoted {{math|Π}}, given by^[151]

If {{mvar|π}} and {{math|Π}} are the standard representations on

and if the action is restricted to

then the two above representations are the adjoint representation of the Lie algebra and the adjoint representation of the group respectively. The corresponding representations (some

) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular.

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

In quantum mechanics this means that if {{math|(π, H)}} or {{math|(Π, H)}} is a representation acting on some Hilbert space {{mvar|H}}, then the corresponding induced representation acts on the set of linear operators on {{mvar|H}}. As an example, the induced representation of the projective spin {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}} representation on {{math|End(H)}} is the non-projective 4-vector ({{sfrac|1|2}}, {{sfrac|1|2}}) representation.^[152]

For simplicity, consider only the "discrete part" of {{math|End(H)}}, that is, given a basis for {{mvar|H}}, the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4-vector representation of above on this simplified {{math|End(H)}} has an invariant 4-dimensional subspace that is spanned by the four gamma matrices.^[153] (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime,

whose complexification is

generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the {{math|(0, 0)}}, a pseudoscalar irrep, also the {{math|(0, 0)}}, but with parity inversion eigenvalue {{math|−1}}, see the next section below, the already mentioned vector irrep, {{math|({{sfrac|1|2}}, {{sfrac|1|2}})}}, a pseudovector irrep, {{math|({{sfrac|1|2}}, {{sfrac|1|2}})}} with parity inversion eigenvalue +1 (not −1), and a tensor irrep, {{math|(1, 0) ⊕ (0, 1)}}.^[154] The dimensions add up to {{math|1=1 + 1 + 4 + 4 + 6 = 16}}. In other words,

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

The {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}} spin representation

The six-dimensional representation space of the tensor {{math|(1, 0) ⊕ (0, 1)}}-representation inside

has two roles. The^[155]

where

are the gamma matrices, the sigmas, only {{math|6}} of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,^[156]

and hence constitute a representation (in addition to spanning a representation space) sitting inside

the {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}} spin representation. For details, see bispinor and Dirac algebra.

The conclusion is that every element of the complexified

in {{math|End(H)}} (i.e. every complex {{gaps|4|×|4}} matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on

making it a space of bispinors.

Reducible representations

There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained by taking direct sums, tensor products, and quotients of the irreducible representations. Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e.g.

and the Poincaré group. These representations are in general not irreducible.

The Lorentz group and its Lie algebra have the complete reducibility property. This means that every representation reduces to a direct sum of irreducible representations. The reducible representations will therefore not be discussed.

Space inversion and time reversal

The (possibly projective) {{math|(m, n)}} representation is irreducible as a representation {{math|SO(3; 1)⁺}}, the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If {{math|1=m = n}} it can be extended to a representation of all of {{math|O(3; 1)}}, the full Lorentz group, including space parity inversion and time reversal. The representations {{math|(m, n) ⊕ (n, m)}} can be extended likewise.^[157]

Space parity inversion

For space parity inversion, the adjoint action {{math|Ad_P}} of {{math|P ∈ SO(3; 1)}} on

is considered, where {{math|P}} is the standard representative of space parity inversion, {{math|1=P = diag(1, −1, −1, −1)}}, given by

It is these properties of {{math|K}} and {{math|J}} under {{mvar|P}} that motivate the terms vector for {{math|K}} and pseudovector or axial vector for {{math|J}}. In a similar way, if {{math|π}} is any representation of

and {{math|Π}} is its associated group representation, then {{math|Π(SO(3; 1)⁺)}} acts on the representation of {{math|π}} by the adjoint action, {{math|π(X) ↦ Π(g) π(X) Π(g)⁻¹}} for

{{math|g ∈ SO(3; 1)⁺}}. If {{math|P}} is to be included in {{math|Π}}, then consistency with {{EquationNote|F1|(F1)}} requires that

holds, where {{math|A}} and {{math|B}} are defined as in the first section. This can hold only if {{math|A_i}} and {{math|B_i}} have the same dimensions, i.e. only if {{math|1=m = n}}. When {{math|m ≠ n}} then {{math|(m, n) ⊕ (n, m)}} can be extended to an irreducible representation of {{math|SO(3; 1)⁺}}, the orthochronous Lorentz group. The parity reversal representative {{math|Π(P)}} does not come automatically with the general construction of the {{math|(m, n)}} representations. It must be specified separately. The matrix {{math|1=β = i γ⁰}} (or a multiple of modulus −1 times it) may be used in the {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}}^[158] representation.

