III Advanced Quantum Field Theory (Full)

Part III — Advanced Quantum Field Theory

Based on lectures by D. B. Skinner

Notes taken by Dexter Chua

Lent 2017

These notes are not endorsed by the lecturers, and I have modified them (often

significantly) after lectures. They are nowhere near accurate representations of what

was actually lectured, and in particular, all errors are almost surely mine.

Quantum Field Theory (QFT) provides the most profound description of Nature we

currently possess. As well as being the basic theoretical framework for describing

elementary particles and their interactions (excluding gravity), QFT also plays a major

role in areas of physics and mathematics as diverse as string theory, condensed matter

physics, topology and geometry, astrophysics and cosmology.

This course builds on the Michaelmas Quantum Field Theory course, using techniques

of path integrals and functional methods to study quantum gauge theories. Gauge

Theories are a generalisation of electrodynamics and form the backbone of the Standard

Mo del — our best theory encompassing all particle physics. In a gauge theory, fields

have an infinitely redundant description; we can transform the fields by a different

element of a Lie Group at every point in space-time and yet still describe the same

physics. Quantising a gauge theory requires us to eliminate this infinite redundancy.

In the path integral approach, this is done using tools such as ghost fields and BRST

symmetry. We discuss the construction of gauge theories and their most important

observables, Wilson Loops. Time permitting, we will explore the possibility that a

classical symmetry may be broken by quantum effects. Such anomalies have many

imp ortant consequences, from constraints on interactions between matter and gauge

fields, to the ability to actually render a QFT inconsistent.

A further major component of the course is to study Renormalization. Wilson’s

picture of Renormalisation is one of the deepest insights into QFT — it explains

why we can do physics at all! The essential point is that the physics we see depends

on the scale at which we look. In QFT, this dependence is governed by evolution

along the Renormalisation Group (RG) flow. The course explores renormalisation

systematically, from the use of dimensional regularisation in p erturbative loop integrals,

to the difficulties inherent in trying to construct a quantum field theory of gravity. We

discuss the various possible behaviours of a QFT under RG flow, showing in particular

that the coupling constant of a non-Abelian gauge theory can effectively become small

at high energies. Known as ”asymptotic freedom”, this phenomenon revolutionised our

understanding of the strong interactions. We introduce the notion of an Effective Field

Theory that describes the low energy limit of a more fundamental theory and helps

parametrise possible departures from this low energy approximation. From a modern

perspective, the Standard Model itself appears to be but an effective field theory.

Pre-requisites

Knowledge of the Michaelmas term Quantum Field Theory course will be assumed.

Familiarity with the course Symmetries, Fields and Particles would be very helpful.

Contents

0 Introduction

0.1 What is quantum field theory

0.2 Building a quantum field theory

1 QFT in zero dimensions

1.1 Free theories

1.2 Interacting theories

1.3 Feynman diagrams

1.4 An effective theory

1.5 Fermions

2 QFT in one dimension (i.e. QM)

2.1 Quantum mechanics

2.2 Feynman rules

2.3 Effective quantum field theory

2.4 Quantum gravity in one dimension

3 Symmetries of the path integral

3.1 Ward identities

3.2 The Ward–Takahashi identity

4 Wilsonian renormalization

4.1 Background setting

4.2 Integrating out modes

4.3 Correlation functions and anomalous dimensions

4.4 Renormalization group flow

4.5 Taking the continuum limit

4.6 Calculating RG evolution

5 Perturbative renormalization

5.1 Cutoff regularization

5.2 Dimensional regularization

5.3 Renormalization of the φ

coupling

5.4 Renormalization of QED

6 Non-abelian gauge theory

6.1 Bundles, connections and curvature

6.2 Yang–Mills theory

6.3 Quantum Yang–Mills theory

6.4 Faddeev–Popov ghosts

6.5 BRST symmetry and cohomology

6.6 Feynman rules for Yang–Mills

6.7 Renormalization of Yang–Mills theory

0 Introduction

0.1 What is quantum field theory

What is Quantum Field Theory? The first answer we might give is — it’s just a

quantum version of a field theory (duh!). We know the world is described by

fields, e.g. electromagnetic fields, and the world is quantum. So naturally we

want to find a quantum version of field theory. Indeed, this is what drove the

study of quantum field theory historically.

But there are other things we can use quantum field theory for, and they

are not so obviously fields. In a lot of condensed matter applications, we can

use quantum field theory techniques to study, say, vibrations of a crystal, and

its quanta are known as “phonons”. More generally, we can use quantum field

theory to study things like phase transitions (e.g. boiling water). This is not so

obvious a place you think quantum field theory will happen, but it is. Some

people even use quantum field theory techniques to study the spread of diseases

in a population!

We can also use quantum field theory to study problems in mathematics!

QFT is used to study knot invariants, which are ways to assign labels to different

types of knots to figure out if they are the same. Donaldson won a fields medal

for showing that there are inequivalent differentiable structures on

, and this

used techniques coming from quantum Yang–Mills theory as well.

These are not things we would traditionally think of as quantum field theory.

In this course, we are not going to be calculating, say, Higgs corrections to

γγ-interactions. Instead, we try to understand better this machinery known as

quantum field theory.

0.2 Building a quantum field theory

We now try to give a very brief outline of how one does quantum field theory.

Roughly, we follow the steps below:

(i)

We pick a space to represent our “universe”. This will always be a manifold,

but we usually impose additional structure on it.

–

In particle physics, we often pick the manifold to be a 4-dimensional

pseudo-Riemannian manifold of signature +

− −−

. Usually, we in

fact pick (M, g) = (R

, η) where η is the usual Minkowski metric.

–

In condensed matter physics, we often choose (

M, g

) = (

, δ

), where

δ is the flat Euclidean metric.

–

In string theory, we have fields living on Riemann surface Σ (e.g.

sphere, torus). Instead of specifying a metric, we only specify the

conformal equivalence class [

] of the metric, i.e. the metric up to a

scalar factor.

–

In QFT for knots, we pick

to be some oriented 3-manifold, e.g.

but with no metric.

–

In this course, we will usually take (

M, g

) = (

, δ

) for some

, where

δ is again the flat, Euclidean metric.

We might think this is a very sensible and easy choice, because we are

all used to working in (

, δ

). However, mathematically, this space is

non-compact, and it will lead to a lot of annoying things happening.

(ii)

We pick some fields. The simplest choice is just a function

M → R

. This is a scalar field. Slightly more generally, we can also have

φ : M → N for some other manifold N . We call N the target space.

For example, quantum mechanics is a quantum field theory. Here we

choose

to be some interval

= [0

1], which we think of as time,

and the field is a map φ : I → R

. We think of this as a path in R

In string theory, we often have fields

: Σ

→ N

, where

is a Calabi–Yau

manifold.

In pion physics,

(

) describes a map

: (

, η

)

→ G/H

, where

and

are Lie groups.

Of course, we can go beyond scalar fields. We can also have fields with

non-zero spin such as fermions or gauge fields, e.g. a connection on a

principal

-bundle, as we will figure out later. These are fields that carry

a non-trivial representation of the Lorentz group. There is a whole load of

things we can choose for our field.

Whatever field we choose, we let

be the space of all field configurations,

i.e. a point

φ ∈ C

represents a picture of our field across all of

. Thus

is some form of function space and will typically be infinite dimensional.

This infinite-dimensionality is what makes QFT hard, and also what makes

it interesting.

(iii)

We choose an action. An action is just a function

C → R

. You tell

me what the field looks like, and I will give you a number, the value of

the action. We often choose our action to be local, in the sense that we

assume there exists some function L(φ, ∂φ, ···) such that

S[φ] =

√

g L(φ(x), ∂φ(x), ···).

The physics motivation behind this choice is obvious — we don’t want

what is happening over here to depend on what is happening at the far

side of Pluto. However, this assumption is actually rather suspicious, and

we will revisit this later.

For example, we have certainly met actions that look like

S[φ] =



(∂φ)



and for gauge fields we might have seen

S[A] =

x F

µν

If we have a coupled fermion field, we might have

S[A, ψ] =

x F

µν

ψ(

D + m)ψ.

But recall when we first encountered Lagrangians in classical dynamics,

we worked with lots of different Lagrangians. We can do whatever thing

we like, make the particle roll down the hill and jump into space etc, and

we get to deal with a whole family of different Lagrangians. But when we

come to quantum field theory, the choices seem to be rather restrictive.

Why can’t we choose something like

S[A] =

+ F

+ cosh(F

) + ···?

It turns out we can, and in fact we must. We will have to work with

something much more complicated.

But then what were we doing in the QFT course? Did we just waste time

coming up with tools that just work with these very specific examples? It

turns out not. We will see that there are very good reasons to study these

specific actions.

(iv)

What do we compute? In this course, the main object we’ll study is the

partition function

Z =

Dφ e

−S[φ]/~

which is some sort of integral over the space of all fields. Note that the

minus sign in the exponential is for the Euclidean signature. If we are not

Euclidean, we get some i’s instead.

We will see that the factor of

−S[φ]/~

means that as

~ →

0, the dominant

contribution to the partition function comes from stationary points of

[

]

over

, and this starts to bring us back to the classical theory of fields.

The effect of

−S[φ]/~

is to try to suppress “wild” contributions to

, e.g.

where φ is varying very rapidly or φ takes very large values.

Heuristically, just as in statistical physics, we have a competition between

the two factors D

and

−S[φ]/~

. The action part tries to suppress crazy

things, but how well this happens depends on how much crazy things are

happening, which is measured by the measure D

. We can think of this

Dφ as “entropy”.

However, the problem is, the measure D

doesn’t actually exist!

Understanding what we mean by this path integral, and what this measure

actually is, and how we can actually compute this thing, and how this has

got to do with the canonical quantization operators we had previously, is

the main focus of the first part of this course.

1 QFT in zero dimensions

We start from the simplest case we can think of, namely quantum field theory in

zero dimensions. This might seem like an absurd thing to study — the universe

is far from being zero-dimensional. However, it turns out this is the case where

we can make sense of the theory mathematically. Thus, it is important to study

0-dimensional field theories and understand what is going on.

There are two reasons for this. As mentioned, we cannot actually define the

path integral in higher dimensions. Thus, if we were to do this “properly”, we will

have to define it as the limit of something that is effectively a zero-dimensional

quantum field theory. The other reason is that in this course, we are not going

to study higher-dimensional path integrals “rigorously”. What we are going

to do is that we will study zero-dimensional field theories rigorously, and then

assume that analogous results hold for higher-dimensional field theories.

One drawback of this approach is that what we do in this section will have

little physical content or motivation. We will just assume that the partition

function is something we are interested in, without actually relating it to any

physical processes. In the next chapter, on one-dimensional field theories, we

are going see why this is an interesting thing to study.

Let’s begin. In

= 0, if our universe

is connected, then the only choice of

{pt}

. There is a no possibility for a field to have spin, because the Lorentz

group is trivial. Our fields are scalar, and the simplest choice is just a single

field

{pt} → R

, i.e. just a real variable. Similarly, we simply have

∼

This is not an infinite-dimensional space.

The action is just a normal function

∼

R → R

of one real variable.

