Universidad de los Andes
Departamento de F´
ısica
Graduation Project
Anomalies in Quantum Field
Theory
Author:
Juan Sebasti´an Cruz
Supervisor:
Dr. Andr´es F. Reyes
Contents
1 Introduction 1
1.1 Classical Fields . . . 1
1.1.1 Calculus of Variations and Lagrangians . . . 2
1.2 Quantization and Scattering . . . 4
1.2.1 Scattering . . . 5
2 First Anomaly Example 9 2.1 Ward Identities . . . 9
2.2 The π0 →γγ decay . . . . 13
2.3 Anomaly example: Experimental remarks. . . 17
3 Algebraic Approach to Anomalies 21 3.1 Mathematical Structures concerning Anomalies . . . 21
3.2 The Schwinger Model (Simplified) . . . 31
3.2.1 Current commutator . . . 33
3.3 Final Remarks. . . 36
Appendix A The Neutral Pion π0 39
Appendix B Noether’s Theorem and Conservation Laws 43
Appendix C The Gamma Matrices 47
Bibliography 53
Acknowledgments
I want to thank Andr´es Reyes for his constant support and time dedication during the making of this document, as well as Juan Carlos Sanabria for proofreading and commenting.
I want to thank last but not least to my mom and dad and Laura who were always supportive and were there when I needed them the most.
Preface
As Bertlmann says (see [Ber00]), “Anomalies are the key to a deeper understand-ing of quantum field theory”, so followunderstand-ing him, the purpose of these dissertation is not only to study anomalies directly but to gain a better understanding of quantum field theory on the way.
So we should start explaining what is meant by quantum field theory (QFT). Such a theory can be described as a framework which uses continuous fields as its main elements to describe quantum mechanics. The principal advantage of QFT, over theories which do not make use of fields, is that systems consisting of an arbitrary number of particles can be treated successfully even in cases where such number is changing. Of particular importance is its application to high energy physics where QFT is able to implement special relativity within it and along with gauge theory they give birth to the standard model of particle physics.
Now turning to the main topic, we can informally describe an anomaly in QFT as a violation of a conservation law in the quantum version of a classical system which has a symmetry implying such law. Anomalies are therefore an important phenomenon of QFT that should be studied carefully given their dual role, in the sense that they sometimes explain experimental data and some other times they render the theory inconsistent.
We start our journey by first showing an explicit example of an anomaly which was one of the first to be identified and treated, the decay of a neutral pion into two photons. In order to gain some insight into what is coming, we introduce first the Ward identities and then proceed to calculate the factors involved in the identities.
The next step will be to consider an algebraic approach to the phenomena. We will introduce the mathematical background concerning Clifford Algebras, Spin
viii CONTENTS
Groups and CAR algebras needed for the treatment to follow and finish by showing the relation between anomalies and the mathematical machinery introduced.
Juan Sebasti´an Cruz January 23, 2014
CHAPTER
1
Introduction
The idea of this chapter is to make a brief exposition of the basic ideas behind quantum field theory (QFT). The reader can find more detailed textbooks on quantum field theory at the end of the document such as [CL88],[IZ80] or [Pok00].
1.1
Classical Fields
In classical mechanics one handles particles as localized on a point of space and time and then considers the trajectory such a point follows according to the laws of physics. In this description one can speak in terms of canonical variables x(t) and p(t) where for example in describing a planetary system we think of a whole planet as a point (its center of mass) located at time t on position x(t). In this context it is possible to describe the system using a point-like particle approach, however if the number of bodies of interest is higher the problem becomes quite complicated.
For the last kind of systems it is useful to suppose that matter can be treated as a continuous substance with general properties at large scales. Under this hypothesis a system with a possibly infinite number of degrees of freedom can be handled by the consideration of a continuous function or classical field. This fields at this point are commutative since they appear on classical situations were there is no need to introduce specific commutation relations.
2 Introduction
Besides the ability to manage a great number of particles, another advantage of the theory of fields is that particles and forces are described on the same footing, meaning that they are both fields which can be coupled to each other. Our model theory at this point would be electromagnetism together with Dirac’s equation where one can formulate a Lagrangian density that leads to the Maxwell’s equations and Dirac’s equation and notice how the electric and magnetic fields interact with fermions. This of course would be a starting model since there has not been any process of quantization.
1.1.1
Calculus of Variations and Lagrangians
The laws of physics take advantage of a mathematical construction named the
Action. The action is then a functional which takes in our case some fields and returns a number. It is commonly denoted by S and has the form
S=
Z
dtL=
Z
d4xL (1.1)
whereLandLare again functionals and are called theLagrangianandLagrangian densityrespectively. In order to obtain a physical theory one must require that the Lagrangian satisfies certain conditions, among them: Poincar´e invariance, locality and invariance under symmetry transformations possessed by the physical system.
Given an action of a physical system we are then able to get the corresponding equations of motion (Euler-Lagrange) and conservation laws through Noether’s Theorem. It is possible as well to employ path integrals or canonical quantization to obtain a quantum version of the system. As an illustration consider a simple action depending on a real field Φ(x),
S =
Z t1
t0
dt
Z
V d
3x
L(Φ(x), ∂µΦ(x)). (1.2)
This is telling us that the dynamics of the system evolving from t0 to t1 is such that δS must be zero. Considering an arbitrary variation of the field is then
1.1. CLASSICAL FIELDS 3
such that on the boundary Ω of the region of integration the variation vanishes (in analogy to the vanishing of a variation of a curve at the endpoints). Then we have
δL=L(Φ′(x), ∂µΦ′(x))− L(Φ(x), ∂µΦ(x))
= ∂L
∂ΦδΦ +
∂L ∂(∂νΦ)
δ(∂νΦ)
= ∂L
∂ΦδΦ +
∂L ∂(∂νΦ)
∂ν(δΦ)
= ∂L
∂Φ −∂ν
∂L ∂(∂νΦ)
!
δΦ +∂ν
∂L ∂(∂νΦ)
δΦ
!
We get then that
δS =
Z
d4x ∂L
∂Φ −∂ν
∂L ∂(∂νΦ)
!
