Quantum Mechanics for Mathematicians: The Poincar´e Group and its Representations

7/27/2019 Quantum Mechanics for Mathematicians: The Poincare Group and its Representations

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Quantum Mechanics for Mathematicians: The

Poincare Group and its Representations

Peter Woit

Department of Mathematics, Columbia University

[email protected]

March 31, 2013

In the previous chapter we saw that one can take the semi-direct productof spatial translations and rotations and that the resulting group has infinite-dimensional unitary representations on the state space of a quantum free parti-cle. The free particle Hamiltonian plays the role of a Casimir operator: to getirreducible representations one fixes the eigenvalue of the Hamiltonian (the en-ergy), and then the representation is on the space of solutions to the Schrodingerequation with this energy. This is a non-relativistic procedure, treating time andspace (and correspondingly the Hamiltonian and the momenta) differently. Fora relativistic analog, we will use instead the semi-direct product of space-timetranslations and Lorentz transformations. Irreducible representations of thisgroup will be labeled by a continuous parameter (the mass) and a discrete pa-rameter (the spin or helicity), and these will correspond to possible relativistic

elementary particles.In the non-relativistic case, the representation occurred as a space of solu-

tions to a differential equation, the Schrodinger equation. There is an analogousdescription of the irreducible Poincare group representations as spaces of solu-tions of relativistic wave equations, but we will put off that story until succeedingchapters.

1 The Poincare group and its Lie algebra

Definition (Poincare group). The Poincare group is the semi-direct product

P = R4 SO(1, 3)

with double-coverP = R4 SL(2,C)

The action of SO(1, 3) or SL(2,C) onR4 is the action of the Lorentz group onMinkowski space.

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We will refer to both of these groups as the Poincare group, with thedouble-cover the meaning only when we need it because spinor representations

of the Lorentz group are involved. The two groups have the same Lie algebra,so the distinction is not needed in discussions that only need the Lie algebra.Elements of the group P will be written as pairs (a, ), with a R4 and SO(1, 3) and the group law is

(a1, 1)(a2, 2) = (a1 + 1a2, 12)

The Lie algebra LieP = LieP has dimension 10, with basis

P0, P1, P2, P3, J1, J2, J3, K1, K2, K3

where the first four elements are a basis of the Lie algebra of the translationgroup (note that we are following physicists in using the same notation for Liealgebra basis elements and the corresponding operators in a representation ofthe Lie algebra), and the next six are a basis of so(1, 3), with the Jj givingthe subgroup of spatial rotations, the Kj the boosts. We already know thecommutation relations for the translation subgroup, which is commutative so

[Pj , Pk] = 0

We have seen that the commutation relations for so(1, 3) are

[J1, J2] = J3, [J2, J3] = J1, [J3, J1] = J2

[K1, K2] = J3, [K3, K1] = J2, [K2, K3] = J1

and that the commutation relations between the Jj and Kj correspond to thefact that the Kj transform as a vector under spatial rotations, so for example

commuting the Kj with J1 gives an infinitesimal rotation about the 1-axis and

[K1, J1] = 0, [K2, J1] = K3, [K3, J1] = K2

To find the Lie algebra relations between translation and so(1, 3) basis ele-ments, recall that the Lie algebra commutators are related to the group multi-plication by

[X, Y] =d

dt(etXY etX)|t=0

Using the semi-direct product relation

(n1k1)(n2k2) = n1k1(n2)k1k2

to compute the conjugation of a translation (a, 1) P by a Lorentz transfor-

mation (0, ) P one finds

(0, )(a, 1)(0, )1 =(0, )(a, 1)(0, 1)

=(0, )(a + 10, 1) = (0, )(a, 1)

=(a, 1)

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Taking = etJ1 one finds

d

dta|t=0 =

d

dt

1 0 0 00 1 0 00 0 cos t sin t0 0 sin t cos t

a0a1a2a3

|t=0

=

00

a3a2

This is a relation on finite translations (a, 1), but it also hold for infinitesimaltranslations, so we have

[J1,

P0P1P2P3

] = ddt

1 0 0 00 1 0 00 0 cos t sin t0 0 sin t cos t

P0P1P2P3

|t=0

=

00

P3P2

which shows that the Pj transform as vectors under infinitesimal rotations aboutthe 1-axis, so