The path integral measure D

can be taken to just be the standard (Lebesgue)

measure dφ on R. So our partition function is just

Z =

dφ e

−S(φ)/~

where we assume

is chosen so that this converges. This happens if

grows

sufficiently quickly as φ → ±∞.

More generally, we may wish to compute correlation functions, i.e. we pick

another function f (φ) and compute the expectation

hf(φ)i =

dφ f(φ)e

−S(φ)/~

Again, we can pick whatever

we like as long as the integral converges. In this

case,

−S(φ)/~

is a probability distribution on

, and as the name suggests,

(

)

is just the expectation value of

in this distribution. Later on, when we

study quantum field theory in higher dimensions, we can define more complicated

by evaluating

at different points, and we can use this to figure out how the

field at different points relate to each other.

Our action is taken to have a series expansion in

, so in particular we can

write

S(φ) =

n=3

We didn’t put in a constant term, as it would just give us a constant factor in

We could have included linear terms, but we shall not. The important thing is

that

has to be even, so that the asymptotic behaviour of

is symmetric in

both sides.

Now the partition function is a function of all these terms:

(

, g

Similarly,

hfi

is again a function of

and

, and possibly other things used

to define f itself.

Note that nothing depends on the field, because we are integrating over all

possible fields.

1.1 Free theories

We consider the simplest possible QFT. These QFT’s are free, and so

(

) is at

most quadratic. Classically, this implies the equations of motions are linear, and

so there is superposition, and thus the particles do not interact.

Let φ : {pt} → R

be a field with coordinates φ

, and define

S(φ) =

M(φ, φ) =

where

× R

→ R

is a positive-definite symmetric matrix. Then the

partition function Z(M) is just a Gaussian integral:

Z(M) =

φ e

−

M(φ,φ)

(2π~)

n/2

√

det M

Indeed, to compute this integral, since

is symmetric, there exists an orthogonal

transformation

→ R

that diagonalizes it. The measure d

is invariant

under orthogonal transformations. So in terms of the eigenvectors of

, this

just reduces to a product of

1D Gaussian integrals of this type, and this is a

standard integral:

dχ e

−mχ

/2~

2π~

In our case,

m >

0 runs over all eigenvalues of

, and the product of eigenvalues

is exactly det M .

A small generalization is useful. We let

S(φ) =

M(φ, φ) + J(φ),

where

→ R

is some linear map (we can think of

as a (co)vector, and

also write

(

) =

J · φ

is a source in the classical case. Then in this theory,

we have

Z(M, J) =

φ exp



−



M(φ, φ) + J(φ)



To do this integral, we complete the square by letting

−1

. In other

words,

= φ

+ (M

−1

)

The inverse exists because

is assumed to be positive definite. We can now

complete the square to find

Z(M, J) =

φ exp



−1

φ,

φ) +

−1

(J, J)



= exp



−1

(J, J)



φ exp



−1

φ,

φ)



= exp



−1

(J, J)



(2π~)

n/2

√

det M

In the long run, we really don’t care about the case with a source. However, we

will use this general case to compute some correlation functions.

We return to the case without a source, and let

→ R

be a polynomial.

We want to compute

hP (φ)i =

Z(M)

φ P (φ) exp



−

M(φ, φ)



By linearity, it suffices to consider the case where P is just a monomial, so

P (φ) =

i=1

(φ)),

for

→ R

linear maps. Now if

is odd, then clearly

(

)

= 0, since

this is an integral of an odd function. When m = 2k, then we have

hP (φ)i =

Z(M)

φ (`

· φ) ···(`

· φ) exp



−

M(φ, φ) −

J · φ



Here we are eventually going to set

= 0, but for the time being, we will be

silly and put the source there. The relevance is that we can then think of our

factors `

· φ as derivatives with respect to J:

hP (φ)i =

(−~)

Z(M)

i=1



∂

∂J



exp



−

M(φ, φ) −

J · φ



Since the integral is absolutely convergent, we can move the derivative out of

the integral, and get

(−~)

Z(M)

i=1



∂

∂J



φ exp



−

M(φ, φ) −

J · φ



= ~

i=1



∂

∂J



exp



−1

(J, J)



When each derivative `

∂

∂J

acts on the exponential, we obtain a factor of

−1

(J, `

in front. At the end, we are going to set

= 0. So we only get contributions if

and only if exactly half (i.e.

) of the derivatives act on the exponential, and the

other k act on the factor in front to get rid of the J.

We let

denote a (complete) pairing of the set

{

, ··· ,

, and Π

be the

set of all such pairings. For example, if we have

= 2, then the possible pairings

are {(1, 2), (3, 4)}, {(1, 3), (2, 4)} and {(1, 4), (2, 3)}:

In general, we have

|Π

| =

(2k)!

and we have

Theorem (Wick’s theorem). For a monomial

P (φ) =

i=1

(φ),

we have

hP (φ)i = ~

σ∈Π

i∈{1,···,2k}/σ

−1

, `

σ(i)

where the

{

, ··· ,

k}/σ

says we sum over each pair

{i, σ

(

)

}

only once, rather

than once for (i, σ(i)) and another for (σ(i), i).

This is in fact the version of Wick’s theorem for this 0d QFT, and

−1

plays

the role of the propagator.

For example, we have

(φ)`

(φ)i = ~M

−1

, `

We can represent this by the diagram

1 2M

−1

Similarly, we have

(φ) ···`

(φ)i = ~



−1

, `

−1

, `

)

+ M

−1

, `

−1

, `

) + M

−1

, `

−1

, `

)



Note that we have now reduced the problem of computing an integral for the

correlation function into a purely combinatorial problem of counting the number

of ways to pair things up.

1.2 Interacting theories

Physically interesting theories contain interactions, i.e.

(

) is non-quadratic.

Typically, if we are trying to compute

φ P (φ) exp



−

S(φ)



these things involve transcendental functions and, except in very special circum-

stances, is just too hard. We usually cannot perform these integrals analytically.

Naturally, in perturbation theory, we would thus want to approximate

Z =

φ exp



−

S(φ)



by some series. Unfortunately, the integral very likely diverges if

~ <

0, as

usually

(

)

→ ∞

φ → ±∞

. So it can’t have a Taylor series expansion

around

= 0, as such expansions have to be valid in a disk in the complex plane.

The best we can hope for is an asymptotic series.

Recall that a series

∞

n=0

(~)

is an asymptotic series for

(

) if for any fixed

N ∈ N

, if we write

(

) for

the first N terms on the RHS, then

lim

~→0

|Z(~) − Z

(~)|

→ 0.

Thus as

~ →

, we get an arbitrarily good approximation to what we really

wanted from any finite number of terms. But the series will in general diverge if

try to fix

~ ∈ R

, and include increasingly many terms. Most of the expansions

we do in quantum field theories are of this nature.

We will assume standard results about asymptotic series. Suppose

(

) is

smooth with a global minimum at φ = φ

∈ R

, where the Hessian

∂

∂φ



is positive definite. Then by Laplace’s method/Watson’s lemma, we have an

asymptotic series of the form

Z(~) ∼ (2π~)

n/2

exp



−

S(φ

)



det ∂

∂

S(φ

)



1 + A~ + B~

+ ···



We will not prove this, but the proof is available in any standard asymptotic

methods textbook. The leading term involves the action evaluated on the

classical solution

, and is known as the semiclassical term. The remaining

terms are called the quantum correction.

In Quantum Field Theory last term, the tree diagrams we worked with were

just about calculating the leading term. So we weren’t actually doing quantum

field theory.

Example. Let’s consider a single scalar field φ with action

S(φ) =

where

, λ >

0. The action has a unique global minimum at

= 0. The

action evaluated at

= 0 vanishes, and

∂

. So the leading term in the

asymptotic expansion of Z(~, mλ) is

(2π~)

1/2

Further, we can find the whole series expansion by

Z(~, m, λ) =

dφ exp



−1





√

φ exp



−



exp



−

4λ~

4!m



∼

√

φ e

−

n=0



−4λ~

4!m



√

n=0



−4λ~

4!m



φ e

−

√

n=0



−4λ~

4!m





2n +



The last line uses the definition of the Gamma function.

We can plug in the value of the Γ function to get

Z(~, m, λ) ∼

√

2π~

n=0



−~λ



(4!)

(4n)!

(2n)!

√

2π~



1 −

~λ

384

+ ···



We will get to understand these coefficients much more in terms of Feynman

diagrams later.

Note that apart from the factor

√

2π~

, the series depends on (

~, λ

) only

through the product

~λ

. So we can equally view it as an asymptotic series in

the coupling constant

. This view has the benefit that we can allow ourselves

to set ~ = 1, which is what we are usually going to do.

As we emphasized previously, we get an asymptotic series, not a Taylor series.

Why is this so?

When we first started our series expansion, we wrote

exp

(

−S

(

)

) as a

series in

. This is absolutely fine, as

exp

is a very well-behaved function when

it comes to power series. The problem comes when we want to interchange the

integral with the sum. We know this is allowed only if the sum is absolutely

convergent, but it is not, as the integral does not converge for negative

. Thus,

the best we can do is that for any

, we truncate the sum at

, and then do

the exchange. This is legal since finite sums always commute with integrals.

We can in fact see the divergence of the asymptotic series for finite (

~λ

)

from the series itself, using Stirling’s approximation. Recall we have

n! ∼ e

n log n

So via a straightforward manipulation, we have

(4!)

(4n)!

(2n)!

∼ e

n log n

So the coefficients go faster than exponentially, and thus the series has vanishing

radius of convergence.

We can actually plot out the partial sums. For example, we can pick

= 1,

= 0

1, and let

be the

th partial sum of the series. We can start

plotting this for n up to 40 (the red line is the true value):

0 5 10 15 20 25 30 35 40

2.46

2.48

2.5

2.52

2.54

After the initial few terms, it seems like the sum has converged, and indeed

to the true value. Up to around

= 30, there is no visible change, and indeed,

the differences tend to be of the order

∼

−7

. However, after around 33, the

sum starts to change drastically.

We can try to continue plotting. Since the values become ridiculously large,

we have to plot this in a logarithmic scale, and thus we plot

instead. Do

note that for sufficiently large

, the actual sign flips every time we increment

n! This thing actually diverges really badly.

0 10 20 30 40 50 60

We can also see other weird phenomena here. Suppose we decided that we

have

0 instead (which is a rather weird thing to do). Then it is clear that

(

, λ

) still exists, as the

eventually dominates, and so what happens at

the

term doesn’t really matter. However, the asymptotic series is invalid in

this region.

Indeed, if we look at the fourth line of our derivation, each term in the

asymptotic series will be obtained by integrating

, instead of integrating

−

, and each of these integrals would diverge.

Fundamentally, this is since when

0, the global minimum of

(

) are

now at

= ±

S(φ)

Our old minimum

= 0 is now a (local) maximum! This is the wrong point to

expand about! Fields with

0 are called tachyons, and these are always

signs of some form of instability.

Actions whose minima occur at non-zero field values are often associated

with spontaneous symmetry breaking. Our original theory has a

2 symmetry,

namely under the transformation

x ↔ −x

. But once we pick a minimum

to expand about, we have broken the symmetry. This is an interesting kind of

symmetry breaking because the asymmetry doesn’t really exist in our theory.