δΦ
since the last term of δL was a divergence that vanishes at the boundary of the volume of integration. Since the above equation must hold for all variations we can conclude that the accompanying factor must always be zero, thus we obtain the Euler-Lagrange equation for this Lagrangian. It is worth noticing that the lagrangian density can be changed then by any other density differing only in a divergence term and the Euler-Lagrange equations will not change.
We now present some of the equations we are interested in and refer the reader to appendixB for a brief description of Noether’s theorem.
Our first and main example of an anomaly will be that of the π → 2γ decay so we must be concerned primarily on the following equations:
(+m2)φ= 0 Klein-Gordon’s Equation (1.4)
(i/∂−m)ψ = 0 Dirac’s Equation (1.5)
They can be obtained from the following Lagrangian densities
LKG=
1 2
∂µφ∗∂
µφ−m2φ∗φ
(1.6)
LQED = ¯ψ(i /Dµ−m)ψ−
1 4FµνF
µν
4 Introduction
where D/µ stands for the covariant derivative.
1.2
Quantization and Scattering
We now consider the simplest example where we can express the ideas behind quantization and scattering. That is a real, scalar field. Consider a real scalar field φ(t, ~x) satisfying Klein-Gordon’s equation (1.4). A general solution to such equation has the form
φ(x) = 1
(2π)32
Z d3~p
2Ep
(eip·x|p0=E
pa
∗
p+e
−ip·x
|p0=E
pap) (1.8)
whereapanda∗pdepend on the initial conditions of the specific situation. Denoting
by Π the conjugate momentum of the field, we have
Π = ∂L
∂φ˙ = ˙φ. (1.9)
Now the idea of quantization is introduced by requiring canonical commutation relations
[φ(x),Π(y)]|x0−y0 =iℏδ(~x−~y). (1.10)
By substituting the previous solution and momentum we arrive to the condition for the ap’s which from now on will be treated as operators,
[ap, a†p] = 2Epδ(~x−~y) (1.11)
which leads to a particle interpretation. We can think of one particle states as being generated by a†
p|0iand a Hamiltonian which resembles number operators
H = 1
2
Z d3~p
2Ep
:a†pap+apa†p : (1.12)
Now using (1.11) we can compute the commutation relations satisfied byφ at equal time and get
[φ(x), φ†(y)] =i∆
1.2. QUANTIZATION AND SCATTERING 5
where ∆0 is a distribution solution of the homogeneous Klein-Gordon equation,
commonly referred to as the Pauli-Jordan distribution. Another relevant quantity is the propagator, defined as the vacuum expectation value of the time-ordered product of two field operators:
h0|T(φ(x)φ(y))|0i= i (2π)4
Z
d4k e
−ik(x−y)
k2 −m2+iǫ (1.14)
1.2.1
Scattering
We present in this section the common approach towards scattering which leads eventually to Feynman rules which are extremely useful in computing actual cross-sections and decay rates, however the justification and implementation has many problems at a formal level (see [Joa75]). We start by explaining how one can obtain the Dyson series which is the cornerstone of the perturbative calculations.
We first consider the interaction picture for quantum dynamics. Suppose the Hamiltonian H of a system can be decomposed into an unperturbed and time-independent partH0 plus a perturbation V, namely
H=H0+V. (1.15)
Assume we know the solutions to the eigenvalue equation of the unperturbed part
H0Φα =EαΦα (1.16)
In order to separate the free motion of the system we take the Schr¨odinger state vector Ψs(t) and define
Ψ(t) =eiH0(t−t0)Ψ
s(t) (1.17)
in the interaction picture. Using the Schr¨odinger equation we arrive to the Tomonaga-Schwinger equation
i∂Ψ(t)
6 Introduction
where
V(t) =eiH0(t−t0)V e−iH0(t−t0).
The state Ψ(t) now depends entirely on the interaction, and the observables are then operators which are subject to equations of motion of the unperturbed part only. Observables are related then to their Schr¨odinger picture by
A(t) =eiH0(t−t0)A
se−iH0(t−t0). (1.19)
Now let us define the evolution operator according to the following expression:
Ψ(t) =U(t, t′)Ψ(t′). (1.20)
We have the three following properties
U(t, t) =1, (1.21)
U(t, t) =U(t, t′′)U(t′′, t′), (1.22)
and (1.23)
U−1(t, t′) =U(t′, t). (1.24)
Using the defining equation in (1.18) we get
i∂ ∂tU(t, t
′
)Ψ(t′) =V(t)U(t, t′)Ψ(t′) (1.25)
from which we can conclude that
i∂ ∂tU(t, t
′
) =V(t)U(t, t′). (1.26)
Now using the first property of the list above as an initial condition we can write
U(t, t′) =1−i Z t
t′ V(t1)U(t1, t
′
)dt1 (1.27)
1.2. QUANTIZATION AND SCATTERING 7
Figure 1.1: Sketch of idea behind the interaction picture and the scattering oper-ator
approximation would be
U(0)(t, t′) =1, (1.28)
the next one would be
U(1)(t, t′) =1−i Z t
t′ V(t1)dt1 (1.29)
and the second order would be
U(2)(t, t′) =1−i Z t
t′ V(t1)dt1+ (−i) 2Z t
t′ dt1
Z t
t′ dt2V(t1)V(t2). (1.30)
We assume the convergence of such expansion and write the evolution operator as
U(t, t′) =
∞
X
n=0
Un(t, t′), (1.31)
where we have for n≥1
Un(t, t′) = (−i)n Z t
t′ dt1
Z t1
t′ dt2· · ·
Z tn−1
t′ dtnV(t1)V(t2)· · ·V(tn). (1.32)
8 Introduction
time-ordering operator T is introduced
T[Ai(ti)Aj(tj)· · ·Ak(tk)] = A1(t1)A2(t2)· · ·An(tn) where t1 > t2 >· · ·> tn.
(1.33) Studying the regions of integration a clever observation leads to
Un(t, t′) =
(−i)n
n!
Z t
t′ dt1
Z t
t′ dt2· · ·
Z t
t′ dtnT[V(t1)V(t2)· · ·V(tn)]. (1.34)
and furthermore to (where we have taken the limitst′
→ −∞and t→ ∞)
S=
∞
X
n=0
(−i)n
n!
