Representation theory of the Lorentz group
The Lorentz group of special relativity has a variety of representations. Many of these representation are important in theoretical physics in the description of particles in relativistic quantum mechanics, as well as fields in quantum field theory. This representation theory provides a theoretical ground for the concept of spin (which can be either integer or half-integer in the unit of the reduced Planck constant ℏ); representations with half-integer weights are normally constructed out of spinors.
The group may also 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. An important special case is the subgroup SO(3), where these reduce to the spherical harmonics, and find practical application in the theory of atomic spectra.
According to the general representation theory of Lie groups, one first looks for the representations of the complexification, so(3;1)C of the Lie algebra so(3;1) of the Lorentz group. A convenient basis for so(3;1) is given by the three generators of rotations Ji and the three generators of boosts Ki. First complexify the Lie algebra, and then change basis to the components of A = (J + i K)/2 and B = (J − i K)/2. In this new basis, one checks that the components of A and B satisfy separately the commutation relations of the Lie algebra su(2) and moreover that they commute with each other.1 In other words, one has the isomorphism
where sl(2,C) is the complexification of su(2). The utility of this isomorphism comes from the fact that all irreducible representations of su(2) are, up to isomorphism, known. Every irreducible representation of su(2) is isomorphic to one of the highest weight representations. Moreover, there is a one-to-one correspondence between linear representations of su(2) and complex linear representations of its complexification sl(2,C).3 Via the displayed isomorphism, all irreducible representations of so(3;1)C, and thus those of so(3;1) are known. Since so(3;1) is semisimple,2 all its representations, not necessarily irreducible, 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 (m, n) which fix representations of the su(2) subalgebras spanned by the components of A and B respectively. The m and n are the highest weights, respectively, of the complexified su(2) representations, i.e. sl(2,C) representations, of AC and BC. Let π(m,n): so(3;1) → gl(V), where V is a vector space, denote the irreducible representations of so(3;1) according to the this classification. These are, up to a similarity transformation, uniquely given by
where the J(n) = (J1,J2,J3) are the (2n + 1)-dimensional irreducible spin n representations of so(3;1) and 1n is the n-dimensional unit matrix. Explicit formulas on component form are given at the end of the article.
The highest weight for a sl(2,C) representation coincides with the highest eigenvalue of the representative of the third Pauli matrix σ3 ∈ sl(2,C) and is in quantum mechanics thus related to the maximum value the spin z-component can take on for a particle whose state vector transform under the representation in question.
Given a so(3;1) representation, one may try to construct a representation of SO(3;1)+, the identity component of the Lorentz group, by using the exponential mapping. If X is an element of so(3;1), then
is a Lorentz transformation by general properties of Lie algebras. Motivated by this, let π: so(3;1) → gl(V) for some vector space V be a representation and attempt to define a representation Π of SO(3)+ by first setting
There are at least two potential problems with this definition. The first is that it is not obvious that this yields a group homomorphism, or even a map at all. But a theorem5 based on the inverse function theorem states that the map exp: so(3;1) → SO(3;1)+ is one-to-one for X small enough (A). The qualitative form of the Baker-Campbell-Hausdorff formula then guarantees that it is a group homomorphism, still for X small enough (B).5 Let U⊂SO(3;1)+ denote image under the exponential mapping of the open set in so(3;1) where conditions (A) and (B) both hold.
Another problem is that for a given g ∈ SO(3;1)+ there may not be exactly one X ∈ so(3;1) such that g = eiX. Technically, formula (1) is used to define Π near the identity. For other elements g ∉ U one chooses a path from the identity to g and defines Π along that path by partitioning it finely enough so that formula (G2) can be used again on the resulting factors in the partition. In detail, one sets
where the gi are on the path and the factors on the far right are uniquely defined by (G2) provided that all gi gi + 1−1 ∈ U. By compactness of the path there is an n large enough so that Π(g) is well defined, possibly depending on the partition and/or the path.
It turns out that the result is always independent of the partitioning of the path.5 For simply connected groups, the construction will be independent of the path as well, yielding a well defined representation.5 In that case formula (G2) can unambiguously be used directly. For a group that is connected but not simply connected, such as SO(3;1)+, the result may depend on the homotopy class of the chosen path.6 The result when using (G1) will then depend on which X in the Lie algebra is used to obtain the representative matrix for g.
