Representations of the Lorentz group
Encyclopedia
The Lorentz group
of theoretical physics
has a variety of representations, corresponding to particle
s with integer and half-integer
spins in quantum field theory
. These representations are normally constructed out of spinor
s.
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.
The Lorentz group has no unitary representation
of finite dimension, except for the trivial representation (where every group element is represented by 1).
of Lie group
s, one first looks for the representations of the complexification
of the Lie algebra
of the Lorentz group
. A convenient basis for the Lie algebra of the Lorentz group is given by the three generators of rotation
s Jk=εijkLij and the three generators of boost
s Ki=Lit where i, j, and k run over the three spatial coordinates and ε is the Levi-Civita symbol
for a three dimensional spatial slice of Minkowski space
. Note that the three generators of rotations transform like components of a pseudovector J and the three generators of boosts transform like components of a vector K under the adjoint action of the spatial rotation subgroup
.
This motivates the following construction: first complexify, and then change basis to the components of A = (J + i K)/2 and B = (J – i K)/2. In this basis, one checks that the components of A and B satisfy separately the commutation relations of the Lie algebra sl2 and moreover that they commute with each other. In other words, one has the isomorphism
The utility of this isomorphism comes from the fact that sl2 is the complexification of the rotation algebra, and so its irreducible representations correspond to the well-known representations of the spatial rotation group; for each j in ½Z, one has the (2j+1)-dimensional spin-j representation spanned by the spherical harmonic
s with j as the highest weight. Thus the finite dimensional irreducible representations of the Lorentz group are simply given by an ordered pair of half-integers (m,n) which fix representations of the subalgebra spanned by the components of A and B respectively.
operator is given by J = A + B, the highest weight of the rotation subrepresentation will be m+n. So for example, the (1/2,1/2) representation has spin
1. The (m,n) representation is (2m+1)(2n+1)-dimensional.
which is invariant under parity 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 Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by and , based on an analogue for SL(2,C) of the integration formula of Hermann Weyl
for compact Lie groups. Elementary accounts of this approach can be found in and .
The theory of spherical functions
for the Lorentz group, required for harmonic analysis
on the 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 , , and the posthumous text of .
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:
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 subspaces or directly without using the Lie algebra.
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 π\nu,k (f) defined by
is Hilbert-Schmidt
. We define a Hilbert space H by
where
and HS(L2(C)) denotes the Hilbert space of Hilbert-Schmidt operators on L2(C). 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*, where
, 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.
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
of theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
has a variety of representations, corresponding to particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s with integer and half-integer
Half-integer
In mathematics, a half-integer is a number of the formn + 1/2,where n is an integer. For example,are all half-integers. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer...
spins in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
. These representations are normally constructed out of spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s.
The group may also be represented in terms of a set of functions defined on the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
. 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
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
, and find practical application in the theory of atomic spectra.
The Lorentz group has no unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
of finite dimension, except for the trivial representation (where every group element is represented by 1).
Finding representations
According to general representation theoryRepresentation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, one first looks for the representations of the complexification
Complexification
In mathematics, the complexification of a real vector space V is a vector space VC over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for V over the real numbers serves as a basis for VC over the complex...
of the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
. A convenient basis for the Lie algebra of the Lorentz group is given by the three generators of rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s Jk=εijkLij and the three generators of boost
Boost
-Science, technology and mathematics:* Automotive:** Boost, positive manifold pressure in cars, see Turbocharger#Pressure increase.*** a loose term for turbo or supercharger** A slang term meaning to start a vehicle, see jump start...
s Ki=Lit where i, j, and k run over the three spatial coordinates and ε is the Levi-Civita symbol
Levi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
for a three dimensional spatial slice of Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
. Note that the three generators of rotations transform like components of a pseudovector J and the three generators of boosts transform like components of a vector K under the adjoint action of the spatial rotation subgroup
Rotation group
In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
.
This motivates the following construction: first complexify, and then change basis to the components of A = (J + i K)/2 and B = (J – i K)/2. In this basis, one checks that the components of A and B satisfy separately the commutation relations of the Lie algebra sl2 and moreover that they commute with each other. In other words, one has the isomorphism
The utility of this isomorphism comes from the fact that sl2 is the complexification of the rotation algebra, and so its irreducible representations correspond to the well-known representations of the spatial rotation group; for each j in ½Z, one has the (2j+1)-dimensional spin-j representation spanned by the spherical harmonic
Spherical Harmonic
Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
s with j as the highest weight. Thus the finite dimensional irreducible representations of the Lorentz group are simply given by an ordered pair of half-integers (m,n) which fix representations of the subalgebra spanned by the components of A and B respectively.
