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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
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