It’s just our arbitrary choice of minimum that breaks it.

1.3 Feynman diagrams

We now try to understand the terms in the asymptotic series in terms of Feynman

diagrams. The easy bits are the powers of



~λ



This combination is essentially fixed by dimensional analysis. What we really

want to understand are the combinatorial factors, given by

(4!)

(4n)!

(2n)!

To understand this, we can write the path integral in a different way. We have

Z(~, m, λ) =

dφ exp



−





dφ

∞

n=0



−

4!~



exp



−



∼

∞

n=0

(4!)



−λ



hφ

free

Thus, we can apply our previous Wick’s theorem to compute this! We now see

that there is a factor of

(4!)

coming from expanding the exponential, and

the factor of

(4n)!

(2n)!

came from Wick’s theorem — it is the number of ways to

pair up 4

vertices together. There will be a factor of

coming from Wick’s

theorem when evaluating hφ

i, so the expansion is left with a ~

factor.

Let’s try to compute the order

term. By Wick’s theorem, we should

consider diagrams with 4 external vertices, joined by straight lines. There are

three of these:

But we want to think of these as interaction vertices of valency 4. So instead of

drawing them this way, we “group” the four vertices together. For example, the

first one becomes

⇒

Similarly, at order λ

, we have a diagram

⇒

Note that in the way we are counting diagrams in Wick’s theorem means we

consider the following diagrams to be distinct:

However, topologically, these are the same diagrams, and we don’t want to count

them separately. Combinatorially, these diagrams can be obtained from each

other by permuting the outgoing edges at each vertex, or by permuting the

vertices. In other words, in the “expanded view”, when we group our vertices as

we are allowed to permute the vertices within each block, or permute the blocks

themselves. We let

be the set of all graphs, and

be the group consisting

of these “allowed” permutations. This

is a semi-direct product (

)

o S

Then by some combinatorics,

| =

(4n)!

(2n)!

, |G

| = (4!)

n!.

Recall that

is the number Wick’s theorem gives us, and we now see that

happens to be the factor we obtained from expanding the exponential. Of

course, this isn’t a coincidence — we chose to put

instead of, say,

in front

of the φ

term so that this works out.

We can now write the asymptotic series as

Z(m

, λ)

Z(m

, 0)

∼

n=0



−~λ



It turns out a bit of pure mathematics will allow us to express this in terms

of the graphs up to topological equivalence. By construction, two graphs are

topologically equivalent if they can be related to each other via

. In other

words, the graphs correspond to the set of orbits

under

. For

= 1,

there is only one graph up to topological equivalence, so

= 1. It is not hard

to see that |O

| = 3.

In general, each graph Γ has some automorphisms that fix the graph. For

example, in the following graph:

3 4

we see that the graph is preserved by the permutations (1 3), (2 4) and (1 2)(3 4).

In general, the automorphism group

Aut

(Γ) is the subgroup of

consisting of

all permutations that preserve the graph Γ. In this case,

Aut

(Γ) is generated by

the above three permutations, and |Aut(Γ)| = 8.

Recall from IA Groups that the orbit-stabilizer theorem tells us

Γ∈O

|Aut(Γ)|

Thus, if we write

O =

[

n∈N

then we have

Z(m

, λ)

Z(m

, 0)

∼

Γ∈O

|Aut(Γ)|



−~λ



where the

appearing in the exponent is the number of vertices in the graph Γ.

The last loose end to tie up is the factor of



−~λ



appearing in the sum.

While this is a pretty straightforward thing to come up with in this case, for

more complicated fields with more interactions, we need a better way of figuring

out what factor to put there.

If we look back at how we derived this factor, we had a factor of

−

coming from each interaction vertex, whereas when computing the correlation

functions

hφ

, every edge contributes a factor of

−2

(since, in the language

we expressed Wick’s theorem, we had

). Thus, we can imagine this as

saying we have a propagator

~/m

and a vertex

−λ/~

Note that the negative sign appears because we have

−S

in “Euclidean” QFT.

In Minkowski spacetime, we have a factor of −i instead.

After all this work, we can now expand the partition function as

Z(m

, λ)

Z(m

, 0)

∼

∅

+ + + + + ···

1 +

−λ~

128

+ ···

In generally, if we have a theory with several fields with propagators of value

, and many different interactions with coupling constants λ

, then

Z({λ

})

Z({0})

∼

Γ∈O

|Aut Γ|

(Γ)|

E(Γ)−V (Γ)

where

– e

(Γ) is the number of edges of type i in Γ;

– v

(Γ) is the number of vertices of type α in Γ;

– V (Γ) is the number of vertices in total; and

– E(Γ) the total number of edges.

There are a few comments we can make. Usually, counting all graphs is rather

tedious. Instead, we can consider the quantity

log(Z/Z

This is the sum of all connected graphs. For example, we can see that the term for

the two-figure-of-eight is just the square of the term for a single figure-of-eight

times the factor of

, and one can convince oneself that this generalizes to

arbitrary combinations.

In this case, we can simplify the exponent of

. In general, by Euler’s theorem,

we have

E − V = L − C,

where

is the number of loops in the graph, and

is the number of connected

components. In the case of

log

(

Z/Z

), we have

= 1. Then the factor of

just ~

L−1

. Thus, the more loops we have, the less it contributes to the sum.

It is not difficult to see that if we want to compute correlation functions, say

hφ

, then we should consider graphs with two external vertices. Note that in

this case, when computing the combinatorial factors, the automorphisms of a

graph are required to fix the external vertices. To see this mathematically, note

that the factors in

(4!)

came from Taylor expanding

exp

and the coefficient

. We do not get

these kinds of factors from φ

when computing

dφ φ

−S[φ]

Thus, we should consider the group of automorphisms that fix the external

vertices to get the right factor of

Finally, note that Feynman diagrams don’t let us compute exact answers

in general. Even if we had some magical means to compute all terms in the

expansions using Feynman diagrams, if we try to sum all of them up, it will

diverge! We only have an asymptotic series, not a Taylor series. However, it

turns out in some special theories, the leading term is the exact answer! The tail

terms somehow manage to all cancel each other out. For example, this happens

for supersymmetry theories, but we will not go into details.

1.4 An effective theory

We now play with another toy theory. Suppose we have two scalar fields

φ, χ ∈ R

and consider the action

S(φ, χ) =

For convenience, we will set ~ = 1. We have Feynman rules

1/m

1/M

with a vertex

−λ

We can use these to compute correlation functions and expectation values. For

example, we might want to compute

log(Z/Z

We have a diagram that looks like

log



Z(m

, λ)

Z(m

, 0)



∼ + + + + ···

−λ

16m

+ ···

We can also try to compute

hφ

. To do so, we consider diagrams with two

vertices connected to a solid line:

The relevant diagrams are

hφ

i ∼ + + + + + ···

−λ

+ ···

Let’s arrive at this result in a different way. Suppose we think of

as “heavy”,

so we cannot access it directly using our experimental equipment. In particular,

if we’re only interested in the correlation functions that depend only on

, then

we could try to “integrate out” χ first.

Suppose we have a function f(φ), and consider the integral

dφ dχ f(φ)e

−S(φ,χ)/~

dφ



f(φ)

dχ e

−S(φ,χ)/~



We define the effective action for φ, S

eff

(φ) by

eff

(φ) = −~ log



dχ e

−S(φ,χ)/~



Then the above integral becomes

dφ f(φ)e

−S

eff

(φ)/~

So doing any computation with

only is equivalent to pretending

doesn’t

exist, and using this effective potential instead.

In general, the integral is very difficult to compute, and we can only find

an asymptotic series for the effective action. However, in this very particular

example, we have chosen

such that it is only quadratic in

, and one can

compute to get

dχ e

−S(φ,χ)/~

= e

−m

/2~

2π~

+ λφ

Therefore we have

eff

(φ) =

log



1 +

λφ



log



2π~





~λ



−

~λ

16M

~λ

48M

+ ··· +

log



2π~



eff

+ ··· +

log



2π~



where

= (−1)

k+1

~(2k)!

k+1

We see that once we integrated out

, we have generated an infinite series of

new interactions for φ in S

eff

(φ). Moreover, the mass also shifted as

7→ m

eff

= m

~λ

It is important to notice that the new vertices generated are quantum effects.

They vanish as

~ →

0 (they happen to be linear in

here, but that is just a

coincidence). They are also suppressed by powers of

. So if

is very very

heavy, we might think these new couplings have very tiny effects.

This is a very useful technique. If the universe is very complicated with

particles at very high energies, then it would be rather hopeless to try to account

for all of the effects of the particles we cannot see. In fact, it is impossible to

know about these very high energy particles, as their energy are too high to

reach. All we can know about them is their incarnation in terms of these induced

couplings on lower energy fields.

A few things to note:

–

The original action had a

× Z

symmetry, given by (

φ, χ

)

7→

(

±φ, ±χ

This symmetry is preserved, and we do not generate any vertices with odd

powers of φ.

– The field S

eff

(φ) also contains a field independent term

log



2π~



This plays no role in correlation functions

(

)

. Typically, we are just

going to drop it. However, this is one of the biggest problem in physics.

This term contributes to the cosmological constant, and the scales that are

relevant to this term are much larger than the actual observed cosmological

constant. Thus, to obtain the observed cosmological constant, we must

have some magic cancelling of these cosmological constants.

–

In this case, passing to the effective action produced a lot of new inter-

actions. However, if we already had a complicated theory, e.g. when we

started off with an effective action, and then decided to integrate out

more things, then instead of introducing new interactions, the effect of this

becomes shifting the coupling constants, just as we shifted the mass term.

In general, how can we compute the effective potential? We are still computing

an integral of the form

dχ e

−S(φ,χ)/~

so our previous technique of using Feynman diagrams should still work. We will

treat

as a constant in the theory. So instead of having a vertex that looks like

−λ

we drop all the φ lines and are left with

−λφ

But actually, this would be rather confusing. So instead what we do is that

we still draw the solid

lines, but whenever such a line appears, we have to

terminate the line immediately, and this contributes a factor of φ:

−λφ

For our accounting purposes, we will say the internal vertex contributes a factor

−

(since the automorphism group of this vertex has order 2, and the action

had

), and each terminating blue vertex contributes a factor of φ.

Since we have a “constant” term

as well, we need to add one more

diagram to account for that. We allow a single edge of the form

−m

With these ingredients, we can compute the effective potential as follows:

−S

eff

(φ) ∼ + + + + ···

−m

−λ

16M

−λ

48M

+ ···

These are just the terms we’ve previously found. There are, again, a few things

to note

– The diagram expansion is pretty straightforward in this case, because we

started with a rather simple interacting theory. More complicated examples

will have diagram expansions that are actually interesting.

–

We only sum over connected diagrams, as

eff

is the logarithm of the

integral.

–

We see that the new/shifted couplings in

eff

(

) are generated by loops of

χ fields.

–

When we computed the effective action directly, we had a “cosmological

constant” term

log



2π~



, but this doesn’t appear in the Feynman

diagram calculations. This is expected, since when we developed our

Feynman rules, what it computed for us was things of the form

log

(

Z/Z

and the Z

term is that “cosmological constant”.

