Appendix:Relativistic Invariance

A-1 Free point-particles and the Poincare algebra

In this appendix we shall briefly discuss the relativistic wave functions offree particles. The symmetry operations relevant to relativistic invarianceare translations (in space and time) and Lorentz transformations (includingspatial rotations). To obtain the wave functions, we need a Hilbert spaceon which we can realize translations and Lorentz transformations as unitarytransformations. In other words, we need a unitary irreducible representation(UIR) of the algebra of translations and Lorentz transformations. The re-quirement of unitarity is clear since all physical observables generate unitarytransformations on the Hilbert space in quantum mechanics. The qualifica-tion of irreducibility is a little more subtle, so we take a moment to recallthat in nonrelativistic quantum mechanics we have a similar situation. Forone-particle dynamics in one dimension, the relevant operators are x and pwith the commutation rules

[x, x] = 0, [p, p] = 0, [x, p] = i (A-1)

We need a unitary representation of this algebra of observables. One way torepresent the operators x, p is as follows. We can consider complex-valuedfunctions ψ(x, p) with the normalization condition∫

dpdx ψ(x, p)∗ψ(x, p) = 1 (A-2)

The operators act by

pψ = −i∂

∂xψ

x ψ =(

i∂

∂p+ x

)ψ (A-3)

Clearly, this gives the wrong quantum mechanics since we are specifying thewave functions as functions of x and p. We can also see that this representa-tion is reducible. In fact, we see that we can impose a condition

∂ψ

∂p= 0 (A-4)

534 Appendix:Relativistic Invariance

on the wave functions and still obtain a representation of (A-1). In particular,in this case,

pψ = −i∂

∂xψ, xψ = x ψ (A-5)

The normalization condition can now be taken as∫dx ψ∗ψ = 1 (A-6)

This is the standard Schrodinger representation. Since we are able to ob-tain a representation on this smaller space of functions obeying the condition(A-4), the former representation (A-3) is reducible. The Schrodinger represen-tation can be shown to be irreducible; i.e., there is no smaller function spaceon which the algebra (A-1) can be realized. (Properly speaking, one shouldconsider bounded operators obtained by exponentiation.) We thus see thatquantum mechanics may be identified as a unitary irreducible representation(UIR) of the algebra of observables. (For the algebra (A-1), there is only onerepresentation, the Schrodinger representation, up to unitary equivalence, aresult due to Stone and von Neumann. With an infinite number of degreesof freedom, or on nonsimply connected spaces, there can be many UIR’sand the physical consequences can be different depending on which UIR onechooses. This is not an issue of immediate relevance to our discussion; wehave discussed some of these issues in Chapters 12, 15 and 20.)

Returning to the question of the relativistic point-particle, we consider thesymmetry transformations in some more detail. The action for the motion ofa relativistic point-particle is the mass m times the proper distance, or

S =∫

dt L = −m

∫ √ηµνdxµdxν (A-7)

The metric tensor in the above expression is the Minkowski metric withCartesian components η00 = 1, ηij = −δij and all other components beingzero. The symmetries of the theory are clearly the symmetries of the properdistance ds =

√ηµνdxµdxν . We first consider the continuous symmetries

which can be understood in terms of infinitesimal transformations. We write

xµ → x′µ = xµ + ξµ (A-8)

where ξµ is infinitesimal. The requirement that this be a symmetry transfor-mation of the proper distance gives

ηαν∂ξα

∂xµ+ ηµα

∂ξα

∂xν= 0 (A-9)

The solution to this condition is given by

ξµ = aµ + ωµνxν (A-10)

A-1 Poincare algebra 535

where aµ, ωµν are constant parameters and ωµν = −ωνµ. ωµν are the six pa-rameters of Lorentz transformations, ω0i being the relative velocities of theframes connected by the transformation (A-10). θi = 1

2εijkωjk are the anglesof spatial rotations and aµ are the parameters of translations. For any func-tion of xµ, we can write the generators of the transformations immediatelyfrom (A-8, A-10).

δf(x) = ξµ ∂

∂xµf = aµ ∂f

∂xµ+ ωµνxν

∂f

∂xµ

=(

i aµPµ − i

2ωµνMµν

)f (A-11)

wherePµ = −i

∂

∂xµ, Mµν = xµPν − xµPµ (A-12)

(The factors of i are convenient for the sake of hermiticity; these operatorswill be interpreted later as physical quantities.) The commutation rules ofthese operators are easily worked out to be

[Pµ, Pν ] = 0[Mµν , Pα] = i(ηµαPν − ηανPµ) (A-13)

[Mµν , Mαβ] = i (ηµαMνβ + ηνβMµα − ηµβMνα − ηναMµβ)

This algebra of translations and Lorentz transformations is the Poincare al-gebra. Although we obtained this algebra by considering the infinitesimaltransformations on a scalar function, it is of general validity. One can alsocharacterize particles with spins in terms of this algebra.

For example, for a vector-valued function Aµ(x), the transformationrule can be worked out by treating Aµdxµ as a scalar; i.e., δAµdxµ =Aµ(x′) dx′µ − Aµ(x)dxµ. Explicitly,

δAµ = ξν ∂Aµ

∂xν+

∂ξν

∂xµAν

=(

iaαPα − i

2ωαβMαβ

)ν

µ

Aν (A-14)

where

Pα = i∂

∂xα

Mαβ = (xαPβ − xβPα) + Sαβ

(Sαβ)µν = −i(ηµαηνβ − ηναηµβ) (A-15)

Sαβ is to be thought of as a 4× 4-matrix, α, β specifying the type of Lorentztransformation we are interested in and µ, ν specifying the matrix elements.Equation (A-15) is in matrix notation, i.e., expressions like aαPα, xαPβ −

536 Appendix:Relativistic Invariance

xβPα are proportional to the identity matrix δνµ. One can easily verify that

Pα, Mαβ in (A-15) obey the same commutation rules as in (A-13), withmatrix multiplication understood for Sαβ. Sαβ is the spin contribution to theLorentz generators.