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

Time reversal

By explicitly including a representative for {{math|T}}, as well as one for {{math|P}}, a representation of the full Lorentz group {{math|O(3; 1)}} is obtained. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the {{mvar|P^μ}}, in addition to the {{mvar|Jⁱ}} and {{mvar|Kⁱ}} generate the group. These are interpreted as generators of translations. The time-component {{math|P⁰}} is the Hamiltonian {{math|H}}. The operator {{math|T}} satisfies the relation^[160]

in analogy to the relations above with

replaced by the full Poincaré algebra. By just cancelling the {{mvar|i}}'s, the result {{math|1=THT⁻¹ = −H}} would imply that for every state {{math|Ψ}} with positive energy {{mvar|E}} in a Hilbert space of quantum states with time-reversal invariance, there would be a state {{math|Π(T⁻¹)Ψ}} with negative energy {{math|−E}}. Such states do not exist. The operator {{math|Π(T)}} is therefore chosen antilinear and antiunitary, so that it anticommutes with {{mvar|i}}, resulting in {{math|THT⁻¹ {{=}} H}}, and its action on Hilbert space likewise becomes antilinear and antiunitary.^[161] It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix.^[162] This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, {{math|Π}} is not a representation.

When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, {{nowrap|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}}, is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.

The third discrete symmetry entering in the CPT theorem along with {{math|P}} and {{math|T}}, charge conjugation symmetry {{math|C}}, has nothing directly to do with Lorentz invariance.^[163]

Action on function spaces

If {{mvar|V}} is a vector space of functions of a finite number of variables {{mvar|n}}, then the action on a scalar function

given by

produces another function {{math|Πf ∈ V}}. Here {{math|Π_x}} is an {{mvar|n}}-dimensional representation, and {{math|Π}} is a possibly infinite-dimensional representation. A special case of this construction is when {{mvar|V}} is a space of functions defined on the a linear group {{mvar|G}} itself, viewed as a {{mvar|n}}-dimensional manifold embedded in

(with {{mvar|m}} the dimension of the matrices).^[164] 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.^[81] The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g.

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

Euclidean rotations

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

where

are the spherical harmonics. An arbitrary square integrable function {{mvar|f}} one the unit sphere can be expressed as^[165]

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

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

The Möbius group

The identity component of the Lorentz group is isomorphic to the Möbius group {{math|M}}. This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection, the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere.

In the plane, a Möbius transformation characterized by the complex numbers {{math|a, b, c, d}} acts on the plane according to^[166]

since multiplication by a nonzero complex scalar does not change {{mvar|f}}. These are elements of

and are unique up to a sign (since {{math|±Π_f}} give the same {{mvar|f}}), hence

The Riemann P-functions

The Riemann P-functions, solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as^[167]

where the {{math|a, b, c, α, β, γ, α′, β′, γ′}} are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is^[168]

The set of constants {{math|0, ∞, 1}} in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation.^[169] Its exponents, i. e. solutions of the indicial equation, for expansion around the singular point {{math|0}} are {{math|0}} and {{math|1 − c}} ,corresponding to the two linearly independent solutions,^[170] and for expansion around the singular point {{math|1}} they are {{math|0}} and {{math|c − a − b}}.^[171] Similarly, the exponents for {{math|∞}} are {{mvar|a}} and {{mvar|b}} for the two solutions.^[172]

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

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

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

where {{mvar|P}} is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Möbius transformation of the argument is that of shifting the poles to new locations, hence changing the critical points, but there is no change in the exponents of the differential equation the new function satisfies. The new function is expressed as

History

The Lorentz group {{math|SO(3; 1)⁺}} and its double cover

also have infinite dimensional unitary representations, studied independently by {{harvtxt|Bargmann|1947}}, {{harvtxt|Gelfand|Naimark|1947}} and {{harvtxt|Harish-Chandra|1947}} at the instigation of Paul Dirac.^[174]^[175] This trail of development begun with {{harvtxt|Dirac|1936}} where he devised matrices {{math|U}} and {{math|B}} necessary for description of higher spin (compare Dirac matrices), elaborated upon by {{harvtxt|Fierz|1939}}, see also {{harvtxt|Fierz|Pauli|1939}}, and proposed precursors of the Bargmann-Wigner equations.^[176] In {{harvtxt|Dirac|1945}} he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors.^[177] These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinite-dimensional generalization of spinors in his 1947 paper.

The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by {{harvtxt|Harish-Chandra|1951}} and {{harvtxt|Gelfand|Graev|1953}}, based on an analogue for

of the integration formula of Hermann Weyl for compact Lie groups.^[178] Elementary accounts of this approach can be found in {{harvtxt|Rühl|1970}} and {{harvtxt|Knapp|2001}}.

The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3-dimensional hyperbolic space sitting in Minkowski space is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on

This theory is discussed in {{harvtxt|Takahashi|1963}}, {{harvtxt|Helgason|1968}}, {{harvtxt|Helgason|2000}} and the posthumous text of {{harvtxt|Jorgenson|Lang|2008}}.

Principal series for SL(2, C)

The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup {{mvar|B}} of

Since the one-dimensional representations of {{mvar|B}} correspond to the representations of the diagonal matrices, with non-zero complex entries {{mvar|z}} and {{math|z⁻¹}}, they thus have the form

for {{mvar|k}} an integer, {{mvar|ν}} real and with {{mvar|1=z = re^iθ}}. The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when {{mvar|k}} is replaced by {{math|−k}}. By definition the representations are realized on {{math|L²}} sections of line bundles on

which is isomorphic to the Riemann sphere. When {{math|1=k = 0}}, these representations constitute the so-called spherical principal series.