Z
· · ·
Z
dt1dt2. . . dtnT[V(t1)V(t2)· · ·V(tn)]. (1.35)
This last expression is known as the Dyson series and leads to the Feynman rules when it is extended to fields appropriately and complemented with Wick’s theorem.
CHAPTER
2
First Anomaly Example
2.1
Ward Identities
The first thing we should do is to understand the Ward identities involved. In general a Ward identity is a relation which must be satisfied by the Green functions as soon as we impose the validity of the classical conservation laws. In fact it can be shown that such identities are a necessary condition for the theory to be renormalizable.
Consider the following vacuum expectation value for a current jµ and some
operatorsOi:
h0|T(jµ(x0)O1(x1). . . On(x
n))|0i, (2.1)
we recall the definition of the time ordering operator for the case of two operators and we show a formal expression for the case of a finite product
T(A(t1)B(t2)) = θ(t1−t2)A(t1)B(t2)∓θ(t2−t1)B(t2)A(t1),
where θ denotes the Heaviside function and the sign depends on the particle statistics. We can write a formal expression for an arbitrary number of operators in the following way:
T O0(x0). . . On(xn) = X
σ∈Sn+1
n Y
i=0
10 First Anomaly Example
For the case of the expectation value above we can differentiate the expression with respect to the argument of the current and obtain the following expression:
∂x0
µ h0|T(j µ
(x0)O1(x1). . . On(xn))|0i=
=h0|T(∂x0
µ j
µ(x0)O1(x1). . . On(x
n))|0i (2.3)
+
n X
i=1
h0|T[j0(x0), Oi(xi)]δ(x00−x0i)O
1. . . Oi−1Oi+1. . . On
|0i.
Once the classical conservation law for the current is introduced and respecting the commutator algebra the expression obtained from (2.3) is what is called a Ward identity (WI).
Our first specific example from QED relates the vertex function and the prop-agator and was derived by Ward and Takahashi. This identity is used in proving that the renormalization constants of the vertex function and the fermion wave function are equal. We shall start from the 3-point function
τµ(x, y, z) =
h0|T jµ(z)ψ(x) ¯ψ(y)
|0i (2.4)
wherejµ= ¯ψγµψ and ψ satisfy the Dirac equation, namely (i/∂−m+e /A)ψ = 0.
We make use of formula (2.3) to differentiate the 3-point function and we get
∂z µτ
µ(x, y, z) =∂z µh0|T j
µ(z)ψ(x) ¯ψ(y)
|0i =h0|T ∂µzj
µ
(z)ψ(x) ¯ψ(y)|0i
+h0|T[j0(z), ψ(x)]δ(z0 −x0) ¯ψ(y)|0i
+h0|T[j0(z),ψ¯(y)]δ(z0−y0)ψ(x)|0i.
Our task is then to calculate the commutators
[j0(z), ψ(x)]δ(z0−x0) = [ψ†(z)ψ(z), ψ(x)]δ(z0 −x0)
= (ψ†(z)ψ(z)ψ(x)−ψ(x)ψ†(z)ψ(z))δ(z0−x0) =−(ψ(x)ψ†(z) +ψ†(z)ψ(x))ψ(z)δ(z0−x0)
=−{ψ(x), ψ†(z)}ψ(z)δ(z0 −x0) =−ψ(z)δ4(z−x),
2.1. WARD IDENTITIES 11
where the fermion commutation relations
{ψ(x), ψ(y)}= 0 {ψ(x), ψ†(y)}|x0=y0 =δ
4(x
−y) (2.5)
have been used.
Analogously we obtain
[j0(z),ψ¯(y)]δ(z0−x0) = ¯ψ(z)δ4(z−y),
so returning to our expression of the differentiated 3-point function and substi-tuting the commutators, we have:
∂z
µτµ(x, y, z) =h0|T ∂µzjµ(z)ψ(x) ¯ψ(y)|0i
− h0|T ψ(z) ¯ψ(y)|0iδ4(z−x) +h0|T ψ(x) ¯ψ(z)|0iδ4(z−y).
To this last equation we apply the conservation of the vector current ∂µjµ = 0
(see AppendixB) and we arrive to the vector Ward identity (VWI) inx-space
∂µzτ µ
(x, y, z) =−iSF(z−y)δ4(z−x) +iSF(x−z)δ4(z−y) (2.6)
where SF is the fermion propagator, which is
SF(x−y) :=ih0|T ψ(x) ¯ψ(y)|0i=
1 (2π)4
Z
d4p (/p+m)e
−ip·(x−y)
p2−m2+iǫ (2.7)
Consider the equation (2.6) now in momentum space
(pµ−p′µ)τµ(p, p′) =SF(p)−SF(p′) (2.8)
recalling the definition of the vertex function
Γµ(p, p′) :=− τ
µ(p, p′)
SF(p)SF(p′)
(2.9)
and substituting on the VWI we get
(pµ−p′µ)Γ
µ(p, p′) = S−1
F (p)−S
−1
F (p
12 First Anomaly Example
and hence
Γ(p, p′) = S
−1
F (p)−S
−1
F (p′)
pµ−p′µ
= ∆S
−1
F (p)
∆pµ
. (2.11)
Rewriting our result we obtain the usual form of the Ward identity
Γ(p, p′) = ∂
∂pµ
SF−1(p). (2.12)
The case shown above is just a simple example of how such identities can be obtained. We now turn to consider the 3-point functions that we will need for the case study that follows:
h0|T jµ(x)jν(y)jλ5(z)|0i (2.13)
h0|T jµ(x)jν(y)P(z)|0i. (2.14)
Herej5
λ denotes an axial current andP(z) a pseudoscalar one. We use the following
notation for the momentum space version of the 3-point functions above
Tµνλ(k1, k2, q) := i Z
d4xd4yd4z eik1x+ik2y−iqz
h0|T jµ(x)jν(y)jλ5(z)|0i (2.15)
Tµν(k1, k2, q) := i Z
d4xd4yd4z eik1x+ik2y−iqz
h0|T jµ(x)jν(y)P(z)|0i (2.16)
We now follow the same procedure, so we first differentiate the 3-point function (2.13), however we take advantage of the Fourier transform properties to write
qλT µνλ =
Z
d4xd4yd4z eik1x+ik2y−iqz∂λ
zh0|T jµ(x)jν(y)jλ5(z)|0i, (2.17)
and using the expression (2.3) we arrive at the following formula (where we have ignored Schwinger terms)
qλT µνλ =
Z
d4xd4yd4z eik1x+ik2y−iqzh0|T j
µ(x)jν(y)∂zλjλ5(z)|0i. (2.18)
Again, applying the classical conservation law for the axial current, we get
qλTµνλ= 2mi Z
d4xd4yd4z eik1x+ik2y−iqz∂λ
2.2. THEπ0→γγ DECAY 13
that is
qλTµνλ = 2mTµν, (2.20)
which is called axial Ward identity (AWI). Analogously we can differentiate the vector current instead and applying the conservation law we get the vector Ward identity (VWI)
k1µTµνλ =k2νTµνλ= 0 (2.21)
2.2
The
π
0→
γγ
decay
Our goal in this section is to calculate the amplitudes Tµνλ and Tµν defined in
(2.15),(2.16) respectively, by using the Feynman rules. We will see that the AWI will not be satisfied after regularizing, this situation is usually known as the ABJ anomaly. We first build our diagrams of study (figure 2.1) so that the expression forTµνλ will come out of applying the Feynman rules to each diagram and adding
both of them.