For a constant vector kµ, the Lorentz transformation is generated en-tirely by Sµν . The finite transformation is given by the composition of manyinfinitesimal transformations as

k′µ =

[lim

N→∞

(1 − i

2ωαβ

NSαβ

)N]

µν

kν

=[exp(− i

2ωαβSαβ

]νµ

kν (A-16)

≡ Lνµ kν

The matrix Lνµ obeys

Lαµ Lβ

ν ηαβ = ηµν (A-17)

and further det L = 1, L00 = 1. (These are sometimes further qualified

as proper orthochronous Lorentz transformations. Improper transformationscan have a determinant equal to −1; they can be understood in terms ofcomposing the proper orthochronous transformations with discrete transfor-mations such as parity and time-reversal. The discrete transformations whichare also symmetries of the proper distance will not be discussed here; they aremost easily understood in the functional integral language and are explainedin chapter 12.)

The Poincare algebra (A-13) is the basic algebra of observables for freepoint-particles. From our understanding of quantum mechanics, we can thusobtain free-particle wave functions by studying the unitary realizations of thePoincare algebra. Pµ, being the generator of translations, will be the four-momentum of the particle, P0 = H being the Hamiltonian. Mij will generatespatial rotations and hence J i = 1

2εijkMjk is the angular momentum. M0i ≡Ki generate Lorentz transformations connecting frames of reference whichare in relative motion; these are the so-called boosts. One can decompose thePoincare algebra in terms of these components as follows:

[Pi, Pj ] = 0, [Pi, H ] = 0[Ji, Pj ] = i εijkP k, [Ji, H ] = 0[Ki, Pj ] = i δijH, [Ki, H ] = iPi

[Ji, Jj ] = i εijkJk (A-18)[Ji, Kj] = i εijkKk

[Ki, Kj] = −iεijkJk

In arriving at these equations, it is useful to remember that εijk is the nu-merical Levi-Civita symbol, so that εijkεiab = δjaδkb − δjbδka.

A-2 Unitary representations of the Poincare algebra 537

A-2 Unitary representations of the Poincare algebra

We begin the construction of unitary realizations of the Poincare algebra bynoticing that the Pµ commute among themselves and so can be simultane-ously diagonalized. We can thus define a set of states, |p〉, depending on afour-vector pµ such that

Pµ|p〉 = pµ|p〉 (A-19)

Now, from the commutation rule (A-13), we can work out the finite Lorentztransformation of Pµ as

P ′µ = L(ω)ν

µ Pν (A-20)

(A-13) is the infinitesimal version of this result with P ′µ = U(ω)PµU−1(ω),

U(ω) = exp(− i2ωµνMµν). For the eigenvalue four-momentum pµ, we thus

have p′µ = Lνµpν .

From the transformation law for Pµ, it is clear that P 2 is invariant underLorentz transformations, and since the P ’s are commuting operators, P 2

commutes with all Pµ and Mµν ; i.e., it is a Casimir operator for the Poincarealgebra. The possible representations can therefore be specified by the valueof P 2. The possibilities are

1. P 2 > 0, say, P 2 = m2

2. P 2 < 03. P 2 = 0.

P 2 > 0 will describe massive particles of mass m. (In fact the conditionP 2 = m2 may be taken as the definition of the mass m.) P 2 < 0 is unphysical,corresponding to propagation faster than light. P 2 = 0 will describe masslessparticles such as the photon.

There is also another Casimir operator for the Poincare algebra. This isgiven as W 2, where Wµ is defined as

Wµ = εµναβPνMαβ (A-21)

Wµ is called the Pauli-Lubanski spin vector. We can use W 2 to characterizethe representations further. However, instead of analyzing the possible valuesof W 2 in general, we shall simply analyze the representations of interest andcalculate W 2 for these.

Given that p2 = m2 or zero, for the states |p〉, we have p0 = ±√

p2 + m2

(or p0 = ±√p2). It is easy to see from the Lorentz transformation of pµ that

Lνµ do not change the sign of p0. Therefore we further characterize represen-

tations by the sign of p0; for the physically interesting cases (with positiveenergy), we have

p2 = m2, p0 > 0p2 = 0, p0 > 0. (A-22)

538 Appendix:Relativistic Invariance

The states |p〉 are thus labeled by the three-momentum p, with p0 =√p2 + m2 (or p0 =

√p2 for the massless case). These states for all of p-

space obeying the conditions (A-22) should give us a complete set of states.Because (A-22) are Lorentz-invariant, we choose a Lorentz-invariant measureof p-integration as

dµ(p) =d4p

(2π)3δ(p2 − m2)θ(p0) (A-23)

(with m = 0 for the massless case). The p0-part of the integration with thismeasure is trivial; carrying out this integration, we get, instead of (A-23),

dµ(p) =d3p

(2π)31

2ωp(A-24)

where ωp is the positive square root solution for p0, ωp =√

p2 + m2; weintegrate over all of p. The completeness condition for the states |p〉 can bewritten as ∫

|p〉 d3p

(2π)31

2ωp〈p| = 1 (A-25)

Since Pi is hermitian, we must have orthogonality of states of different val-ues of p. The orthonormality condition corresponding to the completenessrelation (A-25), is

〈p|p′〉 = (2π)32ωp δ(3)(p − p′) (A-26)

A-3 Massive particles

So far, we have only specified the action of the momentum operators Pµ. Wemust now specify how Lorentz transformations act on |p〉. For this we followthe procedure of Wigner. The strategy is to go to a special frame, suitablychosen, construct a representation there and bring it back to a general frameby appropriate Lorentz transformations. To begin with we must thus choosea special pµ obeying (A-22). For the massive case, which we shall discuss first,we can take

pµ = p(0)µ = (m, 0, 0, 0) (A-27)

This corresponds to the rest frame of the particle. There is a special Lorentztransformation L(ω0) = B(p), which gives pµ from p

(0)µ , i.e.,

pµ = B(p)νµ p(0)

ν (A-28)

Actually there are many choices for B. Explicitly, one choice is given by

B νµ = δν

µ − (pµ + mηµ)(pν + mην)m(p0 + m)

+2pµην

m(A-29)

where ηµ is a fixed vector with η0 = 1, ηi = 0.