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

Using the identification between the Riemann sphere minus a point and

the principal series can be defined directly on

by the formula^[179]

Complementary series for SL(2, C)

The for {{math|0 < t < 2}}, the complementary series is defined on

for the inner product^[183]

The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of {{mvar|K}}, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of {{math|1=K = SU(2)}}. Irreducibility can be proved by analyzing the action of

on the algebraic sum of these subspaces^[181]^[182] or directly without using the Lie algebra.^[186]^[187]

Plancherel theorem for SL(2, C)

The only irreducible unitary representations of

are the principal series, the complementary series and the trivial representation.

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

To decompose the left regular representation of {{mvar|G}} on

only the principal series are required. This immediately yields the decomposition on the subrepresentations

the left regular representation of the Lorentz group, and

the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with {{math|1=k = 0}}.)

The left and right regular representation {{mvar|λ}} and {{mvar|ρ}} are defined on

and

denotes the Hilbert space of Hilbert–Schmidt operators on

^[188] Then the map {{mvar|U}} defined on {{math|C_c(G)}} by

The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively.

The Plancherel formula extends to all

By a theorem of Jacques Dixmier and Paul Malliavin, every smooth compactly supported function on

is a finite sum of convolutions of similar functions, the inversion formula holds for such {{mvar|f}}. It can be extended to much wider classes of functions satisfying mild differentiability conditions.^[81]

Classification of representations of SO(3, 1)

The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to assume they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinite-dimensional representation {{math|Π_H}} on a Hilbert space {{mvar|H}} of {{math|SO(3; 1)⁺}} is at hand.^[190] Since {{math|SO(3)}} is a subgroup, {{math|Π_H}} is a representation of it as well. Each irreducible subrepresentation of {{math|SO(3)}} is finite-dimensional, and the {{math|SO(3)}} representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of {{math|SO(3)}} if {{math|Π_H}} is unitary.^[191]

Step 1

If this were a finite-dimensional representation, then {{math|j₀}} would correspond the lowest occurring eigenvalue {{math|j(j + 1)}} of {{math|J²}} in the representation, equal to {{math|{{!}}m − n{{!}}}}, and {{math|j₁}} would correspond to the highest occurring eigenvalue, equal to {{math|m + n}}. In the infinite-dimensional case, {{math|j₀ ≥ 0}} retains this meaning, but {{math|j₁}} does not.^[90] For simplicity, it is assumed that a given {{mvar|j}} occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown^[194] 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 {{math|J₁, J₂, J₃}} and {{math|K₁, K₂, K₃}} forming the basis of the Lie algebra of

The matrix elements of

and

(the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by^[195]^[196]

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

the triple {{math|(K_i, K_i, K_i) ≡ K}} is a vector operator^[197] and the Wigner–Eckart theorem^[198] applies for computation of matrix elements between the states represented by the chosen basis.^[199] The matrix elements of

where the superscript {{math|(1)}} signifies that the defined quantities are the components of a spherical tensor operator of rank {{math|k {{=}} 1}} (which explains the factor {{math|{{sqrt|2}}}} as well) and the subscripts {{math|0, ±1}} are referred to as {{mvar|q}} in formulas below, are given by^[200]

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

Step 3

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 {{math|j₀}} and {{math|ξ_j}}. Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning

where {{math|β_j}} is the angle of {{math|B_j}} on polar form. For {{math|{{!}}B_j{{!}} ≠ 0}} follows

and

is chosen by convention. There are two possible cases:

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

Explicit formulas

Conventions and Lie algebra bases

The metric of choice is given by {{math|η {{=}} diag(−1, 1, 1, 1)}}, and the physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of basis for the Lie algebra is, in the 4-vector representation, given by:

The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol {{mvar|J}} above and in the sequel should be observed.

Weyl spinors and bispinors

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

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

These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) {{math|V_L}} and {{math|V_R}}, whose elements {{math|Ψ_L}} and {{math|Ψ_R}} are called left- and right-handed Weyl spinors respectively. Given

This is, up to a similarity transformation, the {{math|({{sfrac|1|2}},0) ⊕ (0,{{sfrac|1|2}})}} Dirac spinor representation of

It acts on the 4-component elements {{math|(Ψ_L, Ψ_R)}} of {{math|(V_L ⊕ V_R)}}, called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of

Expressions for the group representations are obtained by exponentiation.

See also

Remarks

1. ^The way in which it enters may take many shapes depending on the theory at hand. While not being the present topic, some details will be provided in footnotes labeled "nb", and in the section applications.
2. ^{{harvnb|Weinberg|2002|p=1}} "If it turned out that a system could not be described by a quantum field theory, it would be a sensation; if it turned out it did not obey the rules of quantum mechanics and relativity, it would be a cataclysm."
3. ^{{harvnb|Bargmann|Wigner|1948}}
4. ^{{harvnb|Bekaert|Boulanger|2006}}
5. ^In 1945 Harish-Chandra came to see Dirac in Cambridge. He became convinced that he was not suitable for theoretical physics. Harish-Chandra had found an error in a proof by Dirac in his work on the Lorentz group. Dirac said "I am not interested in proofs but only interested in what nature does."