Example. We can use the effective potential to compute hφ

hφ

i ∼ + + ···

eff

−λ

eff

+ ···

We can see that this agrees with our earlier calculation correct to order λ

At this moment, this is not incredibly impressive, since it takes a lot of work

to compute

eff

(

). But the computation of

eff

(

) can be reused to compute

any correlation involving

. So if we do many computations with

, then we we

can save time.

But the point is not that this saves work. The point is that this is what we

do when we do physics. We can never know if there is some really high energy

field we can’t reach with our experimental apparatus. Thus, we can only assume

that the actions we discover experimentally are the effective actions coming from

integrating out some high energy fields.

1.5 Fermions

So far, we have been dealing with Bosons. These particles obey Bose statistics,

so different fields

commute with each other. However, we want to study

Fermions as well. This would involve variables that anti-commute with each

other. When we did canonical quantization, this didn’t pose any problem to our

theory, because fields were represented by operators, and operators can obey

any commutation or anti-commutation relations we wanted them to. However,

in the path integral approach, our fields are actual numbers, and they are forced

to commute.

So how can we take care of fermions in our theory? We cannot actually use

path integrals, as these involves things that commute. Instead, we will now treat

the fields as formal symbols.

The theory is rather trivial if there is only one field. So let’s say we have many

fields

, ··· , θ

. Then a “function” of these fields will be a formal polynomial

expression in the symbols

, ··· , θ

subject to the relations

−θ

. So,

for example, the action might be

S[θ

, θ

] =

= −

Now the path integral is defined as

dθ

···dθ

−S[θ

,...,θ

]

There are two things we need to make sense of — the exponential, and the

integral.

The exponential is easy. In general, for any analytic function

R → R

(or

f : C → C), we can write it as a power series

f(x) =

∞

i=0

Then for any polynomial p in the fields, we can similarly define

f(p) =

∞

i=0

Crucially, this expression is a finite sum, since we have only finitely many fields,

and any monomial in the fields of degree greater than the number of fields must

vanish by anti-commutativity.

Example.

+θ

= 1 + θ

+ θ

(θ

+ θ

)

+ ···

= 1 + θ

+ θ

(θ

+ θ

)

= 1 + θ

+ θ

How about integration? Since our fields are just formal symbols, and do not

represent real/complex numbers, it doesn’t make sense to actually integrate it.

However, we can still define a linear functional from the space of all polynomials

in the fields to the reals (or complexes) that is going to act as integration.

If we have a single field

, then the most general polynomial expression in

is a + bθ. It turns out the correct definition for the “integral” is

dθ (a + bθ) = b.

This is known as the Berezin integral.

How can we generalize this to more fields? Heuristically, the rule is that

is an expression that should anti-commute with all fields. Thus, for example,

dθ

(3θ

+ 2θ

) =

dθ



dθ

+ 2

dθ



dθ



3 − 2θ

dθ



dθ

(3 − 2θ

)

= −2.

On the other hand, we have

dθ

(3θ

+ 2θ

) =

dθ



dθ

+ 2

dθ



dθ



3 + 2



dθ



dθ

(3 − 2θ

)

= 2.

Formally speaking, we can define the integral by

dθ

···dθ

···θ

= 1,

and then sending other polynomials to 0 and then extending linearly.

When actually constructing a fermion field, we want an action that is “sensi-

ble”. So we would want a mass term

in the action. But for anti-commuting

variables, this must vanish!

The solution is to imagine that our fields are complex, and we have two

fields

and

. Of course, formally, these are just two formal symbols, and bear

no relation to each other. However, we will think of them as being complex

conjugates to each other. We can then define the action to be

θ, θ] =

θθ.

Then the partition function is

Z =

dθ d

θ e

−S(

θ,θ)

Similar to the previous computations, we can evaluate this to be

Z =

dθ d

θ e

−S(

θ,θ)

dθ



1 −

θθ



dθ



θ −





dθ



−



= −

We will need the following formula:

Proposition.

For an invertible

n × n

matrix

and

, ¯η

, θ

independent

fermionic variables for i = 1, . . . , n, we have

Z(η, ¯η) =

θ d

θ exp



+ ¯η



= det B exp



¯η

−1

)



In particular, we have

Z = Z(0, 0) = det B.

As before, for any function f, we can again define the correlation function

hf(

θ, θ)i =

Z(0, 0)

θ d

θ e

−S(

θ,θ)

θ, θ).

Note that usually,

contains even-degree terms only. Then it doesn’t matter

if we place f on the left or the right of the exponential.

It is an exercise on the first example sheet to prove these, derive Feynman

rules, and investigate further examples.

2 QFT in one dimension (i.e. QM)

2.1 Quantum mechanics

= 1, there are two possible connected, compact manifolds

— we can

have M = S

or M = I = [0, 1].

M = I

We will mostly be considering the case where

. In this case, we need to

specify boundary conditions on the field in the path integral, and we will see that

this corresponds to providing start and end points to compute matrix elements.

We let

t ∈

1] be the worldline coordinate parametrizing the field, and we

write our field as

x : I → N

for some Riemannian manifold (

N, g

), which we call the target manifold. If

U ⊆ N

has coordinates

with

= 1

, ··· , n

dim

(

), then we usually write

(t) for the coordinates of x(t). The standard choice of action is

S[x] =



g( ˙x, ˙x) + V (x)



dt,

where ˙x is as usual the time derivative, and V (x) is some potential term.

We call this theory a non-linear

-model. This is called

-model because

when people first wrote this theory down, they used

for the name of the field,

and the name stuck. This is non-linear because the term

(

˙x, ˙x

) can be very

complicated when the metric

if it is not flat. Note that +

(

) is the correct

sign for a Euclidean worldline.

Classically, we look for the extrema of

[

] (for fixed end points), and the

solutions are given by the solutions to

+ Γ

˙x

= g

∂V

∂x

The left-hand side is just the familiar geodesic equation we know from general

relativity, and the right hand term corresponds to some non-gravitational force.

In the case of zero-dimensional quantum field theory, we just wrote down

the partition function

Z =

−S

and assumed it was an interesting thing to calculate. There wasn’t really a

better option because it is difficult to give a physical interpretation to a zero-

dimensional quantum field theory. In this case, we will see that path integrals

naturally arise when we try to do quantum mechanics.

To do quantum mechanics, we first pick a Hilbert space

. We usually take

it as

H = L

(N, dµ),

the space of square-integrable functions on N.

To describe dynamics, we pick a Hamiltonian operator

H → H

, with the

usual choice being

H =

∆ + V,

where the Laplacian ∆ is given by

∆ =

√

∂

(

√

∂

As usually, the

√

g refers to the square root of the determinant of g.

We will work in the Heisenberg picture. Under this set up, the amplitude for

a particle to travel from x ∈ N to y ∈ N in time T is given by the heat kernel

(y, x) = hy|e

−HT

|xi.

Note that we have

−HT

instead of

−iHT

because we are working in a Euclidean

world. Strictly speaking, this doesn’t really make sense, because

|xi

and

|yi

are not genuine elements of

(

). They are

-functions, which aren’t

functions. There are some ways to fix this problem. One way is to see that the

above suggests K

satisfies

∂

∂t

(y, x) + HK

(y, x) = 0,

where we view

as a fixed parameter, and

is a function of

, so that

can

act on K

. The boundary condition is

lim

t→0

(y, x) = δ(y −x),

and this uniquely specifies

. So we can define

to be the unique solution to

this problem.

We can reconnect this back to the usual quantum mechanics by replacing

the Euclidean time

with

, and the above equation gives us the Schr¨odinger

equation

∂K

∂t

(y, x) = HK

(y, x).

We first focus on the case where

= 0. In the case where (

N, g

) = (

, δ

), we

know from, say, IB Methods, that the solution is given by

(y, x) =

(2πt)

n/2

exp



−

|x − y|



For an arbitrary Riemannian manifold (

N, g

), it is in general very hard to write

down a closed-form expression for

. However, we can find the asymptotic form

lim

t→0

(y, x) ∼

a(x)

(2πt)

n/2

exp



−

d(y, x)



where

(

x, y

) is the geodesic distance between

and

, and

(

) is some invariant

of our manifold built from (integrals of) polynomials of the Riemann curvature

that isn’t too important.

Here comes the magic. We notice that since

I =

z |zihz|

is the identity operator, we can write

(y, x) = hy|e

−T H

|xi

z hy|e

−t

|zihz|e

−t

|xi

z K

(y, z)K

(z, x).

For flat space, this is just reduces to the formula for convolution of Gaussians.

This is the concatenation property of the heat kernel.

Using this many times, we can break up our previous heat kernel by setting

∆t =

for some large N ∈ N. Then we have

(y, x) =

N−1

i=1

∆t

, x

i−1

where we conveniently set x

= x and x

= y.

The purpose of introducing these ∆

is that we can now use the asymptotic

form of K

(y, t). We can now say

−HT

= lim

N→∞



2π∆t



nN/2

N−1

i=1

a(x

) exp

−

∆t



d(x

i+1

, x

)

∆t



This looks more-or-less like a path integral! We now dubiously introduce the

path integral measure

≡ lim

N→∞



2π∆t



nN/2

N−1

i=1

a(x

and also assume our map

(

) is at least once-continuously differentiable, so that

lim

N→∞

N−1

i=1

exp

−

∆t



d(x

i+1

, x

)

∆t



≡ exp



−

dt g( ˙x, ˙x)



= exp(−S[x]).

Assuming these things actually make sense (we’ll later figure out they don’t),

we can write

−HT

i =

]

Dx e

−S[x]

where

[

, y

] is the space of “all” maps

I → N

such that

(0) =

and

x(1) = y

Now consider an arbitrary V 6= 0. Then we can write

H = H

+ V (x),

where H

is the free Hamiltonian. Then we note that for small t, we have

−Ht

= e

−H

−V (x)t

+ o(t).

Thus, for small t, we have

(y, x) = hy|e

−Ht

|xi

≈ hy|e

−H

−V (x)t

|xi

∼

a(x)

(2πt)

n/2

exp

−



d(y, x)



+ V (x)

Then repeating the above derivations will again give us

−HT

i =

]

Dx e

−S[x]

Before we move on to express other things in terms of path integrals, and

then realize our assumptions are all wrong, we make a small comment on the

phenomena we see here.

Notice that the states

i ∈ H

and

| ∈ H

∗

we used to evaluate our

propagator here arise as boundary conditions on the map

. This is a general

phenomenon. The co-dimension-1 subspaces (i.e. subspaces of

of one dimen-

sion lower than

) are associated to states in our Hilbert space

. Indeed, when

we did quantum field theory via canonical quantization, the states corresponded

to the state of the universe at some fixed time, which is a co-dimension 1 slice of

spacetime.

The partition function

We can naturally interpret a lot of the things we meet in quantum mechanics via

path integrals. In statistical physics, we often called the quantity

(

−HT

)

the partition function. Here we can compute it as

−HT

) =

y hy|e

−HT

|yi.

Using our path integral formulation, we can write this as

−HT

) =

[y,y]

Dx e

−S

Dx e

−S

where we integrate over all circles. This is the partition function

(

N, g, V

))

of our theory. If one is worried about convergence issues, then we would have some

problems in this definition. If we work in flat space, then

(

y, y

) =

hy|e

−HT

|yi

is independent of

. So when we integrate over all

, the answer diverges (as

long as

(

y, y

) is non-zero). However, we would get a finite result if we had a

compact manifold instead.