A-3 Massive particles 539

There are still some transformations we can do in the rest frame. Thetransformations L(ω) such that L(ω)p(0) = p(0) form the little group or theisotropy group of p(0). (Here (Lp(0))µ = Lν

µp(0)ν ; we use an obvious matrix

notation.) In our case, it is evident that these are the spatial rotations R(ω) =L(ω), with ω0i = 0. (Clearly such rotations are an ambiguity in the choice ofB also.) Unitary representation of these rotations is well known; it is standardangular momentum theory. A representation is characterized by the highestvalue, denoted s, of a component of angular momentum, say, J3. There are(2s + 1) states and we have the transformation rule

U(θ) |s, n〉 =∑n′

D(s)nn′(θ) |s, n′〉 (A-30)

Here θi = 12εijkωjk. The D(s)

nn′(θ) are the standard Wigner D-matrices of angu-lar momentum theory; they obey the unitarity condition D†D = 1, in matrixnotation. We may write them out as D(s)

nn′(θ) = 〈s, n|eiθkJk |s, n′〉 where Jk areangular momentum generators and |s, n〉 are the spin-sangular momentumstates.

The crucial observation is that the representation for the full Poincaregroup can be obtained on states of the form |p, s, n〉. In other words, a basisfor the Hilbert space is given by products of the form |p〉⊗ |s, n〉, where |s, n〉provide a unitary representation of the little group, in this case, rotations ofp(0). The action of a general Lorentz transformation can be obtained fromthe action of rotations as in (A-30). For this one constructs, for every Lorentztransformation, a pure spatial rotation, called a Wigner rotation, given by

R(θW ) = B−1(L(ω)p) L(ω) B(p) (A-31)

Since R(θW )p(0) = p(0), we see that the combination on the right-hand sideof equation (A-31) is indeed a pure rotation. Thus for every ωµν , we canassociate rotation angles θi

W given by (A-31). The action of a general Lorentztransformation with parameters ωµν can then be defined as

U(ω)|p, s, n〉 =∑n′

D(s)nn′(θW (p, ω)) |Lp, s, n′〉 (A-32)

Since the action of rotations on |s, n〉 is unitary, i.e., D†D = 1, we see that(A-32) defines a unitary realization of Lorentz transformations. This can beexplicitly checked with the scalar product (A-26). A representation such as(A-32) is called an induced representation since the action of Lorentz trans-formations is induced from the action of rotations.

The nature of massive one-particle states is now clear. They are char-acterized by the spin s with (2s + 1) polarization states. s determines theD-matrices to be used. The action of translations on such states is givenby e−iP ·a, which is simply a factor e−ip·a on the momentum eigenfunctions.Lorentz transformations, including rotations, act on the states as in (A-32).

540 Appendix:Relativistic Invariance

It is instructive to work out the Wigner rotation explicitly for an infinites-imal Lorentz transformation. From (A-31)

R ≈ 1 − B−1ωB + (δB−1) B (A-33)

Working out the components, we find that

R00 = 1, Ri

0 = 0 = R0i , Rj

i = δji −ωj

i −1

(p0 + m)(ω0ip

j−ω0jpi)+· · · (A-34)

This identifies the Wigner rotation parameter as

θiW (p, ω) ≈ 1

2εijk

(ωjk +

2(p0 + m)

ω0jpk

)(A-35)

The Pauli-Lubanski spin operator, for p(0)µ , reduces to the angular mo-

mentum operator, W 0 = 0, W i = mJ i. This explains why Wµ is called thespin operator. W 2 is given by m2 s(s+1) for the representations appropriateto massive particles.

A-4 Wave functions for spin-zero particles

For a particle of spin zero, since s = 0, we do not have any nontrivial D-matrices for the transformations. The action of a finite translation by aµ onthe state |p〉 is given by

e−iP ·a |p〉 = e−ipa |p〉 (A-36)

This shows that the x-space wave function for a particle of momentum p canbe taken as

ψp(x) = 〈x|p〉 = e−ipx, (A-37)

since x → x + a is reproduced by the action of e−iP ·a as in (A-36). (Here pxdenotes p · x = p0x0 − p · x.) ψp(x) is a Lorentz scalar. The scalar productfor the x-space wave functions is taken as∫

d3x [ψp(x)∗(i∂0ψp′(x)) − (i∂0ψp(x)∗)ψp′(x)] = 2p0(2π)3δ(3)(p − p′)

(A-38)The choice is dictated by the requirement of consistency with the normaliza-tion condition (A-26). The scalar product

〈1|2〉 =∫

d3x [ψ∗1(i∂0ψ2) − (i∂0ψ

∗1)ψ2] (A-39)

is easily checked to be Lorentz-invariant.