Harish-Chandra later wrote "This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move over to mathematics."

Dirac did however suggest the topic of his thesis, the classification of the irreducible infinite-dimensional representations of the Lorentz group.

See {{harvnb|Dalitz|Peierls|1986}}
6. ^It is tacitly always assumed that each inertial frame has a dedicated Lorentz observer, I. e. someone who has, in principle, a complete record (i. e. coordinates!) of every event as observed in that frame.
7. ^It isn't actually required that the mapping is one-to-one. It is merely required that the mapping is a group homomorphism, i. e. {{math|Π(gh) {{=}} Π(g)Π(h)}} into some {{math|GL(V)}} the general linear group of some vector space {{mvar|V}}. (The vector space {{mvar|V}} is allowed to be infinite-dimensional, e.g a Hilbert space {{math|H}}, in which case one speaks of {{math|B(H)}}, linear operators on {{mvar|H}} instead of {{math|GL(V)}}.
8. ^In an open set of spacetime, there are only two degrees of freedom. Introduction of the four-potential yields nominally four degrees of freedom. The field equations and gauge invariance each remove one degree of freedom.
9. ^This transformation is usually expressed differently, see e.g. transformation of the electromagnetic field. That method can be traced converted to applying a {{math|6 × 6}}-matrix like here and vice versa since the field tensor has {{math|6}} independent components.
10. ^¹{{harvnb|Rossmann|2002|loc=Section 6.1.}}
11. ^There may be representations of uncountable dimension. These are less well behaved (in particular non-unitary) and are not covered here.
12. ^It may happen that the multiplication of two representative infinite-dimensional matrices is ill-defined. The right composition rule can be verified by other means though.
13. ^{{harvnb|Misner|Thorne|Wheeler|1973}}
14. ^{{harvnb|Weinberg|2002|loc=Section 2.5, Chapter 5.}}
15. ^{{harvnb|Tung|1985|loc=Sections 10.3, 10.5.}}
16. ^{{harvnb|Tung|1985|loc=Section 10.4.}}
17. ^{{harvnb|Dirac|1945}}
18. ^{{harvnb|Zwiebach|2004|loc=Section 12.8.}}
19. ^¹{{harvnb|Bekaert|Boulanger|2006|p=48.}}
20. ^{{harvnb|Zwiebach|2004|loc=Section 18.8.}}
21. ^¹{{harvnb|Greiner|Reinhardt|1996|loc=Chapter 2.}}
22. ^{{harvnb|Weinberg|2002|loc=Foreword and introduction to chapter 7.}}
23. ^{{harvnb|Weinberg|2002|loc=Introduction to chapter 7.}}
24. ^{{harvnb|Tung|1985|loc=Definition 10.11.}}
25. ^{{harvtxt|Greiner|Müller|1994|loc=Chapter 1}}
26. ^{{harvtxt|Greiner|Müller|1994|loc=Chapter 2}}
27. ^{{harvnb|Tung|1985|p=203.}}
28. ^{{harvnb|Delbourgo|Salam|Strathdee|1967}}
29. ^{{harvtxt|Weinberg|2002|loc=Section 3.3}}
30. ^See formula (1) in S-matrix#From free particle states for how free multi-particle states transform.
31. ^{{harvtxt|Weinberg|2002|loc=Section 7.4.}}
32. ^{{harvnb|Tung |1985| loc=Introduction to chapter 10.}}
33. ^{{harvnb|Tung|1985|loc=Definition 10.12.}}
34. ^{{harvnb|Tung|1985|loc=Equation 10.5-2.}}
35. ^{{harvnb|Weinberg|2002|loc=Equations 5.1.6–7.}}
36. ^¹{{harvnb|Tung|1985|loc=Equation 10.5–18.}}
37. ^{{harvnb|Weinberg|2002|loc=Equations 5.1.4–5.}} Weinberg deduces the necessity of creation and annihilation operators from another consideration, the cluster decomposition principle, {{harvtxt|Weinberg|2002|loc=Chapter 4.}}
38. ^{{harvnb|Weinberg|2002|loc=Equations 5.1.11–12.}}
39. ^{{harvnb|Tung|1985|loc=Section 10.5.3.}}
40. ^A prescription for how the particle should behave under CPT symmetry may be required as well.
41. ^For instance, there are versions (free field equations, i.e. without interaction terms) of the Klein–Gordon equation, the Dirac equation, the Maxwell equations, the Proca equation, the Rarita–Schwinger equation, and the Einstein field equations that can systematically be deduced by starting from a given representation of the Lorentz group. In general, these are collectively the quantum field theory versions of the Bargmann–Wigner equations.

See {{harvtxt|Weinberg|2002|loc=Chapter 5}}, {{harvtxt|Tung|1985|loc=Section 10.5.2}} and references given in these works.