Properties of the (m,n) irrep
Since the angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
operator is given by J = A + B, the highest weight of the rotation subrepresentation will be m+n. So for example, the (1/2,1/2) representation has spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
1. The (m,n) representation is (2m+1)(2n+1)-dimensional.
Common representations
- (0,0) the Lorentz scalar representation. This representation is carried by relativistic scalar field theories.
- (1/2,0) is the left-handed Weyl spinor and (0,1/2) is the right-handed Weyl spinor representation.
- (1/2,0) ⊕ (0,1/2) is the bispinorBispinorIn physics, bispinor is a four-component object which transforms under the ⊕ representation of the covariance group of special relativity...
representation (see also Dirac spinor). - (1/2,1/2) is the four-vectorFour-vectorIn the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
representation. The electromagnetic vector potentialVector potentialIn vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
lives in this rep. It is a 1-form field. - (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. The electromagnetic field tensor transforms under this representation.
- (1,1/2) ⊕ (1/2,1) is the Rarita-Schwinger field representation.
- (1,1) is the spin-2 representation of the traceless metric tensor.
Full Lorentz group
The (m,n) representation is irreducible under the restricted Lorentz group (the identity component of the Lorentz group). When one considers the full Lorentz group, which is generated by the restricted Lorentz group along with time and parity reversal, not only is this not an irreducible representation, it is not a representation at all, unless m=n. The reason is that this representation is formed in terms of the sum of a vector and a pseudovector, and a parity reversal changes the sign of one, but not the other. The upshot is that a vector in the (m,n) representation is carried into the (n,m) representation by a parity reversal. Thus (m,n)⊕(n,m) gives an irrep of the full Lorentz group. When constructing theories such as QEDQuantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
which is invariant under parity reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors.
History
The Lorentz group SO(3,1) and its double cover SL(2,C) also have infinite dimensional unitary representations, first studied independently by , and (at the instigation of Paul DiracPaul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
). The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by and , based on an analogue for SL(2,C) of the integration formula of Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
for compact Lie groups. Elementary accounts of this approach can be found in and .
The theory of spherical functions
Zonal spherical function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G...
for the Lorentz group, required for harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
on the 3-dimensional hyperboloid in Minkowski space, or equivalently 3-dimensional hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
, 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 , , and the posthumous text of .
Principal series
The principal series, or unitary principal series, are the unitary representations inducedInduced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
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
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
. 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 torusMaximal torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.A torus in a Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...
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 BruhatFrançois BruhatFrançois Georges René Bruhat was a French mathematician who worked on algebraic groups. The Bruhat order of a Weyl group, the Bruhat decomposition, and the Schwartz–Bruhat functions are named after him....
and George MackeyGeorge MackeyGeorge Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...
, relying on the Bruhat decompositionBruhat decompositionIn mathematics, the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases...
G = B
B s B where s is the Weyl groupWeyl groupIn mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
element
.
- 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.
Complementary series
The for 0 < t < 2, the complementary series is defined on L2 functions f on C for the inner productwith 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 thealgebraic sum of these subspaces or directly without using the Lie algebra.
Plancherel theorem
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 π\nu,k (f) defined by
is Hilbert-Schmidt
Hilbert-Schmidt
In mathematics, Hilbert-Schmidt may refer to* a Hilbert–Schmidt operator;** a Hilbert–Schmidt integral operator;* the Hilbert–Schmidt theorem....
. We define a Hilbert space H by
where
and HS(L2(C)) denotes the Hilbert space of Hilbert-Schmidt operators on L2(C). 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 convolutionConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of f1 and f2*, where
, thenThe 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
Jacques Dixmier
Jacques Dixmier is a French mathematician. He worked on operator algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace. He received his Ph.D. in 1949 from the University of Paris, and his students include Alain Connes.-Publications:*J. Dixmier,...
and Paul Malliavin
Paul Malliavin
Paul Malliavin was a French mathematician. He was Professor Emeritus at the Pierre and Marie Curie University. He was a member of the Academy of Sciences since 1979.-Domains of research:...
,
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.