A-4 Wave functions for spin-zero particles 541

In our calculations in text, for simplicity of interpretation of creation andannihilation of particles, we have considered the particles to be in a cubicalbox of volume V = L3, with the limit V → ∞ taken at the end of thecalculation. (In this limit, we will recover the full Lorentz symmetry as well.)In this case, the wave functions for a particle of momentum p can be takenas

up(x) =e−ipx√2ωpV

(A-40)

We shall use periodic boundary conditions on the wave functions. Ultimately,of course, physical results should not be sensitive to the boundary behaviorin the limit of V → ∞; so this convenient choice should be fine. With periodicboundary conditions, i.e., up(x + L) = up(x) for translation by L along anyspatial direction, the values of p are given by

pi =2πni

L(A-41)

(n1, n2, n3) are integers. The wave functions up(x) obey the orthonormalityrelation ∫

V

d3x[u∗

p(i∂0up′) − (i∂0u∗p)up′]

= δp,p′ (A-42)

where δp,p′ denotes the Kronecker δ’s of the corresponding values of ni’s, i.e.,

δp,p′ = δn1,n′1δn2,n′

2δn3,n′

3(A-43)

In the limit of V → ∞, we have

δp,p′ → (2π)3

Vδ(3)(p − p′)∑

p

→∫

Vd3p

(2π)3(A-44)

The wave functions ψp (or up) are obviously solutions of the equation

i∂ψ

∂t=√−∇2 + m2 ψ (A-45)

The differential operator on the right-hand side is not a local operator; it hasto be understood in the sense of√

−∇2 + m2f(x) ≡∫

d3p

(2π)3eip·x√p2 + m2f(p) (A-46)

where

f(x) =∫

d3p

(2π)3eip·xf(p) (A-47)

542 Appendix:Relativistic Invariance

One can, of course, define a local differential equation whose solutions arethe wave functions (A-40). It is the Klein-Gordon equation

( + m2)ψ = 0 (A-48)

where is the d’Alembertian operator, = ∂µ∂µ = (∂0)2 − ∇2. Equation(A-48), however, has, in addition to (A-40), solutions of the form e−ipx withp0 = −

√p2 + m2. At the level of one-particle wave functions, it is difficult

to interpret such solutions. However, in quantum field theory, they can beinterpreted consistently and the Klein-Gordon equation becomes the basisfor the discussion of spin-zero particles.

A-5 Wave functions for spin-12

particles

For a spin- 12 particle, s = 1

2 , we have two spin states. Translations act as in(A-36) and the wave functions have two components, viz.,

ψp,r(x) = 〈x|p, s = 12 , r〉 (A-49)

(r = 1, 2.) There are now nontrivial D-matrices in the transformation law. Inthis case, they are the 2 × 2 rotation matrices

D(12 )(θ) = exp

(iσa

2θa)

(A-50)

where σa are the Pauli matrices

σ1 =(

0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)(A-51)

The wave functions are solutions to the equation

i∂ψr(x)

∂t=√−∇2 + m2 ψr(x) (A-52)

For many examples of spin- 12 particles of mass m, we also have antiparticles

of the same mass. The antiparticle wave functions φr(x) obey equation (A-52), with φr in place of ψr. One can then define a combined wave functionΨ(x) by

Ψ(x) =(

ψr(x)φ∗

r(x)

)(A-53)

Ψ(x) is a four-component column vector. It obeys the equation

i∂Ψ

∂t= γ0

√−∇2 + m2 Ψ (A-54)

where

A-6 Spin-1 particles 543

γ0 =

⎛⎜⎝1 0 0 00 1 0 00 0 −1 00 0 0 −1

⎞⎟⎠ (A-55)

This is one version of the Dirac equation for spin-12 particles, the so-called

Foldy-Wouthuysen representation. As with spin-zero particles and the Klein-Gordon equation, one can seek a local differential equation for Ψ(x). The localequation is the usual version of the Dirac equation. Instead of transforming(A-54) to a local form, it is easier to show that (A-54) follows from the Diracequation.

A-6 Spin-1 particles

For spin s = 1, we have three components and the wave functions have theform

ψp,r(x) = 〈x|p, s = 1, r〉 = e−ipxεr(p) (A-56)

(r = 1, 2, 3.) εr transforms as a vector under rotations, i.e., infinitesimally

δεr = ωrkεk (A-57)

εr behaves like the spatial components of a four-vector. A local spin-1 analogof the Klein-Gordon equation might look like

( + m2)ψr(x) = 0 (A-58)

The functions (A-56) are evidently solutions to (A-58) for any εr(p). Thisequation is, however, not manifestly covariant since only spatial componentsof a four-vector are involved. A local, manifestly covariant equation would be

( + m2)ψµ(x) = 0 (A-59)

(µ = 0, 1, 2, 3.) However, ψµ has one more component, namely ψ0, more thanwe need. Thus if one would like to have a local manifestly covariant equation,one can use (A-59), but must impose additional constraints on ψµ to eliminatethe unwanted degree of freedom. Alternatively, one may choose an equationdifferent from (A-59) with extra symmetries which help us to eliminate theunwanted degree of freedom. These extra symmetries are gauge symmetries.They arise, from the particle point of view, because of the mismatch of thenumber of physical spin states, namely, (2s+1), and the number required for asuitable Lorentz vector or tensor in terms of which a local manifestly covariantequation can be constructed. (Gauge symmetries also have a deep geometricinterpretation, as discussed in text.) Gauge symmetries are required for allspins s ≥ 1.

544 Appendix:Relativistic Invariance

A-7 Massless particles

Although we derived the Poincare algebra as the symmetry of the action for amassive point-particle, it holds for a massless particle as well. This can be seenby considering the symmetry of the equations of motion rather than the actionor by noting that massless particles obey ds = 0. (They follow null geodesicson a general spacetime.) Thus massless particles are also described by theUIR’s of the Poincare algebra. In this case, we have p2 = 0 and p0 > 0. Wedo not have the possibility of going to the rest frame of the particle. A generalsolution to p2 = 0, p0 > 0 can be constructed by Lorentz transformations ofa special vector p

(0)µ = (1, 0, 0, 1). In our discussion of massive particles, we

considered the transformations which left p(0)µ = (m, 0, 0, 0) invariant, viz.,

rotations. From the action of the rotations, via the use of Wigner rotations,we could obtain the action of a general Lorentz transformation on the states.One can do a similar construction for massless particles by considering thetransformations which leave the vector p

(0)µ = (1, 0, 0, 1) invariant or in other

words the isotropy group of this vector. Infinitesimally, these transformationsare given by those ωµν which obey ωµνp