Correlation functions

More interestingly, we can do correlation functions. We will begin by considering

the simplest choice — local operators.

Definition

(Local operator)

A local operator

(

) is one which depends on

the values of the fields and finitely many derivatives just at one point t ∈ M.

We further restrict to local operators that do not depend on derivatives.

These are given by functions

N → R

, and then by pullback we obtain an

operator O(x(t)).

Suppose the corresponding quantum operator is

(

ˆx

), characterized by

the property

O|xi = O(x) |xi.

If we want to evaluate this at time t, then we would compute

O(t) |y

i = hy

−H(T −t)

−Ht

But, inserting a complete set of states, this is equal to

x hy

−H(T −t)

|xihx|

−Ht

|yi

x O(x) hy

−H(T −t)

|xihx|e

−Ht

|yi.

Simplifying notation a bit, we can write

O(t) |y

i =

x O(x(t))

[T,t]

]

Dx e

−S[x]

[t,0]

]

Dx e

−S[x]

But this is just the same as

[T,0]

]

Dx O(x(t))e

−S[x]

More generally, suppose we have a sequence of operators

, ··· , O

we want

to evaluate at times

T > t

> t

n−1

> ··· > t

0, then by the same argument,

we find

−HT

) ···

) |y

i =

[0,T ]

]

Dx O

(x(t

)) ···O(x(t

))e

−S[x]

Note that it is crucial that the operators are ordered in this way, as one would

see if they actually try to prove this. Indeed, the

are operators, but the

objects in the path integral, i.e. the

(

)) are just functions. Multiplication

of operators does not commute, but multiplication of functions does. In general,

if {t

} ∈ (0, T ) are a collection of times, then we have

i=1

(x(t

))e

−S[x]

= hy

−HT

i=1

) |y

where T denotes the time ordering operator. For example, for n = 2, we have

T [

)

)] = Θ(t

− t

)

) + Θ(t

− t

)

where Θ is the step function.

It is interesting to note that in this path integral formulation, we see that

we get non-trivial correlation between operators at different times only because

of the kinetic (derivative) term in the action. Indeed, for a free theory, the

discretized version of the path integral looked like

kin

[x] =



i+1

− x

∆t



∆t,

Now if we don’t have this term, and the action is purely potential:

pot

[x] =

V (x

then the discretized path integral would have factorized into a product of integrals

at these sampling points

. It would follow that for any operators

}

that

depend only on position, we have

(x(t

))

(x(t

))i,

and this is incredibly boring. When we work with higher dimensional universes,

the corresponding result shows that if there are no derivative terms in the

potential, then events at one position in spacetime have nothing to do with

events at any other position.

We have already seen this phenomena when we did quantum field theory

with perturbation theory — the interactions between different times and places

are given by propagators, and these propagators arise from the kinetic terms in

the Lagrangian.

Derivative terms

We now move on to consider more general functions of the field and its derivative.

Consider operators

(

x, ˙x, ···

). We might expect that the value of this operator

is related to path integrals of the form

Dx O

(x, ˙x, ···)|

−S[x]

But this can’t be right. We were told that one of the most important properties

of quantum mechanics is that operators do not commute. In particular, for

= ˙x

, we had the renowned commutator relation

[ˆx

, ˆp

] = δ

But in this path integral formulation, we feed in functions to the path integral,

and it knows nothing about how we order

and

˙x

in the operators

. So what

can we do?

The answer to this is really really important. The answer is that path

integrals don’t work.

The path integral measure

Recall that to express our correlation functions as path integrals, we had to take

the limits

= lim

N→∞

(2π∆t)

nN/2

N−1

i=1

a(x

and also

S[x]

= lim

N→∞

N−1

n=1



n+1

− x

∆t



∆t.

Do these actually make sense?

What we are trying to do with these expressions is that we are trying to

regularize our path integral, i.e. find a finite-dimensional approximation of the

path integral. For quantum field theory in higher dimensions, this is essentially

a lattice regularization.

Before we move on and try to see if this makes sense, we look at another way

of regularizing our path integral. To do so, we decompose our field into Fourier

modes:

(t) =

k∈Z

−2πikt/T

and then we can obtain a regularized form of the action as

[x] =

k=−N

Under this decomposition, we can take the regularized path integral measure to

x =

|k|≤N

This is analogous to high-energy cutoff regularization. Now the natural question

to ask is — do the limits

lim

N→∞

x, lim

N→∞

[x]

exist?

The answer is, again: NO! This is in fact not a problem of us not being able

to produce limits well. It is a general fact of life that we cannot have a Lebesgue

measure on an infinite dimensional inner product space (i.e. vector space with

an inner product).

Recall the following definition:

Definition

(Lebesgue measure)

A Lebesgue measure d

on an inner product

space V obeys the following properties

– For all non-empty open subsets U ⊆ R

, we have

vol(U ) =

dµ > 0.

– If U

is obtained by translating U, then

vol(U

) = vol(U ).

–

Every

x ∈ V

is contained in at least one open neighbourhood

with

finite volume.

note that we don’t really need the inner product structure in this definition.

We just need it to tell us what the word “open” means.

We now prove that there cannot be any Lebesgue measure on an infinite

dimensional inner product space.

We first consider the case of a finite-dimensional inner product space. Any

such inner product space is isomorphic to

for some

. Write

(

) for an

open hypercube of side length

. By translation in variance, the volume of any

two such hypercubes would be the same.

Now we note that given any such hypercube, we can cut it up into 2

hypercubes of side length L/2:

Then since

(

) contains 2

disjoint copies of

(

2) (note that it is not exactly

the union of them, since we are missing some boundary points), we know that

vol(C(L)) ≥

i=1

vol(C(L/2)) = 2

vol(C(L/2)),

Now in the case of an infinite dimensional vector space,

(

) will contain

infinitely many copies of

(

2). So since

vol

(

2)) must be non-zero, as it

is open, we know

vol

(

)) must be infinite, and this is true for any

. Since

any open set must contain some open hypercube, it follows that all open sets

have infinite measure, and we are dead.

Theorem.

There are no Lebesgue measures on an infinite dimensional inner

product space.

This means whenever we do path integrals, we need to understand that we

are not actually doing an integral in the usual sense, but we are just using a

shorthand for the limit of the discretized integral

lim

N→∞



2π∆t



nN/2

N−1

i=1

exp

−



i+1

− x

∆t



∆t

as a whole. In particular, we cannot expect the familiar properties of integrals

to always hold for our path integrals.

If we just forget about this problem and start to do path integrals, then we

would essentially be writing down nonsense. We can follow perfectly logical steps

and prove things, but the output will still be nonsense. Then we would have

to try to invent some new nonsense to make sense of the nonsense. This was,

in fact, how renormalization was invented! But as we will see, that is not what

renormalization really is about.

Note that we showed that the measure D

doesn’t exist, but what we really

need wasn’t Dx. What we really needed was

Dx e

−S[x]

This is no longer translation invariant, so it is conceivable that it exists. Indeed,

in the case of a 1D quantum field theory, it does, and is known as the Wiener

measure.

In higher dimensions, we are less certain. We know it doesn’t exist for QED,

and we believe it does not exist for the standard model. However, we believe

that it does exist for Yang–Mills theory in four dimensions.

Non-commutativity in QM

Now we know that the path integral measure doesn’t exist, and this will solve

our problem with non-commutativity. Indeed, as we analyze the discretization

of the path integral, the fact that

[ˆx, ˆp] 6= 0

will fall out naturally.

Again, consider a free theory, and pick times

T > t

> t > t

−

> 0.

We will consider

Dx x(t) ˙x(t

−

−S

= hy

−H(T −t)

ˆxe

−H(t−t

−

)

ˆpe

−Ht

−

Dx x(t) ˙x(t

−S

= hy

−H(t−t

)

ˆpe

−H(t

−t)

ˆxe

−Ht

As we take the limit t

→ t, the difference of the right hand sides becomes

−H(t−t)

[ˆx, ˆp]e

−Ht

i = hy

−HT

i 6= 0.

On the other hand, in the continuum path integral, the limit seems to give the

same expression in both cases, and the difference vanishes, naively. The problem

is that we need to regularize. We cannot just bring two operators together in

time and expect it to behave well. We saw that the path integral was sensitive to

the time-ordering of the operators, so we expect something “discrete” to happen

when the times cross. This is just like in perturbation theory, when we bring two

events together in time, we have to worry that the propagators become singular.

Normally, we would have something like

x(t) ˙x(t

−

) − x(t) ˙x(t

) = x



−

− x

−

−δt

δt



− x



+δt

− x

δt



In the regularized integral, we can keep increasing

−

and decreasing

, until

we get to the point



− x

t−∆t

∆t



− x



t+∆t

− x

∆t



Now that

have hit

, we need to look carefully what happens to the individual

heat kernels. In general, we stop taking the limit as soon as any part of the

discretized derivative touches

. The part of the integral that depends on

looks like

∆t

t+∆t

, x



− x

t−∆t

∆t

−

t+∆t

− x

∆t



∆t

, x

t−∆t

Using the fact that

∆t

, x

t−∆t

) ∼ exp



−

− x

t−∆t

)

2∆t



we can write the integral as

∂

∂x



∆t

t+∆t

, x

∆t

, x

t−∆t

)



Now integrating by parts, we get that this is equal to

∆t

t+∆t

, x

∆t

, x

t−∆t

) = K

2∆t

t+∆t

, x

t−∆t

So we get the same as in the operator approach.

2.2 Feynman rules

Consider a theory with a single field x : S

→ R, and action

S[x] =



˙x



We pick

as our universe so that we don’t have to worry about boundary

conditions. Then the path integral for the partition function is

Dx x(t

) ···x(t

) e

−S[x]

∼

n=0

i=1

Dx e

−S

free

[x]

(4!)

i=1

x(t

)

n=0

i=1

(4!)

i=1

x(t

)

free

So we have again reduced the problem to computing the correlators of the free

theory.

Instead of trying to compute these correlators directly, we instead move to

momentum space. We write

x(t) =

k∈Z

−ikt

For the sake of brevity (or rather, laziness of the author), we shall omit all

factors of 2π. Using orthogonality relations, we have

S[x] =

k∈Z

+ m

)|x

∈Z

δ(k

+ k

)

Note that here the

is the “discrete” version, so it is 1 if the argument vanishes,

and 0 otherwise.

Thus, we may equivalently represent

∼

n=0

(i)

}

i=1

δ(k

(i)

+ k

(i)

+ k

(i)

+ k

(i)

)

i=1

j=1

(i)

free

This time, we are summing over momentum-space correlators. But in momentum

space, the free part of the action just looks like countably many free, decoupled

0-dimensional fields! Moreover, each correlator involves only finitely many of

these fields. So we can reuse our results for the 0-dimensional field, i.e. we can

compute these correlators using Feynman diagrams! This time, the propagators

have value

If we are working over a non-compact space, then we have a Fourier transform

instead of a Fourier series, and we get

∼

n=0

i=1

j=1

(i)

i=1

δ(k

(i)

)

i=1

j=1

x(k

(i)

)

free

2.3 Effective quantum field theory

We now see what happens when we try to obtain effective field theories in 1

dimension. Suppose we have two real-valued fields

x, y

→ R

. We pick the

circle as our universe so that we won’t have to worry about boundary conditions.