[J1, P0] = [J1, P1] = 0, [J1, P2] = P3, [J1, P3] = P2

with similar relations for the other axes.For boosts along the 1-axis, one does the same calculation with = etK1 to

find

d

dta|t=0 =

d

dt

cosh t sinh t 0 0sinh t cosh t 0 0

0 0 1 00 0 0 1

a0a1a2a3

|t=0

=

a2a100

so

[K1,

P0P1P2P3

] =

P1P000

and in general one will have

[Kj , P0] = Pj, [Kj , Pj ] = P0, [Kj , Pk] = 0 ifj = k, k = 0

2 Representations of the Poincare group

We want to find unitary irreducible representations of the Poincare group. Thesewill be infinite dimensional, so given by operators (g) on a Hilbert space H,

which will have an interpretation as a single-particle relativistic quantum statespace. The standard physics notation for the operators giving the represen-tation is U(a, ), with the U emphasizing their unitarity. To classify theserepresentations, we recall from the last chapter that irreducible representationsof semi-direct products NK are associated with pairs of a K-orbit O in thespace N and an irreducible representation of the corresponding little group K.

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For the Poincare group, N = R4 is the space of characters of the translationgroup of Minkowski space, so functions

p(x) = eipx = ei(p0x0p1x1p2x2p3x3)

labeled by a four-dimensional energy-momentum vector p = (p0, p1, p2, p3). TheLorentz group acts on this R4 by

p p

and, restricting attention to the p0 p3 plane, the picture of the orbits lookslike this

Unlike the Euclidean group case, here there are several different kinds oforbits O. Well examine them and the corresponding stabilizer groups Keach in turn, and see what can be said about the associated representations.One way to understand the equations describing these orbits is to note thatthe different orbits correspond to different eigenvalues of the Poincare groupCasimir operator

P2 = P20 P21 P

22 P

23

This operator commutes with all the generators of the Lie algebra of the Poincaregroup, so by Schurs lemma it must act as a scalar times the identity on an

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irreducible representation (recall that the same phenomenon occurs for SU(2)representations, which can be characterized by the eigenvalue j(j +1) of the Cas-

mir operator J2 for SU(2)). At a point p = (p0, p1, p2, p3) in energy-momentumspace, the Pj operators are diagonalized and P

2 will act by the scalar

p20 p21 p

22 p

23

which can be positive, negative, or zero, so given by m2, m2, 0 for variousm. The value of the scalar will be the same everywhere on the orbit, so inenergy-momentum space orbits will satisfy one of the three equations

p20 p21 p

22 p

23 =

m2

m2

0

Note that in this chapter we are just classifying Poincare group representa-tions, not actually constructing them. It is possible to construct these represen-tations using the data we will find that classifies them, but this would requireintroducing some techniques (for so-called induced representations) that gobeyond the scope of this course. In later chapters we will explicitly constructthese representations in certain specific cases as solutions to certain relativisticwave equations.

2.1 Positive energy time-like orbits

One way to get positive values m2 of the Casimir P2 is to take the vectorp = (m, 0, 0, 0), m > 0 and generate an orbit Om,0,0,0 by acting on it withthe Lorentz group. This will be the upper, positive energy, hyperboloid of thehyperboloid of two sheets

p20 p21 p

22 p

23 = m

2

so

p0 =

p21 +p22 +p

23 + m

2

The stabilizer group of Km,0,0,0 is the subgroup of SO(1, 3) of elements ofthe form

1 00

where SO(3), so Km,0,0,0 = SO(3). Irreducible representations are classi-fied by the spin. For spin 0, points on the hyperboloid can be identified withpositive energy solutions to a wave equation called the Klein-Gordon equationand functions on the hyperboloid both correspond to the space of all solutionsof this equation and carry an irreducible representation of the Poincare group.In the next chapter we will study the Klein-Gordon equation, as well as thequantization of the space of its solutions by quantum field theory methods.

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We will later study the case of spin 12 , where one must use the double coverSU(2) of SO(3). The Poincare group representation will be on functions on

the orbit that take values in two copies of the spinor representation of SU(2).These will correspond to solutions of a wave equation called the massive Diracequation.

For choices of higher spin representations of the stabilizer group, one canagain find appropriate wave equations and construct Poincare group represen-tations on their space of solutions, but we will not enter into this topic.