(0)ν = 0. One can check that a general

Lorentz transformation which preserves p(0)µ is of the form

U ≈ 1 + i(ω01(M01 + M13) + ω02(M02 + M23) + ω12M12

)≈ 1 + i

(ω01T1 + ω02T2 + ω12M12

)(A-60)

Ti = M0i + Mi3, i = 1, 2, and M12 generates rotations around the directionof the spatial momentum p, in this case the z-axis. The commutation rulesamong these operators are

[M12, T±] = ±T±, [T+, T−] = 0 (A-61)

where T± = T1 ± iT2. We can also check that for the special choice of p(0)µ ,

W 2 is given byW 2 = T+T− (A-62)

One class of states and representations is obtained by

T+|p, λ〉 = 0, T−|p, λ〉 = 0, M12|p, λ〉 = λ|p, λ〉 (A-63)

This has W 2 = 0. In this case, since both W 2 and p2 are zero, and W · p = 0from (A-21), so we must have Wµ proportional to pµ. In fact, Wµ = λpµ.This equation gives an invariant definition of λ. We may write

λ =W 0

p0=

p · Jp0

(A-64)

λ is called the helicity of the particle. Since M12 generates rotations, in-variance under ω12 → ω12 + 2π gives λ = 0, ± 1, . . ., for single-valued

A-9 Isometries, anyons 545

representations and λ = ± 12 , ± 3

2 , . . ., for the double-valued representations.The photon (λ = ±1) and the graviton (λ = ±2) are examples of the realiza-tion of these representations. Just as in the massive case, one can constructa suitable Bν

µ such that pµ = Bνµp

(0)ν and Wigner rotations and the explicit

realization of a general Lorentz transformation on these states. We do notdiscuss these matters, since it is a little easier for these cases to obtain thewave functions and the transformations by solving local equations of motionfrom a field theoretic approach.

One can also construct representations for which T± are not representedby zero. However, such representations have an infinite number of polarizationstates and do not seem to be of any physical significance.

A-8 Position operators

We have not discussed the position yet as an operator to be included in thealgebra of observables. From the point of view of obtaining the representa-tions, this does not make much difference, xµ appear explicitly in the x-spacewave functions. However, xµ is not appropriate as a position operator. Forexample, x0 denotes time. It is, in the case of point-particles, just a vari-able parametrizing the path of the particle; to consider it as an operatorwould lead to difficulties of interpretation. Newton and Wigner have defineda proper notion of position operator and calculated it for various cases. Weshall not discuss this in detail. For the case of massive spin-zero particles,one can see quite easily that x is not appropriate. It is not self-adjoint withthe scalar product (A-26). A modification which gives a self-adjoint operatoris

x = i∇p − ip

2E(A-65)

This is appropriate in the sense that it coincides with the position operator ofnonrelativistic theory for small velocities; it is the center of mass for localizedwave packets. Equation (A-65) is the Newton-Wigner (NW) operator for thiscase. For particles with spin, the NW position operator has spin-dependentterms in general.

A-9 Isometries, anyons

It is interesting to follow the logic of the previous sections, viz., of under-standing the one-particle states in terms of representations of the symmetryalgebra, in some unusual situations. As an example, let us consider free parti-cle motion in a spacetime which is not necessarily flat Minkowski space. Theaction is given by S = −m

∫ √gµνdxµdxν , where gµν is the metric tensor.

The requirement that xµ → xµ + ξµ be a symmetry gives

546 Appendix:Relativistic Invariance

gαν∂ξα

∂xµ+ gµα

∂ξα

∂xν+ ξα ∂gµν

∂xα= 0 (A-66)

We can write this equation as

∇µξν + ∇νξµ = 0 (A-67)

where

∇µξν = ∂µξν − Γ αµνξα

Γ αµν = 1

2gαβ

(−∂gµν

∂xβ+

∂gβν

∂xµ+

∂gµβ

∂xν

)(A-68)

Γ αµν is the Christoffel symbol and ∇µξν is the covariant derivative of ξν .

Equation (A-67) is the Killing equation, the solutions ξµ give the isometriesor transformations which leave the distance ds (and hence the particle ac-tion) invariant. The isometries will form a group, the isometry group, of thespacetime of metric ds2 = gµνdxµdxν . From our general discussion of quan-tum mechanics, the Hilbert space for one-particle motion on such a spacetimewill be given by a UIR of the isometry group. A simple concrete example isprovided by anti-de Sitter spacetime, which has the metric

ds2 = dz20 − dz2

1 − dz22 − dz2

3 + dz24 ≡ ηµνdzµdzν + dz2

4 (A-69)

withηµνzµzν + z2

4 = R2 (A-70)

One can solve (A-70) explicitly in terms of local coordinates, valid in somecoordinate patch, as

z0 = R sin t, z4 = R cos t coshχ

(z1, z2, z3) = cos t sinh χ (cos θ, sin θ cosϕ, sin θ sin ϕ) (A-71)

Substitution of these in (A-69) will give a more standard four-dimensionalway of writing the metric. Using the presentation of the space in terms of(A-69), (A-70) with the auxiliary variable z4, we see that the isometries aretransformations which leave the quadratic form (ηµνzµzν + z2

4) invariant.These are the pseudo-orthogonal transformations forming the group SO(3, 2)(and some discrete symmetry transformations). Thus UIR’s of SO(3, 2) willdescribe possible types of particle motion on anti-de Sitter spacetime.

As another example, consider three-dimensional spacetime, with a metricof the form

ds2 = dt2 − gijdxidxj , i, j = 1, 2 (A-72)

For simplicity the nontriviality of the metric is restricted to the spatial di-mensions. For flat space, gij = δij , and the rotation δxi = θεijx

j is clearlya symmetry. The states are of the form |p, s〉 with the action of rotations inthe rest frame, for which pµ = p

(0)µ , given by

A-9 Isometries, anyons 547

U(θ)|p(0), s〉 = eisθ|p(0), s〉 (A-73)

If we allow multivalued representations, s need not be an integer. Particleswith any value of spin are generically called anyons. The quasi-particles rele-vant to the fractional quantum Hall effect are of this type. (For this system,the physics is essentially planar and so a two-dimensional description is rea-sonably accurate. Also a nonrelativistic approximation is quite adequate.)