q
p−q k1
p−k1
k2 p
π0
γν γ
γµ γ
γλγ5
(a) Primer diagrama del de-caimiento
q π0
γν γ, k2
γµ γ, k
1
(b) Segundo diagrama del de-caimiento
Figure 2.1: Feynmann diagram corresponding to the pion (π0) decay into two
photons (γ)
We obtain the following expressions for both Tµνλ from the diagram and for
14 First Anomaly Example
one,
Tµνλ =−i
Z d4p
(2π)4tr i
/p−mγλγ5 i
/p−/q−mγν i
/p−k/1−mγµ
!
+
k1 ↔k2
µ↔ν
,
Tµν =−i
Z d4p
(2π)4tr i
/p−mγ5
i
/p−/q−mγν i /p−/k1−m
γµ !
+
k1 ↔k2
µ↔ν
whereq=k1+k2 and the parenthesis denote the same first term but changing the indicated variables. To manage this expressions we use the fact that{γ5, γµ}= 0
(see Appendix C) so that for any slashed four-vector we have /qγ5 = −γ5/q. We employ the following identity that can be easily verified
/qγ5 = 2mγ5+ (/p−m)γ5+γ5(/p−/q−m). (2.22)
Since we want to check the AWI we multiply our expression for Tµνλ by qλ and
using the fact that the trace is linear, we get the factor qλγ
λγ5 = /qγ5 inside the
trace and we can use the identity. This procedure leaves us with three different terms in the integral forTµνλ (plus another three interchanging the variables). We
can identify the first one as 2mTµν so let us write
qλT
µνλ = 2mTµν +R1µν+R
2
µν, (2.23)
where
R1µν =
Z d4p
(2π)4tr
1
/p−/k2−mγ5γν
1
/p−/q−mγµ
− 1
/p−mγ5γν
1
/p−k/1−mγµ
! (2.24)
R2
µν =
Z d4p
(2π)4tr
1
/p−/k1−m γ5γµ
1
/p−/q−mγν
− 1
/p−mγ5γµ
1
/p−k/2−mγν
!
.
(2.25)
To check if the AWI is satisfied or not we must study the behavior of equations
2.2. THEπ0→γγ DECAY 15
by
p→p+k2 and p→p+k1,
which would cancel out the integrals, however they are in fact linearly divergent, so the variables cannot be shifted.
We now consider an example of a linearly divergent integral in one dimension and then generalize it to higher dimensions. Let
∆(a) =
Z ∞
−∞dx[f(x+a)−f(x)]
=
Z ∞
−∞dx[af
′(x) + a2
2!f
′′(x) +
· · ·] (2.26)
=a[f(∞)−f(−∞)] + a
2
2![f
′
(∞)−f′(−∞)] +· · ·.
If ∆(a) is at most a logarithmically divergent integral then we have thatf(±∞) =
f′(
±∞) = · · · = 0 and ∆(a) = 0 (in these cases the )variable can be shifted). If on the contrary the integral is linearly divergent, it is possible that a contribution from the surface term appears, namely
∆(a) =a[f(∞)−f(−∞)] (2.27)
may be nonzero. Now consider the n-dimensional case
∆(a) =
Z
dnx[f(x+a)
−f(x)]
=
Z
dnx[aµ∂
µf(x) +aµaν∂µ∂νf(x) +· · ·]. (2.28)
By applying Gauss divergence theorem and considering only the first term in the expansion above, we can write, for the case n = 4,
Z
d4x∂µ(aµf(x)) = lim R→∞
Z
S3(a
µf(x R))
Rµ
R dΩ
3 =aµ lim R→∞
Rµ
R
Z
S3f(xR)dΩ
3
=aµlim R→∞
Rµ
R S
3f(x
R)
whereS3 = 2π2R3 is the surface of a 3 dimensional sphere of radius R andf(x
R)
16 First Anomaly Example
last expression in (2.28) and taking only the first term, we arrive to
∆(a) =i2π2aµ lim
R→∞RµR
2f(x
R). (2.29)
(the extraicomes from a wick rotation (x4 =ix0)). Recognizing that the expres-sions (2.24) and (2.25) are of the type described above, we get
R1µν = Z
d4p[f(p−k2)−f(p)],
R2µν =
Z
d4p[f(p−k1)−f(p)],
with
f(p) = 1
(2π)4tr
1
/p−mγ5γν
1
/p−k/1−mγµ
!
= 1
(2π)4tr
(/p+m) (p2−m2)γ5γν
(/p−/k1+m) ((p−k1)2−m2)γµ
!
. (2.30)
The properties of the γ-matrices (see Appendix C) can now be exploited to find that the only term that is not zero is the one containing both momentum factors. We write term by term of the numerator
pβpαtr(γ
βγ5γνγαγµ) = 4iǫβναµpβpα = 0,
pβk1αtr(γβγ5γνγαγµ) = 4iǫβναµpβkα1,
pβtr(γ
βγ5γνγµ) = 0,
(pα
−kα
1)tr(γ5γνγαγµ) = 0,
tr(γ5γνγµ) = 0.