2.2 Negative energy time-like orbits

Starting instead with the energy-momentum vector p = (m, 0, 0, 0), m > 0,the orbit Om,0,0,0 one gets is the lower, negative energy component of thehyperboloid

p20 p21 p

22 p

23 = m

2

satisfying

p0 =

p21 +p22 +p

23 + m

2

Again, one has the same stabilizer group Km,0,0,0 = SO(3) and the same con-stuctions of wave equations of various spins and Poincare group representationson their solution spaces as in the positive energy case. Since negative energieslead to unstable, unphysical theories, we will see that these representations aretreated differently under quantization, corresponding physically not to particles,but to anti-particles.

2.3 Space-like orbits

One can get negative values m2

of the Casimir P2

by considering the orbitO0,0,0,m of the vector p = (0, 0, 0, m). This is a hyperboloid of one sheet,satisfying the equation

p20 p21 p

22 p

23 = m

2

It is not too difficult to see that the stabilizer group of the orbit is K0,0,0,m =SO(2, 1). This is isomorphic to the group SL(2,R), and it has no finite-dimensional unitary representations. These orbits correspond physically totachyons, particles that move faster than the speed of light, and there isno known way to consistently incorporate them in a conventional theory.

2.4 The zero orbit

The simplest case where the Casimir P2 is zero is the trivial case of a pointp = (0, 0, 0, 0). This is invariant under the full Lorentz group, so the orbitO0,0,0,0 is just a single point and the stabilizer group K0,0,0,0 is the entire Lorentzgroup SO(1, 3). For each finite-dimensional representation of SO(1, 3), one getsa corresponding finite dimensional representation of the Poincare group, with

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translations acting trivially. These representations are not unitary, so not usablefor our purposes.

2.5 Positive energy null orbits

One has P2 = 0 not only for the zero-vector in momentum space, but for athree-dimensional set of energy-momentum vectors, called the null-cone. Bythe term cone one means that if a vector is in the space, so are all productsof the vector times a positive number. Vectors p = (p0, p1, p2, p3) are calledlight-like or null when they satisfy

|p|2 = p20 p21 p

22 p

23 = 0

One such vector is p = (1, 0, 0, 1) and the orbit of the vector under the actionof the Lorentz group will be the upper half of the full null-cone, the half with

energy p0 > 0, satisfyingp0 =

p21 +p

22 +p

23

The stabilizer group K1,0,0,1 of p = (1, 0, 0, 1) includes rotations about thex3 axis, but also boosts in the other two directions. It is isomorphic to theEuclidean group E(2). Recall that this is a semi-direct product group, and ithas two sorts of irreducible representations

Representations such that the two translations act trivially. These areirreducible representations ofSO(2), so one-dimensional and characterizedby an integer n (half-integers when one uses the Poincare group doublecover).

Infinite dimensional irreducible representations on a space of functions ona circle of radius r

The first of these two gives irreducible representations of the Poincare groupon certain functions on the positive energy null-cone, labeled by the integer n,which is called the helicity of the representation. We will in later chaptersconsider the cases n = 0 (massless scalars, wave-equation the Klein-Gordonequation), n = 12 (Weyl spinors, wave equation the Weyl equation), and n =1 (photons, wave equation the Maxwell equations).

The second sort of representation ofE(2) gives representations of the Poincaregroup known as continuous spin representations, but these seem not to cor-respond to any known physical phenomena.

2.6 Negative energy null orbitsLooking instead at the orbit of p = (1, 0, 0, 1), one gets the negative energypart of the null-cone. As with the time-like hyperboloids of non-zero massm, these will correspond to anti-particles instead of particles, with the sameclassification as in the positive energy case.

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3 For Further Reading

The Poincare group and its Lie algebra is discussed in pretty much any quantumfield theory textbook. Weinberg [3] (Chapter 2) has some discussion of therepresentations of the Poincare group on single particle state spaces that we haveclassified here. Folland [2] (Chapter 4.4) and Berndt [1] (Chapter 7.5) discussthe actual construction of these representations using the induced representationmethods that we have chosen not to try and explain here.

References

[1] Berndt, R., Representations of Linear Groups, Vieweg, 2007.

[2] Folland, G., Quantum Field Theory: A tourist guide for mathematicians,AMS, 2008.

[3] Weinberg, S., The Quantum Theory of Fields I, Cambridge UniversityPress, 1995.

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Quantum Mechanics for Mathematicians: The Poincar´e Group and its Representations