We pick the action

S[x, y] =



˙x

˙y



dt.

As in the zero-dimensional case, we have Feynman rules

1/(k

+ m

) 1/(k

+ M

)

−λ

As in the case of zero-dimensional QFT, if we are only interested in the correla-

tions involving

(

), then we can integrate out the field

(

) first. The effective

potential can be written as

Dy exp



−



−

+ M

λx



y dt



where we integrated by parts to turn ˙y

to y ¨y.

We start doing dubious things. Recall that we previously found that for a

bilinear operator M : R

× R

→ R, we have

x exp



−

M(x, x)



(2π)

n/2

√

det M

Now, we can view our previous integral just as a Gaussian integral over the

operator

(y, ˜y) 7→



−

+ M

λx



˜y dt (∗)

on the vector space of fields. Thus, (ignoring the factors of (2

)

n/2

) we can

formally write the integral as

det



−

+ M

λx



−1/2

eff

[x] thus looks like

eff

[x] =

( ˙x

+ m

) dt +

log det



−

+ M

λx



We now continue with our formal manipulations. Note that

log det

tr log

since

det

is the product of eigenvalues and

is the sum of them. Then if we

factor our operators as

−

+ M

λx



−

+ M



1 − λ



−

+ M



−1

then we can write the last term in the effective potential as

tr log



−

+ M



tr log

1 − λ



− M



−1

The first term is field independent, so we might as well drop it. We now look

carefully at the second term. The next dodgy step to take is to realize we know

what the inverse of the differential operator

− M

is. It is just the Green’s function! More precisely, it is the convolution with the

Green’s function. In other words, it is given by the function G(t, t

) such that



− M



G(t, t

) = δ(t − t

Equivalently, this is the propagator of the

field. If we actually try to solve this,

we find that we have

G(t, t

) =

n∈Z

exp



−M



t − t





We don’t actually need this formula. The part that will be important is that it

is ∼

We now try to evaluate the effective potential. When we expand

log



1 − λG(t, t

)



the first term in the expansion is

−λG(t, t

)

What does it mean to take the trace of this? We pick a basis for the space we

are working on, say {δ(t −t

) : t

∈ S

}. Then the trace is given by

∈S

t∈S

dt δ(t − t

)



∈S

(−λ)G(t, t

)

δ(t

− t

)



We can dissect this slowly. The rightmost integral is nothing but the definition

of how

acts by convolution. Then the next

integral is the definition of how

bilinear forms act, as in (

∗

). Finally, the integral over

is summing over all

basis vectors, which is what gives us the trace. This simplifies rather significantly

−

t∈S

G(t, t)x

(t) dt.

In general, we find that we obtain

tr log



1 − λG(t, t

)



= −

G(t, t)x

(t) dt −

×S

dt dt

G(t

, t)x

(t)G(t, t

) ···

These terms in the effective field theory are non-local! It involves integrating

over many different points in

. In fact, we should have expected this non-

locality from the corresponding Feynman diagrams. The first term corresponds

x(t)

Here G(t, t) corresponds to the y field propagator, and the −

comes from the

vertex.

The second diagram we have looks like this:

x(t)

x(t

)

x(t

)

We see that the first diagram is local, as there is just one vertex at time

. But

in the second diagram, we use the propagators to allow the

at time

to talk

to x at time t

. This is non-local!

Non-locality is generic. Whenever we integrate out our fields, we get non-local

terms. But non-locality is terrible in physics. It means that the equations of

motion we get, even in the classical limit, are going to be integral differential

equations, not just normal differential equations. For a particle to figure out

what it should do here, it needs to know what is happening in the far side of the

universe!

To make progress, we note that if

is very large, then we would expect

(

t, t

) could be highly suppressed for

t 6

. So we can try to expand around

t = t

. Recall that the second term is given by

dt dt

G(t, t

)

(t)x

)

We can write out x

) as

) = x

(t) + 2x(t) ˙x(t)(t

− t) +



˙x

(t) +

x(t) ˙x(t)



(t − t

)

+ ··· .

Using the fact that

(

t, t

) depends on

only through

(

−t

), by dimensional

analysis, we get an expansion that looks like

(t) +



˙x

¨x



(4-derivative terms) + ···

Here α, β, γ are dimensionless quantities.

Thus, we know that every extra derivative is accompanied by a further power

. Thus, provided

(

) is slowly varying on scales of order

, we may hope

to truncate the series.

Thus, at energies

E  M

, our theory looks approximately local. So as long

as we only use our low-energy approximation to answer low-energy questions, we

are fine. However, if we try to take our low-energy theory and try to extrapolate

it to higher and higher energies, up to

E ∼ M

, it is going to be nonsense. In

particular, it becomes non-unitary, and probability is not preserved.

This makes sense. By truncating the series at the first term, we are ignoring

all the higher interactions governed by the

fields. By ignoring them, we are

ignoring some events that have non-zero probability of happening, and thus we

would expect probability not to be conserved.

There are two famous examples of this. The first is weak interactions. At

very low energies, weak interactions are responsible for

-decay. The effective

action contains a quartic interaction

nν

p G

weak

This coupling constant

weak

has mass dimensional

−

1. At low energies, this is

a perfectly well description of beta decay. However, this is suspicious. The fact

that we have a coupling constant with negative mass dimension suggests this

came from integrating some fields out.

At high energies, we find that this 4-Fermi theory becomes non-unitary, and

weak

is revealed as an approximation to a

-boson propagator. Instead of an

interaction that looks like this:

what we really have is

There are many other theories we can write down that has negative mass

dimension, the most famous one being general relativity.

2.4 Quantum gravity in one dimension

In quantum gravity, we also include a (path) integral over all metrics on our

spacetime, up to diffeomorphism (isometric) invariance. We also sum over all

possible topologies of

. In

= 1 (and

= 2 for string theory), we can just do

this.

= 1, a metric

only has one component

(

) =

(

). There is no

curvature, and the only diffeomorphism invariant of this metric is the total

length

T =

e(t) dt.

So the instruction to integrate over all metrics modulo diffeomorphism is just

the instruction to integrate over all possible lengths of the worldline

T ∈

, ∞

which is easy. Let’s look at that.

The path integral is given by

[0,T ]

[y,x]

Dx e

−S[x]

where as usual

S[x] =

˙x

dt.

Just for fun, we will include a “cosmological constant” term into our action, so

that we instead have

S[x] =

˙x

dt.

The reason for this will be revealed soon.

We can think of the path integral as the heat kernel, so we can write it as

∞

dT hy|e

−HT

|xi =

∞

p d

(2π)

hy|qihq|e

−HT

|pihp|xi

∞

p d

(2π)

ip·x−iq·y

−T (p

)/2

(p − q)

∞

(2π)

ip·(x−y)

−T (p

)/2

= 2

(2π)

ip·(x−y)

+ m

= 2D(x, y),

where

(

x, y

) is the propagator for a scalar field on the target space

with

action

S[Φ] =

(∂Φ)

So a 1-dimensional quantum gravity theory with values in

is equivalent to

(or at least has deep connections to) a scalar field theory on R

How about interactions? So far, we have been taking rather unexciting

1-dimensional manifolds as our universe, and there are only two possible choices.

If we allow singularities in our manifolds, then we would allow graphs instead of

just a line and a circle. Quantum gravity then says we should not only integrate

over all possible lengths, but also all possible graphs.

For example, to compute correlation functions such as

Φ(

)

···

Φ(

)

theory on

, say, we consider all 4-valent with

external legs with one

endpoint at each of the x

, and then we proceed just as in quantum gravity.

For example, we get a contribution to hΦ(x)Φ(y)i from the graph

x x

The contribution to the quantum gravity expression is

z∈R

[0,∞)

[z,x]

Dx e

−S

[x]

[z,z]

Dx e

−S

[x]

[y,z]

Dx e

−S

[x]

where

[x] =

dt +

dt.

We should think of the second term as the “cosmological constant”, while the

1D integrals over

’s are the “1d quantum gravity” part of the path integral

(also known as the Schwinger parameters for the graph).

We can write this as

z dT

hz|e

−HT

|xihz|e

−HT

|zihy|e

−HT

|zi.

Inserting a complete set of eigenstates between the position states and the time

evolution operators, we get

p d

` d

(2π)

ip·(x−z)

+ m

iq·(y−z)

+ m

i`·(z−z)

+ m

p d

(2π)

ip·(x−y)

+ m

)

+ m

This is exactly what we would have expected if we viewed the above diagram as

a Feynman diagram:

p p

This is the worldline perspective to QFT, and it was indeed Feynman’s original

approach to doing QFT.

3 Symmetries of the path integral

From now on, we will work with quantum field theory in general, and impose no

restrictions on the dimension of our universe. The first subject to study is the

notion of symmetries.

We first review what we had in classical field theory. In classical field theory,

Noether’s theorem relates symmetries to conservation laws. For simplicity, we

will work with the case of a flat space.

Suppose we had a variation

δφ = εf(φ, ∂φ)

of the field. The most common case is when

(

φ, ∂φ

) depends on

only locally,

in which case we can think of the transformation as being generated by the

vector

x f(φ, ∂φ)

δφ(x)

acting on the “space of fields”.

If the function

[

] is invariant under

when

is constant, then for general

ε(x), we must have

δS =

x j

(x) ∂

ε.

for some field-dependent current

(

) (we can actually find an explicit expression

for j

). If we choose ε(x) to have compact support, then we can write

δS = −

x (∂

) ε(x).

On solutions of the field equation, we know the action is stationary under

arbitrary variations. So δS = 0. Since ε(x) was arbitrary, we must have

∂

= 0.

So we know that j

is a conserved current.

Given any such conserved current, we can define the charge

[

] associated

to an (oriented) co-dimension 1 hypersurface N as

Q[N] =

d−1

where

is the normal vector to

. Usually,

is a time slice, and the normal

points in the future direction.

Now if

and

are two such hypersurfaces bounding a region

⊆ M

then by Stokes’ theorem, we have

Q[N

] − Q[N

] =



−



n−1

x =

(∂

) d

x = 0.

So we find

Q[N

] = Q[N

This is the conservation of charge!

3.1 Ward identities

The derivation of Noether’s theorem used the classical equation of motion. But in

a quantum theory, the equation of motion no longer holds. We must re-examine

what happens in the quantum theory.

Suppose a transformation φ 7→ φ

of the fields has the property that

Dφ e

−S[φ]

= Dφ

−S[φ

]

In theory, the whole expression D

φ e

−S[φ]

is what is important in the quantum

theory. In practice, we often look for symmetries where D

and

−S[φ]

are

separately conserved. In fact, what we will do is that we look for a symmetry

that preserves

[

], and then try to find a regularization of D

that is manifestly

invariant under the symmetry.

Often, it is not the case that the obvious choice of regularization of D

manifestly invariant under the symmetry. For example, we might have a

[

]

that is rotationally invariant. However, if we regularize the path integral measure

by picking a lattice and sampling

on different points, it is rarely the case that

this lattice is rotationally invariant.