This leaves us with the following expression after considering that in the limit onlyp4 matters in the denominator, the other possible terms diverge, (2.29)
R1µν =
1
2π2ǫβναµk
α
1kλ2plim→∞ gβσp
λpσ
p2 (2.31)
2.3. ANOMALY EXAMPLE: EXPERIMENTAL REMARKS 17
directions). Performing such a limit one obtains
lim
p→∞
pλpσ
p2 =
gλσ
4 (2.32)
and at last we get for both integrals (just by interchanging characters)
R1µν =R2µν =−
1
8π2ǫµναβk
α
1k
β
2. (2.33)
So that the anomalous AWI is
qλTµνλ = 2mTµν−
1
4π2ǫµναβk
α
1k
β
2. (2.34)
2.3
Anomaly example: Experimental remarks
In order to calculate the explicit decay rate we recall the transition matrix element for the decay in question
hγ(ǫ1, k1), γ(ǫ2, k2)|π0(q)i= (2π)4δ4(q−k1−k2)ǫµ1(k1)ǫν2(k2)Γµν(k1, k2, q) (2.35)
with
Γµν(k1, k2, q) =e2i Z
d4x d4yeik1x+ik2y
h0|T[jµ(x)jν(y)]|π0(q)i. (2.36)
According to the negative parity of the pion (see Appendix A) the amplitude above has a general structure of the form
Γµν(k1, k2, q) = Γ(q2)ǫµναβkα1k2β, (2.37)
which will be used later on to find the decay rate we look for. Using the LSZ reduction formula (Lehmann, Symanzik and Zimmermann) (see [Ber00]) we get
Γµν(k1, k2, q) = e2 Z
d4yeik2y
Z
18 First Anomaly Example
which can be turned into
Γµν(k1, k2, q) =e2(−q2 +m2π) Z
d4yeik2y
Z
d4ze−iqzh0|T[φπ(z)jµ(x)jν(y)]|π0(q)i
by integration by parts. At this point we compare the last expression with the one we have being dealing with, namely
qλT µνλ =
Z
d4y d4zeik2y−iqz
h0|T[∂λ
zjλ5(z)jµ(x)jν(y)]|π0(q)i. (2.38)
It turns out that at the time Steinberger was doing his computations the successful hypothesis (so far) was the partially conserved axial current (PCAC) which coupled the pion field to the non-conserved axial current through
∂λjλ5a =fπm2πφ a
π(z), (2.39)
where fπ = 93 MeV is the pion decay constant which can be measured in the
decay π+ →+ν
µ and a = 3 corresponds to the SU(2) index of the neutral pion.
Using (2.39) we have then that
qλTµνλ =
fπm2π
e2(−q2 +m2
π)
Γµν. (2.40)
and we encounter ourselves with the Sutherland-Veltman paradox (See [Vel67] and [Sut67]). Since the left hand side has no poles in the limit q → 0 it has to vanish, therefore the transition element Γ must go to zero as well, so that
hγγ|π0i ∼Γ(q2 =m2
π)≈Γ(q2 = 0) = 0 (2.41)
implying that the pion shouldn’t decay into two photons, but it does!
By considering the anomaly in the AWI Adler was able to explain the obser-vations by modifying the PCAC hypothesis[Adl69],[Adl70], for the neutral pion
∂µjµ5(z) =fπm2πφπ(z) +
α
8πǫ
µναβ
2.3. ANOMALY EXAMPLE: EXPERIMENTAL REMARKS 19
where the current is (using doublet notation)
jµ5(3) = ¯ψγµγ5 σ3
2 ψ. (2.43)
Correcting the anomaly relation we derived earlier we arrive to
qλT µνλ =
fπm2π
e2(−q2+m2
π)
Γµν −
c
2π2ǫµναβk
α
1k
β
2 (2.44)
It is possible to get the value of c = 1
2 (see [GB01]) using the quark model
and the hypothesis that there are 3 colors of quarks. We can finally compute the corrected decay rate. First note that for a soft pion we have
lim
q→0Γµν(k1, k2, q) = e2c
2π2f
π
ǫµναβkα1k2β. (2.45)
This implies, comparing with (2.37), that
Γ(q2 = 0) = e
2c
2π2f
π
(2.46)
So
Γ(π0 → γγ) = 1 2mπ
Z d3k
1
(2π)32k0 1
d3k 2
(2π)32k0 2
(2π)4δ4(q−k1−k2)
· X
polarizations
|ǫµ1ǫν2ǫµναβk1αk2βΓ(q2)|2
= α
2m3
π
64π3f2
π
= 7.63 eV.
CHAPTER
3
Algebraic Approach to Anomalies
The following chapter shows how certain algebraic structures appear when dealing with second quantization, quantum field theory and in general many particle systems and how this structures can lead to an understanding of the anomalies. The mathematical background is briefly explained followed by the exposition of a particular example in which we can demonstrate the use of such a structure. The main concepts are reviewed based principally on [GBVF01] and [GBV94]. The reader is also referred to [Bla06],[Gar11] for more formal details and proofs.
3.1
Mathematical Structures concerning
Anoma-lies
When working with fermions, we start by considering a real vector space V with a bilinear symmetric form g. We should have in mind that later on V will be identified in the next section with the space of solutions to the Dirac equation, considered as a real vector space. In this section we introduce the terminology needed to understand the bigger scheme of quantization and treatment of the anomalies.
22 Algebraic Approach to Anomalies
orthogonal complex structure means a linear operatorJ :V →V satisfying
J2 =−I, and g(Ju, Jv) = g(u, v) for u, v ∈V (3.1)
By introducing the rule (α+iβ)v :=αv+βJv and the following inner product
hu|viJ :=g(u, v) +ig(Ju, v) (3.2)
we make (V, g, J) a complex Hilbert space.
Definition 3.1.2 (Orthogonal group).
O(V) :={m∈GLR(V) : ∀u, v ∈V g(mu, mv) =g(u, v)} (3.3)
Let us denote byJ(V) the set of all orthogonal complex structures forV. We note J(V)⊆o(V), where o(V) ={X :V →V real-linear| ∀v, w∈V g(v, Xw) +
g(Xv, w) = 0} is the Lie algebra of O(V). We can view the complexification
VC = V ⊕iV as a complex Hilbert space under the following positive definite
form:
hhw1|w2ii:= 2g(w1∗, w2) (3.4)
We define now UJ(V) ⊆ J(V) as the isotropy group of J in (V;g, J) that is
the set of vectors such that
hhJu|Jvii=hu|vi (3.5)
or equivalently the unitary group of VJ.