The Lorentz group is doubly connected so that its fundamental group π1(SO(3,1)+), whose elements are the path homotopy classes, has two members. Thus not all representations of the Lie algebra will yield representations of the group, but some will instead yield projective representations.5 Once these conclusions has been reached, and once one knows whether a representation is projective, there is no need not be concerned about paths and partitions. Formula (1) applies to all representations, including the projective ones.
For a projective representation Π of SO(3;1)+ it holds that
since any loop in SO(3;1)+ traversed twice, due to the double connectedness, is contractible to a point so that its homotopy class is that of a constant map. It follows that Π is a double-valued function. One cannot consistently chose sign to obtain a continuous representation of all of SO(3;1)+, but this is possible to do locally around any point.8
Let πg denote the set of path homotopy classes pg of paths pg(t), 0 ≤ t ≤ 1, from 1 ∈ SO+(3;1) to g ∈ SO+(3;1) and define the set
and endow it with the multiplication operation
With this multiplication, G is a group and G ≈ SL(2,C),9 the universal covering group of SO+(3;1). By the above construction, there is, since each πg has two elements, a 2:1 covering map p:SL(2.C)→SO(3,1)+. According to covering group theory, the Lie algebras so(3;1) and sl(2,C) are isomorphic and the Π(m,n) representations (whether projective or not) correspond to non-projective representations of the simply connected group SL(2,C).
self-dual 2-form2-form field
Purple: (m, n) complex irreps Black: (m, n) ⊕ (n, m)
Since for any irrep where m ≠ n it is essential to operate over the field of complex numbers, the direct sum of representations (m, n) and (n, m) has a particular relevance to physics, since it permits to use linear operators over real numbers.
- (0, 0) is the Lorentz scalar representation. This representation is carried by relativistic scalar field theories.
- (1, 0) is the left-handed Weyl spinor and (0, 1) is the right-handed Weyl spinor representation.
- (1, 0) ⊕ (0, 1) is the bispinor representation. (See also Dirac spinor and Weyl spinors and bispinors below.)
- (1, 1) is the four-vector representation. The four-momentum of a particle (either massless or massive) transforms under this representation.
- (1, 0) is the self-dual 2-form field representation and (0, 1) is the anti-self-dual 2-form field representation.
- (1, 0) ⊕ (0, 1) is the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation.
- (1, 1) ⊕ (1, 1) is the Rarita–Schwinger field representation.
- (1, 1) is the spin 2 representation of the traceless metric tensor.
The irreducible (m, n) representations are (2m + 1)(2n + 1)-dimensional, and they are the only irreducible representations.
Since the angular momentum operator is given by J = A + B, the highest weight (or spin modulo ℏ in quantum mechanics) of the rotation subrepresentation will be m + n.
The (m, n) Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary. This is due to the non-compactness of the Lorentz group. It can also be seen directly from the definitions. The representations of A and B used in the construction are Hermitian. This means that J is Hermitian, but K is anti-Hermitian. The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.4
The (m, n) representation is, however, unitary when restricted to the rotation subgroup SO(3), but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an (m, n) representation have SO(3)-invariant subspaces of highest weight (spin) m + n, m + n − 1, …, | m − n |,4 where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) j is (2j + 1)-dimensional. So for example, the (1, 1) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.
The (m, n) representation is the dual of the (n, m) representation.
In general representation theory, if (π, V) is a representation of a Lie algebra g, then there is an associated representation of g on End V, also denoted π, given by
Likewise, a representation (Π,V) of a group G yields a representation Π on End V of G, still denoted Π, given by
Applying this to the Lorentz group, if (Π, V) is a projective representation, then direct calculation using (2) shows that the induced representation on End V is, in fact, a proper representation, i.e. a representation without phase factors.
The (possibly projective) (m, n) representation is irreducible as a representation SO+(3;1), the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If m = n it can be extended to a representation of all of O(3;1), the full Lorentz group, including space parity inversion and time reversal.4 This follows from considering the adjoint action AdP of P ∈ O(3;1) on so(3;1), where P is the standard representative of space parity inversion, P = diag(1, −1, −1, −1), given by
It is these properties of K and J under P that motivate the terms vector for K and pseudovector or axial vector for J. In a similar way, if π is any representation of so(3;1) and Π is its associated group representation, then Π(SO(3;1)+) acts on the representation of π by the adjoint action, π(X) ↦ Π(g) π(X) Π(g)−1 for X ∈ so(3;1), g ∈ SO(3;1)+. If P is to be included in Π, then consistency with (F1) requires that
holds, where A and B are defined as in the first section. This can hold only if Ai and Bi have the same dimensions, i.e. only if m = n. When m ≠ n then (m, n) ⊕ (n, m) can be extended to an irreducible representation of O+(3;1), the orthocronous Lorentz group. The parity reversal representative Π(P) does not come automatically with the general construction of the (m, n) representations. It must be specified separately. The matrix β = i γ0 may be used in the (1, 0) ⊕ (0, 1)4 representation. If parity is included in the (0,0) representation, it is called a pseudoscalar representation.