If space is a sphere of radius R,

gijdxidxj = R2(dθ2 + sin2θdϕ2) (A-74)

in terms of the usual angular coordinates θ, ϕ on the sphere. The isometriesin this case are

ξ(1) = (−sinϕ,−cosϕ cotθ), ξ(2) = (cosϕ,−sinϕ cotθ), ξ(3) = (0, 1)(A-75)

The generators Lµ = −iξi(µ)(∂/∂xi) obey the angular momentum algebra

[Lµ, Lν ] = iεµναLα, (µ, ν, α = 1, 2, 3). In other words, the isometry groupof the two-dimensional sphere is SU(2), the angular momentum group. Uni-tary representations clearly require integer or half-odd-integer values of spin.Thus a spherical world cannot support anyons, at least with the actionS = −m

∫ds. It is also interesting to investigate what kind of anyons are

possible if the world is a two-dimensional torus of metric ds2 = dθ2 + dϕ2.

References

1. The uniqueness theorem on the representation of the Heisenberg algebrais due to M.H. Stone, Proc. Nat. Acad. Sci. USA, 16, 172 (1930); J. vonNeumann, Math. Ann. 104, 570 (1931).

2. The representation theory of the Poincare group is due to E.P. Wigner,Ann. Math. 40, 149 (1939).

3. The position operator was introduced by T.D. Newton and E.P. Wigner,Rev. Mod. Phys. 21, 400 (1949).

4. Isometries and the Killing equation are discussed in most books on thegeneral theory of relativity, see, for example, S.W. Hawking and G.F.R.Ellis, The Large Scale Structure of Space-time, Cambridge UniversityPress (1973). Anti-de Sitter space is also given here.

5. The Poincare group analysis for anyons in three dimensions is given inB. Binegar, J. Math. Phys. 23, 1511 (1982).

General References

1. Lowell S. Brown, Quantum Field Theory, Cambridge University Press(1992).

2. C. Itzykson and J-B. Zuber, Quantum Field Theory, McGraw Hill Inc.(1980).

3. J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons,Springer-Verlag (1955 & 1976).

4. Michio Kaku, Quantum Field Theory: A Modern Introduction, OxfordUniversity Press, Inc. (1993).

5. F. Mandl and G. Shaw, Quantum Field Theory, John Wiley (1984).6. Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum

Field Theory, Westview Press (1995).7. P. Ramond, Field Theory: A Modern Primer, Addison-Wesley Pub. Co.

Inc. (1990).8. Lewis H. Ryder, Quantum Field Theory, Cambridge University Press

(1985 & 1996).9. S. S. Schweber, An Introduction to Relativistic Quantum Field Theory,

Harper and Row, New York (1961).10. S. S. Schweber, QED and the Men Who Made It, Princeton University

Press (1994).11. Steven Weinberg, The Quantum Theory of Fields: Volume I Foundations,

Cambridge University Press, (1995);The Quantum Theory of Fields: Volume II Modern Applications, Cam-bridge University Press, (1996);The Quantum Theory of Fields: Volume III Supersymmetry, CambridgeUniversity Press, (2000).

12. Anthony Zee, Quantum Field Theory in a Nutshell, Princeton UniversityPress (2003).

13. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Claren-don Press (1996).

Index

A0 = 0 gauge, 78CPT theorem, 229S-matrix

matrix elements, 59operator formula, 72

S-matrix functionaland S-matrix elements, 60effective action, 126functional integral, 111general formula, 58operator formula, 74

ZN symmetry, 453ZN vortices, 453β-function

scalar field, 172Yang-Mills theory, 214

η′-mass, 294θ-parameter, 254, 351, 364, 495, 520’t Hooft algebra, 454’t Hooft effective interaction, 370’t Hooft loop operator, 452’t Hooft tensor, 366’t Hooft-Polyakov monopoles, 455Cerenkov radiation, 96

Gauge choicesunitary gauge, 265

Abelian Higgs model, 447Abrikosov-Nielsen-Olesen vortices, 447Adler-Weisberger sum rule, 276Anomalous dimension, 172Anomalous magnetic moment, 156Anomaly

UA(1) in QCD, 291UA(1) problem, 293’t Hooft’s matching conditions, 382Adler-Bardeen theorem, 287

Adler-Bell-Jackiw result, 285and functional measure, 375anomaly polynomial, 381Bardeen form, 288baryon number, 290Fujikawa’s method, 375global anomaly, 384in 2n dimensions, 381index theorem, 377lepton number, 290mixed anomaly, 381pion decay, 292quantization of coefficient, 380standard model, 288triangle diagram, 282Wess-Zumino condition, 286Wess-Zumino term, 380

Anti-de Sitter space, 544Anyons, 351, 508, 545Asymptotic freedom, 214Atiyah-Singer index theorem, 369

Bargmann states, 499Baryon number violation, 371Belinfante tensor, 36Betti number, 314Black body radiation, 407Bogoliubov’s recursion formula, 160Bogomol’nyi inequality, 366, 459Bogomol’nyi-Prasad-Sommerfield limit,

459Boltzmann equation, 426, 427BRST symmetry

charge operator, 197for action, 195on states, 198quartet fields, 198WT identity, 200