In general, there are two possibilities:

(i)

The symmetry could be restored in the limit. This typically means there

exists a regularized path integral measure manifestly invariant under this

symmetry, but we just didn’t use it. For example, rotational invariance is

not manifestly present in lattice regularization, but it is when we do the

cut-off regularization.

(ii)

It could be that the symmetry is not restored. It is said to be anomalous,

i.e. broken in the quantum theory. In this case, there can be no invariant

path integral measure. An example is scale invariance in QED, if the mass

of the electron is 0.

Sometimes, it can be hard to tell. For now, let’s just assume we are in a situation

where Dφ = Dφ

when ε is constant. Then for any ε(x), we clearly have

Z =

Dφ e

−S[φ]

Dφ

−S[φ

]

since this is just renaming of variables. But using the fact that the measure is

invariant when

is constant, we can can expand the right-hand integral in

and again argue that it must be of the form

Dφ

−S[φ

]

Dφ e

−S[φ]



1 −

∂

ε d



in first order in

. Note that in general,

can receive contributions from

[

]

and D

. But if it doesn’t receive any contribution from D

, then it would just

be the classical current.

Using this expansion, we deduce that we must have

Dφ e

−S[φ]

∂

ε d

x = 0.

Integrating by parts, and using the definition of the expectation, we can write

this as

−

ε∂

(x)i d

for any

with compact support. Note that we dropped a normalization factor

in the definition of

(

)

, because

times zero is still zero. So we know

that hj

(x)i is a conserved current, just as we had classically.

Symmetries of correlation functions

Having a current is nice, but we want to say something about actual observable

quantities, i.e. we want to look at how how symmetries manifest themselves with

correlation functions. Let’s look at what we might expect. For example, if our

theory is translation invariant, we might expect, say

hφ(x)φ(y)i = hφ(x − a)φ(y − a)i

for any a. This is indeed the case.

Suppose we have an operator

(

). Then under a transformation

φ 7→ φ

our operator transforms as

O(φ) 7→ O(φ

By definition, the correlation function is defined by

hO(φ)i =

Dφ e

−S[φ]

O(φ).

Note that despite the appearance of

on the left, it is not a free variable. For

example, the correlation hφ(x

)φ(x

)i is not a function of φ.

We suppose the transformation

φ 7→ φ

is a symmetry. By a trivial renaming

of variables, we have

hO(φ)i =

Dφ

−S[φ

]

O(φ

)

By assumption, the function D

φ e

−S[φ]

is invariant under the transformation.

So this is equal to

Dφ e

−S[φ]

O(φ

)

= hO(φ

)i.

This is, of course, not surprising. To make this slightly more concrete, we look

at an example.

Example.

Consider (

M, g

) = (

, δ

), and consider spacial translation

x 7→ x

x − a for a constant vector a. A scalar field φ transforms as

φ(x) 7→ φ

(x) = φ(x − a).

In most cases, this is a symmetry.

We suppose O(φ) can be written as

O(φ) = O

(φ(x

)) ···O

(φ(x

)),

where

depends only on the value of

. A canonical example is when

O(φ) = φ(x

) ···φ(x

) is an n-point correlation function.

Then the above result tells us that

(φ(x

)) ···O

(φ(x

))i = hO

(φ(x

− a)) ···O

(φ(x

− a))i

So the correlation depends only on the separations

−x

. We can obtain similar

results if the action and measure are rotationally or Lorentz invariant.

Example.

Suppose we have a complex field

, and we have a transformation

φ 7→ φ

iα

for some constant

α ∈ R/

πZ

. Then the conjugate field

transforms as

φ 7→

= e

−iα

φ.

Suppose this transformation preserves the action and measure. For example,

the measure will be preserved if we integrate over the same number of

and

modes. Consider the operators

(φ,

φ) = φ(x

)

φ(x

)

Then the operators transform as

(φ,

φ) 7→ O

(φ

) = e

−α(r

−s

)

(φ,

φ).

So symmetry entails

i=1

)

= exp

iα

i=1

− s

)

i=1

)

Since this is true for all α, the correlator must vanish unless

i=1

So we need the same number of φ and

φ insertions in total.

We can interpret this in terms of Feynman diagrams — each propagator joins

up a

and

. So if we don’t have equal number of

and

, then we can’t draw

any Feynman diagrams at all! So the correlator must vanish.

Ward identity for correlators

What we’ve done with correlators was rather expected, and in some sense trivial.

Let’s try to do something more interesting.

Again, consider an operator that depends only on the value of

at finitely

many points, say

O(φ) = O

(φ(x

)) ···O

(φ(x

)).

As before, we will write the operators as O

) when there is no confusion.

As in the derivation of Noether’s theorem, suppose we have an infinitesimal

transformation, with

φ 7→ φ

εδφ

that is a symmetry when

is constant. Then

for general ε(x), we have

Dφ e

−S[φ]

(φ(x

)) ···O

(φ(x

))

Dφ

−S[φ

]

(φ

)) ···O

(φ

))

Dφ e

−S[φ]



1 −

(x)∂

ε(x) dx







(φ(x

)) ···O

(φ(x

))

i=1

ε(x

)δO

)

j6=i

)





where

δO(x

) =

∂O

∂φ

δφ.

Again, the zeroth order piece of the correlation function cancels, and to lowest

non-trivial order, we find that we must have

∂

ε(x)

(x)

i=1

)

x =

i=1

ε(x

)

δO

)

j6=i

)

On the left, we can again integrate by parts to shift the derivative to the current

term. On the right, we want to write it in the form

(

)

···

, so that we

can get rid of the

term. To do so, we introduce some

-functions. Then we

obtain

ε(x)∂

(x)

i=1

)

= −

i=1

ε(x)δ

(x − x

)

δO

)

j6=i

)

Since this holds for arbitrary ε(x) (with compact support), we must have

∂

(x)

)

= −

i=1

(x − x

)

δO

)

j6=i

)

This is the Ward identity for correlation functions. It says that the vector field

(x, x

) =

(x)

)

is divergence free except at the insertions x

This allows us to recover the previous invariance of correlations. Suppose

is compact without boundary. We then integrate the Ward identity over all

By Stokes’ theorem, we know the integral of any divergence term vanishes. So

we obtain

0 =

∂

(x, x

) d

x = −

i=1

δO

)

j6=i

)

= δ

i=1

)

Of course, this is what we would obtain if we set ε(x) ≡ 1 above.

That was nothing new, but suppose

⊆ M

is a region with boundary

− N

Let’s see what integrating the Ward identity over

gives us. The left hand

side gives us

∂

(x)

i=1

)

x =

−N

(x)

i=1

)

d−1

Q[N

]

i=1

)

−

Q[N

]

i=1

)

The right hand side of Ward’s identity just gives us the sum over all points inside

Q[N

]

i=1

)

−

Q[N

]

i=1

)

= −

∈M

δO

)

j6=i

)

In particular, if

contains only one point, say

, and we choose the region to

be infinitesimally thin, then in the canonical picture, we have

hΩ|T [

)]

j=2

) |Ωi = −hΩ|T







j=2

)







|Ωi,

where |Ωi is some (vacuum) state. So in the canonical picture, we find that

O = −[

O].

So we see that the change of

under some transformation is given by the

commutator with the charge operator.

3.2 The Ward–Takahashi identity

We focus on an important example of this. The QED action is given by

S[A, ψ] =



µν

+ i

Dψ + m

ψψ



This is invariant under the global transformations

ψ(x) 7→ e

iα

ψ(x), A

(x) 7→ A

(x)

for constant

α ∈ R/

πZ

. The path integral measure is also invariant under this

transformation provided we integrate over equal numbers of

and

modes in

the regularization.

In this case, the classical current is given by

(x) =

ψ(x)γ

ψ(x).

We will assume that D

is invariant under a position-dependent

. This is a

reasonable assumption to make, if we want our measure to be gauge invariant.

In this case, the classical current is also the quantum current.

Noting that the infinitesimal change in

is just (proportional to)

itself,

the Ward identity applied to hψ(x

)

ψ(x

)i gives

∂

(x)ψ(x

)

ψ(x

)i = −δ

(x − x

)hψ(x

)

ψ(x

)i + δ

(x − x

)hψ(x

)

ψ(x

)i.

We now try to understand what these individual terms mean. We first understand

the correlators hψ(x

)

ψ(x

)i.

Recall that when we did perturbation theory, the propagator was defined as

the Fourier transform of the free theory correlator

hψ

(

)

(

)

. This is given

D(k

, k

) =

·x

−ik

·x

hψ(x

)

ψ(x

y d

i(k

−k

)·x

·y

hψ(y)

ψ(0)i

= δ

− k

)

y e

ik·y

hψ(y)

ψ(0)i.

Thus, we can interpret the interacting correlator

hψ

(

)

(

)

as the propa-

gator with “quantum corrections” due to the interacting field.

Definition (Exact propagator). The exact (electron) propagator is defined by

S(k) =

y e

ik·y

hψ(y)

ψ(0)i,

evaluated in the full, interacting theory.

Usually, we don’t want to evaluate this directly. Just as we can compute the

sum over all diagrams by computing the sum over all connected diagrams, then

take exp, in this case, one useful notion is a one-particle irreducible graph.

Definition

(One-particle irreducible graph)

A one-particle irreducible graph for

hψ

ψi

is a connected Feynman diagram (in momentum space) with two external

vertices

and

such that the graph cannot be disconnected by the removal of

one internal line.

This definition is rather abstract, but we can look at some examples to see

what this actually means.

Example. The following are one-particle irreducible graphs:

ψ ψ

while the following is not:

ψ ψ

We will write

1PI

= Σ(

for the sum of all contributions due to one-particle irreducible graphs. This is

known as the electron self-energy. Note that we do not include the contributions

of the propagators connecting us to the external legs. It is not difficult to see

that any Feynman diagram with external vertices

ψ, ψ is just a bunch of 1PI’s

joined together. Thus, we can expand

S(k) ∼

ψ ψ

k + m

+ quantum corrections.

with the quantum corrections given by

ψ ψ

1PI

ψ ψ

1PI 1PI

+ ···

This sum is easy to perform. The diagram with

many 1PI’s has contributions

from

many 1PI’s and

+ 1 many propagators. Also, momentum contribution

forces them to all have the same momentum. So we simply have a geometric

series

S(k) ∼

k + m

Σ(

k + m

Σ(

k + m

Σ(

k + m

+ ···

k + m − Σ(

We can interpret this result as saying integrating out the virtual photons

gives us a shift in the kinetic term by Σ(

k).

We now move on to study the other term. It was the expectation

(x)ψ(x

)

ψ(x

)i.

We note that using the definition of D, our classical action can be written as

S[A, ψ] =

µν

∂ψ + j

+ m

ψψ.

In position space, this gives interaction vertices of the form

Again, we want to consider quantum corrections to this interaction vertex. It

turns out the interesting correlation function is exactly hj

(x)ψ(x

)

ψ(x

)i.