Time reversal T = diag(−1, 1, 1, 1), acts similarly on so(3;1) by
By explicitly including a representative for T, as well as one for P, one obtains a representation of the full Lorentz group O(3;1). The matrix Π(T) is antiunitary and its action on Hilbert space is antilinear.10
When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, (1, 0) ⊕ (0, 1), is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.
The Lorentz group SO(3,1)clarification needed and its double cover SL(2, C) also have infinite dimensional unitary representations, first studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) (at the instigation of Paul Dirac). The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by Harish-Chandra (1951) and Gelfand & Graev (1953), based on an analogue for SL(2, C) of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in Rühl (1970) and Knapp (2001).
The theory of spherical functions for the Lorentz group, required for harmonic analysis on the 3-dimensional hyperboloid in Minkowski space, or equivalently 3-dimensional hyperbolic space, is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on R. This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008).
The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup B of G = SL(2, C). Since the one-dimensional representations of B correspond to the representations of the diagonal matrices, with non-zero complex entries z and z−1, and thus have the form
for k an integer and ν real. The representations are irreducible; the only repetitions occur when k is replaced by −k. By definition the representations are realised on L2 sections of line bundles on G / B = S2, the Riemann sphere. When k = 0, these representations constitute the so-called spherical principal series.
The restriction of a principal series to the maximal compact subgroup K = SU(2) of G can also be realised as an induced representation of K using the identification G / B = K / T, where T = B ∩ K is the maximal torus in K consisting of diagonal matrices with | z | = 1. It is the representation induced from the 1-dimensional representation zk T, and is independent of ν. By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions | k | + 2m + 1 with m a non-negative integer.
Using the identification between the Riemann sphere minus a point and C, the principal series can be defined directly on L2(C) by the formula
Irreducibility can be checked in a variety of ways:
- The representation is already irreducible on B. This can be seen directly, but is also a special case of general results on ireducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition G = B ∪ B s B where s is the Weyl group element .12
- The action of the Lie algebra of G can be computed on the algebraic direct sum of the irreducible subspaces of K can be computed explicitly and the it can be verified directly that the lowest dimensional subspace generates this direct sum as a -module.1314
The for 0 < t < 2, the complementary series is defined on L2-functions f on C for the inner product
with the action given by
The complementary series are irreducible and inequivalent. As a representation of K, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of K = SU(2). Irreducibility can be proved by analysing the action of on the algebraic sum of these subspaces1314 or directly without using the Lie algebra.1516
The only irreducible unitary representations of SL(2, C) are the principal series, the complementary series and the trivial representation. Since −I acts (−1)k on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even.
To decompose the left regular representation of G on L2(G), only the principal series are required. This immediately yields the decomposition on the subrepresentations L2(G/±I), the left regular representation of the Lorentz group, and L2(G/K), the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with k = 0.)
The left and right regular representation λ and ρ are defined on L2(G) by
Now if f is an element of Cc(G), the operator πν,k(f) defined by
is Hilbert–Schmidt. We define a Hilbert space H by
and HS(L2(C)) denotes the Hilbert space of Hilbert–Schmidt operators on L2(C).17 Then the map U defined on Cc(G) by
extends to a unitary of L2(G) onto H.
The map U satisfies
If f1, f2 are in Cc(G) then
Thus if f = f1 ∗ f2* denotes the convolution of f1 and f2*, and , then
The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively. The Plancherel formula extends to all fi in L2(G). By a theorem of Jacques Dixmier and Paul Malliavin, every function f in is a finite sum of convolutions of similar functions, the inversion formula holds for such f. It can be extended to much wider classes of functions satisfying mild differentiability conditions.18
The metric signature is (−1, 1, 1, 1) and the physics convention for Lie algebras is used in this article. The Lie algebra of so(3;1) is in the standard representation given by
The commutation relations of the Lie algebra so(3;1) are
In three-dimensional notation, these are
Let π(m,n): so(3;1) → gl(V), where V is a vector space, denote the irreducible representations of so(3;1) according to the (m,n) classification. In components, with -m ≤ a ≤ m, -n ≤ b ≤ n, the representations are given by 4
Here the J(n) are the (2n + 1)-dimensional irreducible representations of so(3;1).