It should be remarked that high spin theories ({{math|s > 1}}) encounter difficulties. See {{harvtxt|Weinberg|2002|loc=Section 5.8}}, on general {{math|(m, n)}} fields, where this is discussed in some depth, and references therein. High spin particles do without a doubt exist, e.g. nuclei, the known ones are just not elementary.
42. ^For part of their representation theory, see {{harvtxt|Bekaert|Boulanger|2006}}, which is dedicated to representation theory of the Poincare group. These representations are obtained by the method of induced representations or, in physics parlance, the method of the little group, pioneered by Wigner in 1939 for this type of group and put on firm mathematical footing by George Mackey in the fifties.
43. ^{{harvnb|Zwiebach|2004|loc=Section 6.4.}}
44. ^{{harvnb|Zwiebach|2004|loc=Chapter 7.}}
45. ^{{harvnb|Zwiebach|2004|loc=Section 12.5.}}
46. ^¹{{harvnb|Weinberg|2000|loc=Section 25.2.}}
47. ^{{harvnb|Zwiebach|2004|loc=Last paragraph, section 12.6.}}
48. ^These facts can be found in most introductory mathematics and physics texts. See e.g. {{harvtxt|Rossmann|2002}}, {{harvtxt|Hall|2015}} and {{harvtxt|Tung|1985}}.
49. ^{{harvtxt|Hall|2015|loc=Section 4.4.}}{{Paragraph break}}One says that a group has the complete reducibility property if every representation decomposes as a direct sum of irreducible representations.
50. ^{{harvtxt|Hall|2015|loc=Theorem 4.34 and following discussion.}}
51. ^¹²{{harvnb|Wigner|1939}}
52. ^{{harvnb|Hall|2015|loc=Appendix D2.}}
53. ^{{harvnb|Greiner|Reinhardt|1996}}
54. ^{{harvnb|Weinberg|2002|loc=Section 2.6 and Chapter 5.}}
55. ^¹{{harvnb|Coleman|1989|p=30.}}
56. ^{{harvnb|Lie|1888}}, 1890, 1893. Primary source.
57. ^{{harvnb|Coleman|1989|p=34.}}
58. ^{{harvnb|Killing|1888}} Primary source.
59. ^¹{{harvnb|Rossmann|2002|loc=Historical tidbits scattered across the text.}}
60. ^{{harvnb|Cartan|1913}} Primary source.
61. ^{{harvnb|Green|1998|loc=p=76.}}
62. ^{{harvnb|Brauer|Weyl|1935}} Primary source.
63. ^{{harvnb|Tung|1985|loc=Introduction.}}
64. ^{{harvnb|Weyl|1931}} Primary source.
65. ^{{harvnb|Weyl|1939}} Primary source.
66. ^{{harvnb|Langlands|1985|pp=203–205}}
67. ^{{harvnb|Harish-Chandra|1947}} Primary source.
68. ^{{harvnb|Tung|1985|loc=Introduction}}
69. ^{{harvnb|Wigner|1939}} Primary source.
70. ^{{harvnb|Klauder|1999}}
71. ^{{harvnb|Bargmann|1947}} Primary source.
72. ^Bargmann was also a mathematician. He worked as Albert Einsteins assistant at the Institute for Advanced Study in Princeton ({{harvtxt|Klauder|1999}}).
73. ^{{harvnb|Bargmann|Wigner|1948}} Primary source.
74. ^{{harvnb|Dalitz|Peierls|1986}}
75. ^{{harvnb|Dirac|1928}} Primary source.
76. ^Dirac suggested the topic of {{harvtxt|Wigner|1939}} as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in {{harvtxt|Dirac|1945}} ({{harvnb|Langlands|1985}}), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations ({{harvnb|Dalitz|Peierls|1986}}).
77. ^{{harvnb|Weinberg|2002|loc=Equations 5.6.7–8.}}
78. ^{{harvnb|Weinberg|2002|loc=Equations 5.6.9–11.}}
79. ^¹²{{harvnb|Hall|2003|loc=Chapter 6.}}
80. ^{{harvnb|Knapp|2001}} The rather mysterious looking third isomorphism is proved in chapter 2, paragraph 4.
81. ^¹²{{harvnb|Knapp|2001}}
82. ^This is an application of {{harvnb|Rossmann|2002|loc=Section 6.3, Proposition 10.}}
83. ^Tensor products of representations, {{math|π_g ⊗ π_h}} of can, when both factors come from the same Lie algebra either be thought of as a representation of or .
84. ^When complexifying a complex Lie algebra, it should be thought of as a real Lie algebra of real dimension twice its complex dimension. Likewise, a real form may actually also be complex as is the case here.
85. ^¹{{harvnb|Knapp|2001|p=32.}}
86. ^Combine {{harvtxt|Weinberg|2002|loc=Equations 5.6.7–8, 5.6.14–15}} with {{harvtxt|Hall|2015|loc=Proposition 4.18}} about Lie algebra representations of group tensor product representations.