552 Index

Cabibbo-Kobayashi-Maskawa matrix,273

Canonical one-form, 18, 484Canonical transformations, 19, 484

Canonical two-form, 19, 483Cartan-Killing metric, 246Characteristic classes

A-genus, 335

Chern character, 335Chern class, 334Euler class, 336

Charge conjugation, 226Charge operator

complex scalar field, 33

Dirac field, 30, 33gauge theory, 185

Chern-Simons

2n + 1-form, 379action, 181five-form, 379level quantization, 517

theory, quantization, 513three-form, 295, 327, 363

Christoffel symbols, 309

Clifford algebracommutation rules, 3fundamental theorem, 4

Coherent states, 498Cohomology group, 313Cohomology of Lie groups, 314

Coleman-Weinberg potential, 129, 143Completeness

momentum eigenstates, 2plane wave spinors, 5

Complex geometryalmost complex structure, 311Kahler manifold, 312

Kahler potential, 313Newlander-Nirenberg theorem, 312Nijenhuis tensor, 312

Complex projective space, 247, 313,331, 504

Compton scattering

electrons, 92low-energy theorem, 95, 230scalar particles, 85

Confinementarea law, 445

dual superconductivity, 445hypothesis, 444

Conformal transformations, 40Connected Green’s functions, 119Coset manifolds, 245Coulomb Green’s function, 78Covariant derivative, 180, 309Cross section, general, 68Crossing symmetry, 463

d’Alembertian, 2Daisy diagrams, 435Darboux’s theorem, 487Debye screening

Abelian fields, 407nonabelian fields, 413WZW action, 416

Debye-Waller factor, 91Decay rate

general, 67time-dilation effect, 68

Density matrixaction for, 398at equilibrium, 399ensemble average, 397Liouville equation, 398scalar field, 400

Diffeomorphism, 298Differential forms

k-form, 300divergence δ, 306exact, closed, 304exterior algebra, 304exterior derivative, 301Hodge dual, 305integration, 302Lie derivative, 306one-form, 299Poincare lemma, 304pullback, 301Stokes’ theorem, 304volume form, 305

Dilatation current, 41Dirac field

Feynman propagator, 50generating functional, 53

Dirac monopolecharged particle, 506Dirac quantization, 332, 355, 508

Index 553

Dirac string, 324field, potentials, 324

Dolbeault complex, 500Drell-Hearn sum rule, 233

Effective actionS-matrix functional, 1261PI diagrams, 122flavor anomalies, 382for baryons, 258for pseudoscalar mesons, 257for skyrmions, 467low-energy theorem, 230QED, one-loop, 153scalar, one-loop, 129, 143

Effective chargeQED, 155Yang-Mills theory, 214

Einstein equation, 310Electrodynamics

S-matrix functional, 84action, 77BRST symmetry, 198charge conjugation, 226differential forms, 322electron self-energy, 149Euclidean QED, 112Maxwell equations, 77parity, 225photon self-energy, 150, 155QED action, 84quartet mechanism, 198time-reversal, 228vertex correction, 152Ward-Takahashi identity, 222

Electron self-energy, 149Energy-momentum

Belinfante tensor, 35electromagnetic field, 36general relativity, 38Poincare symmetry, 34scalar field, 23

Euclidean propagatorscalar field, 46

Euler number, 314, 337Exclusion principle, 15

Faddeev-Popovdeterminant, 191

ghosts, 192Fermion doubling problem, 477Ferromagnet, 252Feynman formula for denominators,

137Fiber bundles

connection, 329definition, 327homotopy sequence, 330Mobius strip, 328tangent, cotangent, 328

Fine structure constant, 86Flux, formula for, 68Frame fields, 308Frank-Tamm formula, 99Frobenius theorem, 299Functional integral

Dirac field, 108generating functional, 105lattice gauge theory, 475nonequilibrium, 423partition function, 429scalar field, 104Yang-Mills theory, 192

Furry’s theorem, 229, 282

Gamma matrices, traces, 93Gap equation, 435Gauge choices

A0 = 0 gauge, 78Landau gauge, 194radiation gauge, 79

Gauge fieldsdifferential forms, 325field strength, 181gauge potential, 180

Gauge potentialsspace of, 189, 358

Gauge principleelectrodynamics, 82general, 179

Gauge theoryBRST symmetry, 195configuration space, 190

Gauge transformationsspace of, 188, 358

Gauss law, 187Gauss linking number, 451Gauss-Bonnet theorem, 337

554 Index

Gell-Mann-Okubo mass formula, 259Generating functional

free Dirac field, 53free scalar field, 47, 49functional integral, 105, 108interacting scalar field, 56, 58Yang-Mills theory, 192

Geometric quantizationS2 and SU(2) algebra, 501and H1(M,R), 494and H2(M,R), 496Chern-Simons theory, 513coherent states, 498holomorphic polarization, 493polarization, 492prequantum operators, 491

Glueballs, 462Goldberger-Treiman relation, 259Gravity, Einstein equation, 310Gribov ambiguity

gauge fixing, 193nontrivial bundle, 332three dimensions, 334

Haar measure, 475Hard thermal loops, 413Heisenberg equation of motion, 24Helicity, 542Higgs mechanism, 264Higgs particle, 272High temperature

dimensional reduction, 433symmetry restoration, 434

Hodge dual, 305Holonomy, 392Homotopy groups

definition, 317exact sequence, 330, 362, 386, 456,

459Lie groups, 321spheres, 319

In-field operator, 72Instantons

θ-parameter, 522’t Hooft interaction, 370axial U(1) problem, 295baryon number violation, 371Bogomol’nyi inequality, 366

fermion functional integral, 369field configuration, 364index theorem, 367instanton number, 326, 364, 521

Isometry group, 544

Jacobi Θ-function, 520Jacobian variety, 519

Kahler manifolds, 312, 504Killing equation, 544Killing equation

conformal, 40Minkowski metric, 39

Klein-Gordon equation, 2Klein-Nishina formula, 94Kubo formula, 419Kubo-Martin-Schwinger condition, 404