We are interested now in a particular type of elements of the orthogonal group, first the Hilbert-Schmidt norm is defined:
Definition 3.1.3 (Hilbert-Schmidt norm). Let H be a Hilbert space and let
A:H →H be a linear operator. The Hilbert-Schmidt norm is defined by
kAk2HS = tr|(A
∗
A)|=X
i∈I
kAeik2 (3.6)
3.1. MATHEMATICAL STRUCTURES CONCERNING ANOMALIES 23
countable. An operator A whose Hilbert-Schmidt norm is finite is said to be of Hilbert-Schmidt type or simply Hilbert-Schmidt.
Definition 3.1.4 (Restricted Orthogonal Group and Lie Algebra). It is denoted by O′
J(V) and consists of the elements m ∈ O(V) such that [J, m] is of
Hilbert-Schmidt type. Furthermore the restricted orthogonal Lie algebra o′J(V) is the
set of skew-symmetric bounded real-linear operators X on V for which [J, X] is Hilbert-Schmidt.
Analogously we define the restricted set of complex structures
J′(V) := {J′ ∈ J(V) : J −J′ is Hilbert-Schmidt} (3.7)
In this sense we can identify UJ(V) as the set {g ∈O(V) : [J, g] = 0}.
The next step is to build the (fermion) Fock space. Suppose we are given a triple (V, g, J) where V is a real vector space, g a symmetric bilinear form and
J a orthogonal complex structure so that we can regard (V, g, J) as a complex Hilbert space. We recall the definition of the exterior algebra and endow it with an inner product.
Definition 3.1.5 (Exterior Algebra).
^∗
V :=
∞
M
n=0
n ^
V (3.8)
By conventionV0
V =C. Let Ω (the vacuum) be a fixed vector inV0
V of unit norm. An inner product is then defined by
hu1∧ · · · ∧um|v1∧ · · · ∧vni:=δmndet(huk|uli) (3.9)
Definition 3.1.6(Antisymmetric Fock Space). Denoted byFJ(V), it corresponds
to the completion to a Hilbert space of V∗
V.
To give FJ(V) a basis first fix a basis {ei} for (V, g, J). A basis for FJ(V)
consists of ǫk := ek1 ∧ · · · ∧ ekr where K = {k1, . . . , kr} is a finite subset of
24 Algebraic Approach to Anomalies
The fermion Fock space can be split by even and odd components, that is
FJ(V) =F0 ⊕ F1 where F0 is the completion of L∞k=0
V2k
V and F1 is the
com-pletion ofL∞ k=0
V2k+1
V.
We leave aside the complex structure for the moment, and concentrate on another aspect which is important for the example that follows. Let (V, g) be a real vector space with a symmetric bilinear form as before.
Definition 3.1.7 (Field Algebra). The field algebra over (V, g) is defined as Cl(V, g) ⊗ C, completed with respect to the natural C∗-norm induced by the
inner product (defined using inductive limit[Emc09])
ha|bi:=τ(a∗b) =X K
¯
aKbK (3.10)
whereτ is the trace that is equal to the constant term. The field algebra is denoted by A(VC).
We have then a linear map B :VC →A(VC) we interpret as the fermion field,
that satisfies
B(w∗) =B(w)† and
{B(v), B(v′)
}= 2g(v, v′) (3.11)
Remark. AnyC∗-algebra generated by a linear mapB′ that satisfies the properties
above turns out to be isomorphic to that generated by B.
In order to understand the field algebra we first need to review the concepts related with the Clifford algebra.
Definition 3.1.8 (Clifford Algebra (Real)). Let V be a real vector space and consider V∗
V. Define the following linear mappings:
ε :V −→EndR(
^∗
V)
v 7−→ε(v)
3.1. MATHEMATICAL STRUCTURES CONCERNING ANOMALIES 25
and
ι:V −→EndR(
^∗
V)
v 7−→ι(v)
where ι(v)(u1∧ · · · ∧uk) := k X
j=1
(−1)j−1g(v, uj)u1∧ · · · ∧uˆj ∧ · · · ∧uk (3.13)
An easy calculation shows that these operators satisfy ε(v)2 = 0 and ι(v)2 = 0.
The next step is to define the actual generators of the algebra, let it be
B(v) =ε(v) +ι(v). (3.14)
The operator defined above fulfills the properties (3.11) that can be checked using the definition and the properties ofε and ι. Lastly and formally the real Clifford algebra Cl(V, g) is defined to be the sub-algebra of EndR(V∗V) generated by the
set {B(v) : v ∈V}.
Remark. The real Clifford algebra in the caseV is finite dimensional has the same dimension as V∗
V which is 2n where dim(V) =n. This can be proven by fixing a
basis forV and pairing the basis of the exterior algebra with the possible non-zero products of B(v).
The complex Clifford algebraCl(V) is defined similarly but starting withV∗
VC
whereVC
≃V ⊗RC. Furthermore it can be thought ofV∗V with a different
prod-uct. We can abuse of the notation and write v1v2· · ·vk for B(v1)B(v2). . . B(vk).
We can introduce the vector space isomorphismσ :Cl(V)→V∗
VC
(referred usu-ally as the symbol map) determined by σ(a) = c(a)1. Written in this manner the
product on Cl(V) is
uv+vu= 2g(u, v) ∀u, v ∈V (3.15)
26 Algebraic Approach to Anomalies
Q (called quantization map). Then
Q:^∗
VC
−→Cl(V)
Q(v1 ∧ · · · ∧vk) =
1
k!
X
τ∈Sk
(−1)τv
τ(1)· · ·vτ(k) (3.16)
For completeness we state the universal property of Clifford algebras which is useful as a characterization tool.
Theorem 3.1.9(Universal property). LetAbe a unital complex algebra,f :V →
A a real-linear map satisfying f(v)2 = g(v, v)1
A for all v ∈ V. Then Cl(V) is
the unique algebra (modulo isomorphism) such that there exists a unique algebra homomorphism f˜:Cl(V)→ A that extends f that is f = ˜f|V.