By taking , in turn, m = 1, n = 0 and m = 0, n = 1 and by setting
in the general expression (G1), and by using the trivial relations 11 = 1 and J(0) = 0, one obtains
These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) VL and VR, whose elements ΨL and ΨR are called left- and right-handed Weyl spinors respectively. Given (π(1,0),VL) and (π(0,1),VR) one may form their direct sum as representations,
This is, up to a similarity transformation, the (1,0)⊕(0,1) Dirac spinor representation of so(3,1). It acts on the 4-component elements (ΨL, ΨR) of (VL⊕VR), 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 so(3,1). Expressions for the group representations are obtained by exponentiation.
- Weinberg 2003, Chapter 5
- Hall 2003, Chapter 6
- Hall 2003, Chapter 4
- Weinberg 2002, Chapter 5
- Hall 2003, Chapter 3
- Weinberg 2002, Appendix B, Chapter 2
- Weinberg 2002, Section 2.7, Chapter 2
- Wigner 1937
- Wigner 1937, p.27
- Weinberg 2002 Chapter 2
- Weinberg 2002}, Chapter 3
- Knapp 2001, Chapter II
- Harish-Chandra 1947
- Taylor 1986
- Gelfand & Naimark 1947
- Takahashi 1963, p. 343
- Note that for a Hilbert space H, HS(H) may be identified canonically with the Hilbert space tensor product of H and its conjugate space.
- Knapp 2001
- Weinberg 2002, Equations (5.4.19) and (5.4.20)
- Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group", Ann. Of Math. 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129 (the representation theory of SO(2,1) and SL(2, R); the second part on SO(3,1) and SL(2, C), described in the introduction, was never published).
- Dixmier, J.; Malliavin, P. (1978), "Factorisations de fonctions et de vecteurs indéfiniment différentiables", Bull. Sc. Math. 102: 305–330
- Gelfand, I. M.; Naimark, M. A. (1947), "Unitary representations of the Lorentz group", Izvestiya Akad. Nauk SSSR. Ser. Mat. 11: 411–504
- Gelfand, I. M.; M. I. Graev (1953), [On a general method of decomposition of the regular representation of a Lie group into irreducible representations]
|title=(help), Doklady Akademii Nauk SSSR 92: 221–224
- Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. Roy. Soc. London. Ser. A. 189 (1018): 372–401, doi:10.1098/rspa.1947.0047
- Harish-Chandra (1951), "Plancherel formula for complex semi-simple Lie groups", Proc. Nat. Acad. Sci. U. S. A. 37 (12): 813–818, doi:10.1073/pnas.37.12.813
- Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Springer, ISBN 0-387-40122-9
- Helgason, S. (1968), Lie groups and symmetric spaces, Battelle Rencontres, Benjamin, pp. 1–71 (a general introduction for physicists)
- Helgason, S. (2000), Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions (corrected reprint of the 1984 original), Mathematical Surveys and Monographs 83, American Mathematical Society, ISBN 0-8218-2673-5
- Jorgenson, J.; Lang, S. (2008), The heat kernel and theta inversion on SL(2,C), Springer Monographs in Mathematics, Springer, ISBN 978-0-387-38031-5
- Knapp, A. (2001), Representation theory of semisimple groups. An overview based on examples., Princeton Landmarks in Mathematics, Princeton University Press, ISBN 0-691-09089-0 (elementary treatment for SL(2,C))
- Naimark, M.A. (1964), Linear representations of the Lorentz group (translated from the Russian original by Ann Swinfen and O. J. Marstrand), Macmillan
- Paërl, E.R. (1969) Representations of the Lorentz group and projective geometry, Mathematical Centre Tract #25, Amsterdam.
- Rühl, W. (1970), The Lorentz group and harmonic analysis, Benjamin (a detailed account for physicists)
- Takahashi, R. (1963), "Sur les représentations unitaires des groupes de Lorentz généralisés", Bull. Soc. Math. France 91: 289–433
- Taylor, M.E. (1986), Noncommutative harmonic analyis, Mathematical Surveys and Monographs 22, American Mathematical Society, ISBN 0-8218-1523-7, Chapter 9, SL(2, C) and more general Lorentz groups
- Weinberg, S (2002), The Quantum Theory of Fields, vol I, ISBN 0-521-55001-7
- Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics 40 (1): 149–204, doi:10.2307/1968551, MR 1503456.