87. ^{{harvnb|Weinberg|2002|loc=Equations 5.6.16–17.}}
88. ^{{harvnb|Weinberg|2002|loc=Section 5.6.}} The equations follow from equations 5.6.7–8 and 5.6.14–15.
89. ^The "traceless" property can be expressed as {{math|1=S_αβg^αβ = 0}}, or {{math|1=S_α^α = 0}}, or {{math|1=S^αβg_αβ = 0}} depending on the presentation of the field: covariant, mixed, and contravariant respectively.
90. ^¹{{harvnb|Tung|1985}}
91. ^This doesn't necessarily come symmetric directly from the Lagrangian by using Noether's theorem, but it can be symmetrized as the Belinfante–Rosenfeld stress–energy tensor.
92. ^This is provided parity is a symmetry. Else there would be two flavors, {{math|({{sfrac|3|2}}, 0)}} and {{math|(0, {{sfrac|3|2}})}} in analogy with neutrinos.
93. ^{{Harvnb|Weinberg|2002}} See footnote on p. 232.
94. ^{{harvnb|Lie|1888}}
95. ^{{harvnb|Rossmann|2002|loc=Section 2.5.}}
96. ^{{harvnb|Hall|2015|loc=Theorem 2.10.}}
97. ^{{harvnb|Bourbaki|1998|p=424.}}
98. ^{{harvnb|Weinberg|2002|loc=Section 2.7 p.88.}}
99. ^¹²³⁴{{harvnb|Weinberg|2002|loc=Section 2.7.}}
100. ^{{harvnb|Hall|2015|loc=Appendix C.3.}}
101. ^The terminology differs between mathematics and physics. In the linked article term projective representation has a slightly different meaning than in physics, where a projective representation is though of as a local section (a local inverse) of the covering map from the covering group onto the group being covered, composed with a proper representation of the covering group. Since this can be done (locally) continuously in two ways in the case at hand as explained below, the terminology of a double-valued or two-valued representation is natural.
102. ^{{harvnb|Wigner|1939|p=27.}}
103. ^In particular, {{math|A}} commutes with the Pauli matrices, hence with all of {{math|SU(2)}} making Schur's lemma applicable.
104. ^{{harvnb|Gelfand|Minlos|Shapiro|1963}} This construction of the covering group is treated in paragraph 4, section 1, chapter 1 in Part II.
105. ^Meaning the kernel is trivial, to see this recall that the kernel of a Lie algebra homomorphism is an ideal and hence a subspace. Since {{math|p}} is {{math|2:1}} and both and {{math|SO(3; 1)⁺}} are {{nowrap|{{math|6}}-dimensional}}, the kernel must be {{nowrap|{{math|0}}-dimensional}}, hence {{math|{0}.}}
106. ^The exponential map is one-to-one in a neighborhood of the identity in hence the composition where {{mvar|σ}} is the Lie algebra isomorphism, is onto an open neighborhood {{math|U ⊂ SO(3; 1)⁺}} containing the identity. Such a neighborhood generates the connected component.
107. ^{{harvnb|Rossmann|2002|loc=Section 2.1.}}
108. ^{{harvnb|Rossmann|2002}} From Example 4 in section 2.1 : This can be seen as follows. The matrix {{mvar|q}} has eigenvalues {{math|{-1, −1} }}, but it is not diagonalizable. If {{math|q {{=}} exp(Q)}}, then {{mvar|Q}} has eigenvalues {{math|λ, −λ}} with {{math|λ {{=}} iπ + 2πik}} for some {{mvar|k}} because elements of are traceless. But then {{mvar|Q}} is diagonalizable, hence {{mvar|q}} is diagonalizable, which is a contradiction.
109. ^{{harvnb|Hall|2015|loc=First displayed equations in section 4.6.}}
110. ^{{harvnb|Hall|2015|loc=Example 4.10.}}
111. ^¹{{harvnb|Knapp|2001|loc=Chapter 2.}}
112. ^{{harvnb|Knapp|2001}} Equation 2.1.
113. ^{{harvnb|Hall|2015|loc=Equation 4.2.}}
114. ^{{harvnb|Hall|2015|loc=Equation before 4.5.}}
115. ^{{harvnb|Knapp|2001}} Equation 2.4.
116. ^{{harvnb|Knapp|2001}}
117. ^{{harvnb|Knapp|2001|loc=Section 2.3.}}
118. ^{{harvnb|Rossmann|2002|loc=Proposition 10, paragraph 6.3.}} This is easiest proved using character theory.
119. ^{{harvnb|Hall|2015|loc=Theorems 9.4–5.}}
120. ^{{harvnb|Weinberg|2002|loc=Chapter 5.}}
121. ^{{harvnb|Hall|2015|loc= Theorem 10.18.}}
122. ^{{harvnb|Hall|2003|p=235.}}