Lamb shift, 156Landau damping, 421Landau gauge, 194Landau pole, 155Landau-Ginzburg theory, 447Large N expansion

chiral symmetry breaking, 464double line, 460factorization, 462

Lie derivative, 306Linking number, 451Loop expansion, 127Lorentz transformation

spinors, 4vectors, 533

Mandelstam variables, 260Massive vector particle, 205Matsubara frequencies, 430, 432Mermin-Wagner-Coleman theorem, 240Metric, 308Minimal coupling, 81Mott scattering, 87

Nielsen-Ninomiya theorem, 480Nonequilibrium phenomena

Boltzmann equation, 426, 427Kadanoff-Baym equations, 425time-contour, 423

Nonrenormalizable theories, 165

Index 555

One-particle irreducible diagrams, 122Optical theorem, 262Orientable manifold, 303

Parallel transportgauge theory, 182gravity, 310

Parity, 225Partition function

fermions, 402functional integral, 429lattice gauge theory, 475scalar field, 400

Path integralθ-parameter, 363and Π1(C), 349and H2(C), 351evolution kernel, 339for QCD, 362Huygens’ principle, 348Schrodinger equation, 342Wess-Zumino term, 351

Pauli-Lubanski vector, 535PCAC relation, 261Phase space

definition, 18light-cone quantization, 37Liouville measure, 487

Photoncovariant propagator, 81mode expansion, 79polarization vectors, 79self-energy, 150, 155transverse propagator, 80

Pion decay, 83, 95, 292Pion-nucleon interaction, 259Pion-pion scattering, 259Planck mass, 167Plasma oscillations, 411Poincare algebra, 533Poincare duality, 314Poincare lemma, 304Point-particle action, 532Poisson bracket, 21, 485Poisson bracket

light-cone quantization, 513WZW model, 526WZW model, light-cone, 527

Propagator

fermion, 50fermions, thermal, 403ghost, 194gluon, 194gluon, one-loop, 206gluon, pinching term, 213scalar field, 45scalar, thermal, 401thermal, imaginary-time, 430

Quantum chromodynamicsΛ-parameter, 215θ-parameter, 363, 520asymptotic freedom, 214axial U(1) problem, 293axial anomaly, 291chiral symmetry breaking, 255effective action for anomalies, 382path integral, 362

Quark masses, 253

Radiation gauge, 79Reduction formula, 60, 347Regulators

anomalies, 279counterterms, 167definition, 133dimensional regularization, 207Pauli-Villars, 279

RenormalizationZ-factors for scalar field, 134Z-factors, δm, for QED, 148Z-factors, Yang-Mills theory, 204BPHZ, 158counterterms, 157forest formula, 161QED, one-loop, 154recursion formula, 160Weinberg’s theorem, 164

Renormalization groupcritical exponents, 174scalar field, 168

Retarded functionsand finite T , 418

Retarded propagatorand interacting field, 73scalar field, 44

Rho meson decay, 99Ricci tensor, Ricci scalar, 310

556 Index

Riemann curvature, 308Riemannian manifold, 308Rutherford scattering, 89

Scalar fieldβ-function, 172anomalous dimension, 172Euclidean propagator, 46Feynman propagator, 45generating functional, 47, 49, 56, 58partition function, 400retarded propagator, 44thermal propagator, 401

Schrodinger field, 70Schwinger model, 395Schwinger-Dyson equations

ϕ4 theory, 125general form, 124nonequilibrium, 424

Sigma modelscoset spaces, 247functional integral, 114, 118principal chiral model, 247

Skyrmionsand large N , 467baryon number, 468spin and flavor, 472topological current, 467

Slavnov-Taylor identities, 202Spin-statistics theorem, 15, 31Spontaneous symmetry breaking

chiral currents, 260currents, 251effective action, 249Goldstone’s theorem, 242PCAC, 261

Standard modelCP -violation, 275W , Z, masses, 272action, 270baryon number violation, 371CKM matrix, 273mixed anomaly, 381particle content, 269symmetry restoration, 437Weinberg angle, 271, 272Yukawa terms, 272

Stokes’ theorem, 304Stone-von Neumann theorem, 532

Strong CP -problem, 364Structure factor, 92Symmetries

Goldstone realization, 218, 235of QCD, 254Wigner realization, 218

Symmetry restorationhigh T , 434standard model, 437

Symplectic structure, 19, 483

Thomson cross section, 87, 94Time-reversal, 228Topological manifold, 297Torsion tensor, 308

Uehling potential, 156Unitarity

bound for cross section, 263effective theory, 264general relation, 262nonrenormalizable theory, 166optical theorem, 262

Unitary gauge, 265

Vacuum diagrams, 64, 121Vacuum polarization, 150, 155Variational principle, 489Vector fields

definition, 298Hamiltonian, 485

Veneziano-Witten formula, 294Vertex functions, 59Vortices

ZN , 453Abrikosov-Nielsen-Olesen, 447

Ward-Takahashi identityO(N) theory, 221BRST symmetry, 200low-energy theorem, 230QED, 155, 222Yang-Mills effective action, 203

Wave functionChern-Simons theory, 515scalar field vacuum, 345

Weil homomorphism, 317, 335Weinberg angle, 271, 272Weinberg’s theorem, 164

Index 557

Wess-Zumino condition, 286Wess-Zumino-Witten model

action, 388canonical two-form, 525canonical two-form, light-cone, 527Dirac determinant in 2 dim., 394Kac-Moody algebra, 527level number, 389Poisson brackets, 526Poisson brackets, light-cone, 527Polyakov-Wiegmann identity, 389,

395Wick’s theorem, 60Wigner D-functions, 245, 472, 476, 537Wigner rotation, 537Wilson lattice action, 474Wilson loop operator

area law, 445

definition, 184lattice, strong coupling, 477

Winding number, 319, 321, 359

Yang-Mills theoryβ-function, 214action, 181, 326BRST symmetry, 195functional integral, 192

Yukawa interactionpion-nucleon, 259standard model, 272

Zero-point energyDirac field, 29Lorentz invariance of vacuum, 27scalars, 12

Zimmerman’s forest formula, 161