We can build a faithful irreducible representation πJ of A(VC) using the GNS
construction with the state ωJ defined by the property ωJ(B(u)B(v)) = hu|viJ.
We follow the brief exposition of the GNS construction given in [Bla06].
Theorem 3.1.10 (GNS construction). Let A be a C∗-algebra and ω a positive
linear functional on A. A pre-inner product on A can be defined by hx, yiω =
ω(y∗x). Consider
Nω ={x∈A:ω(x∗x) = 0}, (3.17)
it is a closed ideal of A, so that h·,·iω is an inner product when considered over
A/Nω. If a ∈ A, define πω(a) to be multiplication by a on A/Nω i.e. πω(a)(x+
Nω) =ax+Nω. Given that
x∗a∗ax≤ kak2x∗x, (3.18)
πω(a) is a bounded operator and kπω(a)k ≤ kak so that it is possible to extend
it to a bounded operator on the Hilbert space Hω = L2(A, ω) (the completion of
A/Nω). πω is the GNS representation of A with respect to ω.
It is possible to prove the representation associated to ωJ is equivalent to the
standard representation of the canonical anti-commutation relations (CAR) on the fermion Fock space FJ(V). In order to define the creation and annihilation
3.1. MATHEMATICAL STRUCTURES CONCERNING ANOMALIES 27
operators of the fermion field we make use of P±J := 12(I∓iJ) and write
aJ(v) :=πJB(P−Jv) and a†J(v) := πJB(P+Jv) (3.19)
then for allv ∈V aJ(Jv) =−iaJ(v) and a†J(Jv) =ia
†
J(v) and πJB(v) =aJ(v) +
a†J(v). This operators act as we expect, for example
a†J(v)Ω = 1 2v−
i
2Jv =v and aJ(v)Ω = 1 2v+
i
2Jv= 0 (3.20)
and generally for a k-vector
a†J(v1)a
†
J(v2)· · ·a
†
J(vk)Ω =v1∧v2∧ · · · ∧vk (3.21)
and
a†J(v)(u1∧ · · · ∧uk) = v∧u1∧ · · · ∧uk (3.22)
aJ(v)(u1∧ · · · ∧uk) = k X
j=1
(−1)j −1hv|ujiJu1∧ · · · ∧uˆi∧ · · · ∧uk. (3.23)
So a simple calculation leads to the verification of
{πJB(v), πJB(v′)}= 2g(v, v′) (3.24)
Through their definition one can check their canonical anti-commutation rela-tions hold, namely
{aJ(v), aJ(v′)}= 0 and {aJ(v), a†J(v
′
)}=hv|v′i (3.25)
We can say then that there is an identification of the Clifford algebra with the CAR algebra for each complex structure. Thus we should view A(VC) as the
algebra of field operators over FJ(V).
28 Algebraic Approach to Anomalies
operator acting on FJ(V). Let
ΓJ :UJ(V)−→U(FJ(V))
U 7−→U ∧U∧ · · · ∧U (3.26)
that is ΓJ sends U ∈UJ(U) to an operator which acts
ΓJ(U)(v1∧ · · · ∧vk) =Uv1 ∧ · · · ∧UvK
and additionally ΓJ(U)Ω := Ω. This leads us to the intertwining property (using
B in terms of creation and annihilation operators)
ΓJ(U)B(v)ΓJ(U)−1 =B(Uv) (3.27)
Lifting the elements of UJ(V) in this manner is referred to as second quantizing
the element. One can define as well a second quantization for self adjoint operators on (V, g, J) bydΓ where
dΓJ(A) :=
d dt
t=0
ΓJ(exp(itA)) (3.28)
which acts explicitly
dΓJ(A)(v1∧ · · · ∧vn) = n X
k=1
v1∧ · · · ∧vk−1∧Avk∧vk+1∧ · · · ∧vn (3.29)
whenever v1, v2,· · · , vn∈Dom(A). For the vacuum we have dΓJ(A)Ω := 0.
Remark. Observables of the one-particle theory correspond then to skew-symmetric real-linear operatorsX ∈EndR(V), that is the Lie Algebrao′(V) of the restricted
orthogonal group. On the other hand ifAis self adjoint then its quantized version under dΓ is a current. Moreover for anyX ∈o′(V) such that [X, J] = 0, −JX is
selfadjoint on (V, g, J).
To summarize what we have; given (V, g) a real vector space with a symmetric bilinear form, we call (FJ(V), πJB,Ω,Γ) the full quantization of (V, g), where
3.1. MATHEMATICAL STRUCTURES CONCERNING ANOMALIES 29
• πJB(V) is a family of self-adjoint operators on FJ(V) satisfying (3.11)
• Ω is a unit vector in FJ(V) such that πJB(V)Ω = FJ(V) (that is Ω is a
cyclic vector)
• Γ is a unitary representation ofUJ(V) intertwiningπJB(V) (equation (3.27)),
for which Ω is stationary,dΓ(A) is positive on FJ(V) if A is positive on the
Hilbert space (V, g, J).
Infinitesimal Spin Representation
We study the case of a finite vector space just to give intuition on the phenomenon that wants to be described. We first note that the Lie algebras of Spin(V) and
SO(V) are isomorphic, explicitly
so(V) ={A∈EndRV :g(y, Ax) =−g(Ay, x)}. (3.30)
Lemma 3.1.11. If ad(a)denotes the operator x7→[a, x], then b7→ad(b) is a Lie algebra homomorphism from Q(∧2V) onto so(V).
A formula for the inverse isomorphism is the following:
˜
A= 1
4
n X
k,l=1
g(ek, Ael)ekel (3.31)
where A∈so(V) and ad ˜A=A for an orthonormal basis {e1, . . . , en}.