123. ^See any text on basic group theory.
124. ^Any discrete normal subgroup of a path connected group {{math|G}} is contained in the center {{math|Z}} of {{math|G}}.{{Paragraph break}}{{harvnb|Hall|2015|loc=Exercise 11, chapter 1.}}
125. ^A semisimple Lie group does not have any non-discrete normal abelian subgroups. This can be taken as the definition of semisimplicity.
126. ^A simple group does not have any non-discrete normal subgroups.
127. ^{{harvnb|Rossmann|2002}} Propositions 3 and 6 paragraph 2.5.
128. ^{{harvnb|Hall|2003}} See exercise 1, Chapter 6.
129. ^By contrast, there is a trick, also called Weyl's unitarian trick, but unrelated to the unitarian trick of above showing that all finite-dimensional representations are, or can be made, unitary. If {{math|(Π, V)}} is a finite-dimensional representation of a compact Lie group {{mvar|G}} and if {{math|(·, ·)}} is any inner product on {{mvar|V}}, define a new inner product {{math|(·, ·)_Π}} by {{math|(x, y)_Π {{=}} ∫_G(Π(g)x, Π(g)y dμ(g)}}, where {{mvar|μ}} is Haar measure on {{mvar|G}}. Then {{math|Π}} is unitary with respect to {{math|(·, ·)_Π}}. See {{harvtxt|Hall|2015|loc=Theorem 4.28.}}{{Paragraph break}}Another consequence is that every compact Lie group has the complete reducibility property, meaning that all its finite-dimensional representations decompose as a direct sum of irreducible representations. {{harvtxt|Hall|2015|loc=Definition 4.24., Theorem 4.28.}}{{Paragraph break}}It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in {{harvtxt|Greiner|Müller|1994|loc=Section 15.2.}}.
130. ^{{harvnb|Bekaert|Boulanger|2006}} p.4.
131. ^{{harvnb|Hall|2003}} Proposition 1.20.
132. ^{{harvnb|Lee|2003|loc=Theorem 8.30.}}
133. ^{{harvnb|Lee|2003}} Lemma A.17 (c). Closed subsets of compact sets are compact.
134. ^{{harvnb|Lee|2003}} Lemma A.17 (a). If {{math|f : X → Y}} is continuous, {{mvar|X}} is compact, then {{math|f(X)}} is compact.
135. ^The non-unitarity is a vital ingredient in the proof of the Coleman–Mandula theorem, which has the implication that, contrary to in non-relativistic theories, there can exist no ordinary symmetry relating particles of different spin. See {{harvtxt|Weinberg|2000|loch=Chapter 24.}}
136. ^{{harvnb|Weinberg|2002|loc=Section 5.6, p. 231.}}
137. ^{{harvnb|Weinberg|2002|loc=Section 5.6.}}
138. ^{{harvnb|Weinberg|2002|p=231.}}
139. ^{{harvnb|Weinberg|2002|loc=Sections 2.5, 5.7.}}
140. ^{{harvnb|Tung|1985|loc=Section 10.5.}}
141. ^{{harvnb|Weinberg|2002}} This is outlined (very briefly) on page 232, hardly more than a footnote.
142. ^{{harvnb|Hall|2003|loc=Proposition 7.39.}}
143. ^This is one of the conclusions of Cartan's theorem, the theorem of the highest weight.{{Paragraph break}}{{harvtxt|Hall|2015|loc=Theorems 9.4–5.}}
144. ^¹{{harvnb|Hall|2003|loc=Theorem 7.40.}}
145. ^{{harvnb|Hall|2003|loc=Section 6.6.}}
146. ^{{harvnb|Hall|2015|loc=Section 8.2}} The root system is the union of two copies of {{math|A₁}}, where each copy resides in its own dimensions in the embedding vector space.
147. ^{{harvnb|Rossmann|2002}} This definition is equivalent to the definition in terms of the connected Lie group whose Lie algebra is the Lie algebra of the root system under consideration.
148. ^{{harvnb|Hall|2003|loc=Second item in proposition 4.5.}}
149. ^{{harvnb|Hall|2003|p=219.}}
150. ^{{harvnb|Rossmann|2002|loc=Exercise 3 in paragraph 6.5.}}
151. ^{{harvnb|Hall|2003}} See appendix D.3
152. ^{{harvnb|Weinberg|2002|loc=Equation 5.4.8.}}
153. ^¹{{harvnb|Weinberg|2002|loc=Section 5.4.}}
154. ^{{harvnb|Weinberg|2002|pp=215–216.}}
155. ^{{harvnb|Weinberg|2002|loc=Equation 5.4.6.}}
156. ^{{harvnb|Weinberg|2002}} Section 5.4.
157. ^{{harvnb|Weinberg|2002|loc=Section 5.7, pp. 232–233.}}
158. ^{{harvnb|Weinberg|2002|loc=Section 5.7, p. 233.}}
159. ^{{harvnb|Weinberg|2002}} Equation 2.6.5.
160. ^{{harvnb|Weinberg|2002}} Equation following 2.6.6.