Definition 3.1.12. Infinitesimal Spin Representation It is defined to be the real-linear map
˙
µ(B) := c( ˜B) for any B ∈so(V). (3.32)
30 Algebraic Approach to Anomalies
operator T
a†Sa:=X
k,l
huk|SvliJa†kal, (3.33)
aT a :=X
k,l
hT vl|ukiJalak, (3.34)
a†T a† :=X
k,l
huk|T vliJa†ka
†
l. (3.35)
We can split X ∈so(V) into its linear and anti-linear part and write its spin
representation like this:
Lemma 3.1.13. If V has a finite even dimension
˙
µ(X) =dΓ(X+) + 1 2(a
†X
−a†−aX−a)−
1
2trX+ (3.36)
for anyX ∈so′(V), whereX+ = 1
2(X−JXJ)(linear part) andX− = 1
2(X+JXJ)
(anti-linear part).
For the infinite dimensional case it is necessary to suppress the trace term since it is possible forX no to be traceclass. This maintains ˙µbeing a homomorphism but not a Lie algebra homomorphism. At this point we can see how anomalies appear.
Definition 3.1.14 (Infinitesimal spin representation of o′
J(V)). Let (V, g) be an
infinite-dimensional real Hilbert space with an orthogonal complex structure J. The infinitesimal spin representation is the correspondence X 7→µ˙(X) defined by
˙
µ(X) := 1 2(a
†
X−a†+ 2a†X+a−aX−a) (3.37)
which is the same as (3.36) without the trace term.
Anomalies or Schwinger terms are thus the obstructions in the infinite dimen-sional case.
Theorem 3.1.15. If X, Y ∈o′
J(V) then
[ ˙µ(X),µ˙(Y)]−µ˙([X, Y]) = −1
3.2. THE SCHWINGER MODEL (SIMPLIFIED) 31
3.2
The Schwinger Model (Simplified)
We will turn to consider the Schwinger model which is concerned in analyzing “QED” in two dimensions: one time dimension plus one space dimension which is moreover compactified so that we are dealing with R×S1. More details and
discussions concerning this topic can be found in [Ber00],[Mad92]. We make an additional simplification, we take out the electromagnetic field and concentrate only on the Dirac equation, therefore one possible Lagrangian density is
L =i( ¯ψγµ∂µψ)−mψψ¯ (3.39)
The idea is to apply a quantization method based on representations of Clifford algebras. We identify V from the last section to be the vector space of solutions of the Dirac equation. Additionally we make m = 0. Our first task will be to motivate the choice of a particular symmetric bilinear form “g”. First by looking at the extended bilinear form described at the beginning of the previous section
hu|viJ =g(u, v) +ig(Ju, v) (3.40)
we notice it can be inverted to express g in terms of h·,·iJ.
g(u, v) = 1
2(hu, viJ+hu, viJ) (3.41)
The Hilbert space at hand is H = L2(S1,C2) ∼= L2(S1)⊗C2 where we have
a natural inner product. Consider Ψ = (ψ+, ψ−)T with ψ± : S1 → C, then the
inner product of H is
hΨ, χi=
Z
S1Ψ
∗
(θ)χ(θ)dθ =
Z
S1dθ ψ
∗
+(θ)χ+(θ) +ψ
∗
−(θ)χ−(θ). (3.42)
We can look for a J and g such that the above product is exactly h·,·iJ. First
it is required that such definition is invariant with respect to the time coordinate
32 Algebraic Approach to Anomalies
this, expand the Dirac equation
i(γ0∂0 +γ1∂1)Ψ = 0, (3.43)
by multiplying with γ0 from the left we arrive to
i∂
∂tΨ =α
1
i ∂
∂θΨ≡HDΨ (3.44)
with α = γ0γ1, which is the Hamiltonian version of Dirac’s equation. We can
take the following representation of theγ-matrices as an example, but anyone will work the same.
γ0 =σ2 =
0 −i
i 0
, γ1 =iσ1 =
o i
i 0
and γ5=γ0γ1 =σ3 =
1 0
0 −1
Note that the operator 1
i ∂
∂θ is formally selfadjoint and that a representation of the
gamma matrices can be chosen so that α is self adjoint as well. This makes HD
selfadjoint. From the Hamiltonian version for Dirac’s equation one can conclude that the time evolution of the system can be expressed in terms of an evolution operator of the formU(t) =e−itHD which is in turn unitary. So that
Ψ(t, θ) =U(t)Ψ(0, θ). (3.45)
Which leads to
hΨ(t, θ), χ(t, θ)i=hU(t)Ψ(0, θ), U(t)χ(0, θ)i=hΨ(0, θ), χ(0, θ)i (3.46)
as desired. We can take
V :={Ψ∈L2(S1,C2)|i∂0Ψ = αpΨ =HDΨ} (3.47)
and ignore the complex structure for the moment. So that by thinking of V as a real vector space we can implement all the machinery describe in the previous
3.2. THE SCHWINGER MODEL (SIMPLIFIED) 33
section. Motivated by the inner product described above we defineg by
g(Ψ1,Ψ2) =
1 2
Z
S1Ψ
∗
1·Ψ2 dθ+
Z
S1Ψ
∗
2·Ψ1 dθ
(3.48)
where Ψ∗
1·Ψ2 =ψ1+∗ ψ2++ψ1∗−ψ2−.
To keep following this route, we can introduce an orthogonal complex structure
J as being the difference between the projectors P+ and P−, which extract the
positive part of the spectrum or negative part respectively. One can then solve explicitly the Dirac equation for this case to proceed and calculate the energy-momentum tensor and commutators for example. In this way it is possible to obtain the obstruction as in formula3.38and understand it as the chiral anomaly in its commutator version.
3.2.1
Current commutator
Our purpose is now to show how one can evidence the anomaly by calculating the commutator for chiral currents in 1+1 dimension where we haven’t yet compact-ified the space dimension. This will be done following the procedure in [ABH93] in which the Dirac sea is filled incompletely and the limit is taken. Lets start by considering massless spinors of the form
Ψi(x, t) =
1
√
2π
Z
dk exp(ikx)ai,k(t) (3.49)
Ψ∗i(x, t) =
1
√
2π
Z
dk exp(−ikx)a∗i,k(t), (3.50)
with the a’s obeying canonical anti-commutation relations
{a∗
i,k(t), aj,l(t′)}|t=t′ =δ(k−l)δij (3.51)
with this formulation the Hamiltonian is not bounded below en terms of energy so it is necessary to fill the Dirac sea to obtain a positive-definite Hamiltonian. Let us denote the Heaviside step function by Θ(k), and make the following