161. ^{{harvnb|Weinberg|2002|loc=Section 2.6.}}
162. ^For a detailed discussion of the spin 0, {{sfrac|1|2}} and 1 cases, see {{harvnb|Greiner|Reinhardt|1996}}.
163. ^{{harvnb|Weinberg|2002|loc=Chapter 3.}}
164. ^{{harvnb|Rossmann|2002}} See section 6.1 for more examples, both finite-dimensional and infinite-dimensional.
165. ^{{harvnb|Gelfand|Minlos|Shapiro|1963}}
166. ^{{harvnb|Churchill|Brown|2014|loc=Chapter 8 pp. 307–310.}}
167. ^{{cite journal|ref=harv|last1=Gonzalez|first1=P. A.|last2=Vasquez|first2=Y.|title=Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity|year=2014|journal= Eur. Phys. J. C |issn=1434-6044|volume=74:2969|issue=7|arxiv=1404.5371|doi=10.1140/epjc/s10052-014-2969-1|page=3|bibcode=2014EPJC...74.2969G}}
168. ^{{harvnb|Abramowitz|Stegun|1965|loc=Equation 15.6.5.}}
169. ^{{harvnb|Simmons|1972|loc=Sections 30, 31.}}
170. ^See {{harvtxt|Simmons|1972|loc=Section 30.}} for precise conditions under which two Frobenius method yields two linearly independent solutions. If the exponents do not differ by an integer, this is always the case.
171. ^{{harvnb|Simmons|1972|loc=Sections 30.}}
172. ^{{harvnb|Simmons|1972|loc=Section 31.}}
173. ^{{harvnb|Simmons|1972|loc=Equation 11 in appendix E, chapter 5.}}
174. ^{{harvnb|Langlands|1985|p=205.}}
175. ^{{harvnb|Varadarajan|1989|loc=Sections 3.1. 4.1.}}
176. ^{{harvnb|Langlands|1985|p=203.}}
177. ^"This is as close as one comes to the source of the theory of infinite-dimensional representations of semisimple and reductive groups...", {{harvtxt|Langlands|1985|p=204.}}, referring to an introductory passage in Dirac's 1945 paper.
178. ^{{harvnb|Varadarajan|1989|loc=Section 4.1.}}
179. ^{{harvnb|Gelfand|Graev|Pyatetskii-Shapiro|1969}}
180. ^{{harvnb|Knapp|2001|loc=Chapter II.}}
181. ^¹²{{harvnb|Harish-Chandra|1947}}
182. ^¹{{harvnb|Taylor|1986}}
183. ^{{harvnb|Knapp|2001}} Chapter 2. Equation 2.12.
184. ^{{harvnb|Bargmann|1947}}
185. ^{{harvnb|Gelfand|Graev|1953}}
186. ^{{harvnb|Gelfand|Naimark|1947}}
187. ^{{harvnb|Takahashi|1963|page=343.}}
188. ^Note that for a Hilbert space {{mvar|H}}, {{math|HS(H)}} may be identified canonically with the Hilbert space tensor product of {{mvar|H}} and its conjugate space.
189. ^{{harvnb|Knapp|2001|loc=Equation 2.24.}}
190. ^{{harvnb|Folland|2015|loc=Section 3.1.}}
191. ^{{harvnb|Folland|2015|loc=Theorem 5.2.}}
192. ^{{harvnb|Tung|1985|loc=Section 10.3.3.}}
193. ^If finite-dimensionality is demanded, the results is the {{math|(m, n)}} representations, see {{harvtxt|Tung|1985|loc=Problem 10.8.}} If neither is demanded, then a broader classification of all irreducible representations is obtained, including the finite-dimensional and the unitary ones. This approach is taken in {{harvtxt|Harish-Chandra|1947}}.
194. ^{{harvnb|Harish-Chandra|1947|loc=Footnote p. 374.}}
195. ^{{harvnb|Tung|1985|loc=Equations 7.3–13, 7.3–14.}}
196. ^{{harvnb|Harish-Chandra|1947|loc=Equation 8.}}
197. ^{{harvnb|Hall|2015|loc=Proposition C.7.}}
198. ^{{harvnb|Hall|2015|loc=Appendix C.2.}}
199. ^{{harvnb|Tung|1985|loc=Step II section 10.2.}}
200. ^{{harvnb|Tung|1985|loc=Equations 10.3–5. Tung's notation for Clebsch–Gordan coefficients differ from the one used here.}}
201. ^{{harvnb|Tung|1985|loc=Equation VII-3.}}
202. ^{{harvnb|Tung|1985|loc=Equations 10.3–5, 7, 8.}}
203. ^{{harvnb|Tung|1985|loc=Equation VII-9.}}
204. ^{{harvnb|Tung|1985|loc=Equations VII-10, 11.}}
205. ^{{harvnb|Tung|1985|loc=Equations VII-12.}}
206. ^{{harvnb|Tung|1985|loc=Equations VII-13.}}
207. ^{{harvnb|Weinberg|2002|loc=Equation 2.4.12.}}
208. ^{{harvnb|Weinberg|2002|loc=Equations 2.4.18–2.4.20.}}
209. ^{{harvnb|Weinberg|2002|loc=Equations 5.4.19, 5.4.20.}}

Notes

Freely available online references

References

series=Springer Monographs in Mathematics|publisher= Springer|year=2008|isbn=978-0-387-38031-5|postscript=}}