This might seem a bit odd. Why do we not just look at the vertex itself, and

just consider

? Looking at

ψj

instead corresponds to including including

the propagators coming from the external legs. The point is that there can be

photons that stretch across the vertex, looking like

So when doing computations, we must involve the two external electron prop-

agators as well. (We do not include the photon propagator. We explore what

happens when we do that in the example sheet)

We again take the Fourier transform of the correlator, and define

Definition

(Exact electromagnetic vertex)

The exact electromagnetic vertex

, k

) is defined by

(p + k

− k

)S(k

)Γ

, k

)S(k

)

x d

(x)ψ(x

)

ψ(x

)ie

ip·x

·x

−ik

·x

Note that we divided out the

(

) and

(

) in the definition of Γ

(

, k

because ultimately, we are really just interested in the vertex itself.

Can we figure out what this Γ is? Up to first order, we have

hψ(x

(x)

ψ(x

)i ∼ hψ(x

)

ψ(x)iγ

hψ(x)

ψ(x

)i + quantum corrections.

So in momentum space, after dividing out by the exact propagators, we obtain

, k

) = γ

+ quantum corrections.

This first order term corresponds to diagrams that do not include photons going

across the two propagators, and just corresponds to the classical

inside the

definition of j

. The quantum corrections are the interesting parts.

In the case of the exact electron propagator, we had this clever idea of

one-particle irreducible graphs that allowed us to simplify the propagator compu-

tations. Do we have a similar clever idea here? Unfortunately, we don’t. But we

don’t have to! The Ward identity relates the exact vertex to the exact electron

propagator.

Taking the Fourier transform of the Ward identity, and dropping some

δ-functions, we obtain

− k

)

S(k

)Γ

, k

)S(k

) = iS(k

) − iS(k

Recall that

(

) are matrices in spinor space, and we wrote them as

···

. So it is

easy to invert them, and we find

− k

)

, k

) = iS

−1

) − iS

−1

)

= −i(i

+ m − Σ(

) − i

− m + Σ(

))

= (k

− k

)

+ i(Σ(

) − Σ(

)).

This gives us an explicit expression for the quantum corrections of the exact

vertex Γ

in terms of the quantum corrections of the exact propagator S(k).

Note that very little of this calculation relied on what field we actually worked

with. We could have included more fields in the theory, and everything would

still go through. We might obtain a different value of Σ(

), but this relation

between the quantum corrections of Γ

and the quantum corrections of

still

holds.

What is the “philosophical” meaning of this? Recall that the contributions

to the propagator comes from the

∂ψ

term, while the contributions to the

vertex comes from the

Aψ term. The fact that their quantum corrections are

correlated in such a simple way suggests that our quantum theory treats the

Dψ

term as a whole, and so they receive the “same” quantum corrections. In

other words, the quantum theory respects gauge transformations. When we first

studied QED, we didn’t understand renormalization very well, and the Ward

identity provided a sanity check that we didn’t mess up the gauge invariance of

our theory when regularizing.

How could we have messed up? In the derivations, there was one crucial

assumption we made, namely that D

is invariant under position-dependent

transformations

(

)

7→ e

iα(x)

(

). This was needed for

to be the classical

current. This is true if we regularized by sampling our field at different points

in space, as long as we included the same number of ψ and

ψ terms.

However, historically, this is not what we used. Instead, we imposed cutoffs

in the Fourier modes, asking

≤

. This is not compatible with arbitrary

changes

(

)

7→ e

iα(x)

(

), as we can introduce some really high frequency

changes in ψ by picking a wild α.

4 Wilsonian renormalization

4.1 Background setting

We are now going to study renormalization. Most of the time, we will assume we

are talking about a real scalar field ϕ, but the ideas and results are completely

general.

Supposed we did some experiments, obtained some results, and figured that

we are probably working with a quantum field theory described by some action.

But we didn’t test our theory to arbitrarily high energies. We don’t know how

“real physics” looks like at high energy scales. So we can’t really write down a

theory that we can expect is valid to arbitrarily high energies.

However, we have previously seen that “integrating out” high energy particles

has the same effect as just modifying the coupling constants of our theory.

Similarly, even with a single fixed field

, we can integrate out the high energy

modes of the field

, and obtain an effective theory. Suppose we integrate out

all modes with k

≥ Λ

, and obtain an effective action

[ϕ] =

(∂ϕ)

(ϕ, ∂ϕ)

This, by definition, means the partition function of the theory is now given by

Z =

∞

(M)

≤Λ

Dϕ e

−S

[ϕ]

where

∞

(

)

≤Λ

denotes the space of all functions on

consisting of sums

(integrals) of eigenmodes of the Laplacian with eigenvalues

≤

(in “layman”

terms, these are fields with momentum

≤

). This effective action can

answer questions about “low energy physics”, at scales

, which we can use

to test our theory against experiments.

Note that in the case of a compact universe, the Laplacian has discrete

eigenvalues. For example, if we work on a flat torus (equivalently,

with

periodic boundary conditions), then the possible eigenvalues of the Laplacian lie

on a lattice. Then after imposing a cutoff, there are only finitely many energy

modes, and we have successfully regularized the theory into something that

makes mathematical sense.

In the case of a non-compact universe, this doesn’t happen. But still, in

perturbation theory, this theory will give finite answers, not infinite ones. The

loop integrals will have a finite cut-off, and will thus give finite answers. Of

course, we are not saying that summing all Feynman diagrams will give a finite

answer — this doesn’t happen even for 0-dimensional QFTs. (This isn’t exactly

true. There are further subtleties due to “infrared divergences”. We’ll mostly

ignore these, as they are not really problems related to renormalization)

Either way, we have managed to find ourselves a theory that we are reasonably

confident in, and gives us the correct predictions in our experiments.

Now suppose 10 years later, we got the money and built a bigger accelerator.

We can then test our theories at higher energy scales. We can then try to write

down an effective action at this new energy scale. Of course, it will be a different

action, since we have changed the energy scale. However, the two actions are

not unrelated! Indeed, they must give the same answers for our “low energy”

experiments. Thus, we would like to understand how the action changes when

we change this energy scale.

In general, the coupling constants are a function of the energy scale Λ

, and

we will write them as

(Λ

). The most general (local) action can be written as

[ϕ] =

(∂ϕ)

(Λ

)Λ

d−d

(ϕ, ∂ϕ)

where

(

ϕ, ∂ϕ

) are monomials in fields and derivatives, and

the mass di-

mension [

]. Note that this expression assumes that the kinetic term does not

depend on Λ

. This is generally not the case, and we will address this issue later.

We inserted the factor of Λ

d−d

such that the coupling constants

(Λ

) are

dimensionless. This is useful, as we are going to use dimensional analysis a

lot. However, this has the slight disadvantage that even without the effects of

integrating out fields, the coupling constants

must change as we change the

energy scale Λ

To do this, we need to actually figure out the mass dimension of our operators

. Thus, we need to figure out the dimensions of

and

∂

. We know that

itself is dimensionless, since we want to stick it into an exponential. Thus, any

term appearing in the integrand must have mass dimension

(as the measure

has mass dimension −d).

By looking at the kinetic term in the Lagrangian, we deduce that we must

have

[(∂ϕ)

] = d.

as we have to eventually integrate it over space.

Also, we know that [∂

] = 1. Thus, we must have

Proposition.

[∂

] = 1, [ϕ] =

d − 2

4.2 Integrating out modes

Suppose, for some magical reason, we know exactly what the theory at the

energy scale Λ

is, and they are given by the coupling coefficients

(Λ

). What

happens when we integrate out some high energy modes?

We pick some Λ

, and split our field

into “low” and “high” energy

modes as follows:

ϕ(x) =

|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

0≤|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

Λ<|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

We thus define

φ(x) =

0≤|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

χ(x) =

Λ<|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

and so

ϕ(x) = φ(x) + χ(x).

Let’s consider the effective theory we obtain by integrating out

. As before, we

define the scale Λ effective action

[φ] = −~ log

∞

(M)

Λ<|p|<Λ

Dχ e

−S

[ϕ,χ]/~

. (∗)

Of course, this can be done for any Λ

, and so defines a map from [0

] to

the “space of all actions”. More generally, for any

, this procedure allows us to

take a scale Λ action and produce a scale Λ

− ε

effective action from it. This is

somewhat like a group (or monoid) action on the “space of all actions”, and thus

the equation (∗) is known as the Wilsonian renormalization group equation.

Just as we saw in low-dimensional examples, when we do this, the coupling

constants of the interactions will shift. For each Λ

, we can define the

shifted coefficients g

(Λ), as well as Z

and δm

, by the equation

[φ] =

(∂φ)

d−d

(Λ)O

(φ, ∂φ)

where n

is the number of times φ or ∂φ appears in O

Note that we normalized the

(Λ) in terms of the new Λ and

. So even if,

by some miracle, our couplings receive no new corrections, the coefficients still

transform by

(Λ) =





d−d

(Λ

The factor

account from the fact that there could be new contributions to

the kinetic term for

. This is called wavefunction renormalization. The factor

is not to be confused with the partition function, which we denote by a

calligraphic Z instead. We will explore these in more detail later.

We define

Z(Λ, g

(Λ)) =

∞

(M)

≤Λ

Dϕ e

−S

[ϕ]/~

Then by construction, we must have

Z(Λ

, g

(Λ

)) = Z(Λ, g

(Λ))

for all Λ

. This is a completely trivial fact, because we obtained

(Λ

, g

(Λ))

simply by doing part of the integral and leaving the others intact.

We will assume that

varies continuously with Λ (which is actually not the

case when the allowed modes are discrete, but whatever). It is then convenient

to write the above expression infinitesimally, by taking the derivative. Instead

of the usual

dΛ

, it is more convenient to talk about the operator Λ

dΛ

instead,

as this is a dimensionless operator.

Differentiating the above equation, we obtain

dΛ

(Λ, g

(Λ)) = Λ

∂Z

∂Λ



∂Z

∂g



∂g

∂Λ

= 0. (†)

This is the Callan-Symanzik equation for the partition function.

It is convenient to refer to the following object:

Definition (Beta function). The beta function of the coupling g

) = Λ

∂g

∂Λ

As mentioned before, even if our coupling constants magically receive no

corrections, they will still change. Thus, it is convenient to separate out the

boring part, and write

) = (d

− d)g

+ β

quantum

({g

}).

Notice that perturbatively, the

quantum

(

}

) come from loops coming from

integrating out diagrams. So generically, we expect them to depend on all other

coupling constants.

We will later need the following definition, which at this point is rather

unmotivated:

Definition (Anomalous dimension). The anomalous dimension of φ by

= −

∂ log Z

∂Λ

Of course, at any given scale, we can absorb, say,

by defining a new field

ϕ(x) =

so as to give

(

) canonically normalized kinetic terms. Of course, if we do

this at any particular scale, and then try to integrate out more modes, then the

coefficient will re-appear.

4.3 Correlation functions and anomalous dimensions

Let’s say we now want to compute correlation functions. We will write

[φ, g

] =

(∂φ)

d−d

(φ, ∂φ)

where, as before, we will assume

is one the of the

. Note that the action

of the

we produced by integrating out modes is not

[

φ, g

(Λ)], because

we had the factor of

sticking out in the action. Instead, it is given by

1/2

φ, g

(Λ)].

Now we can write a general n-point correlation function as

hφ(x

) ···φ(x

)i =

≤Λ

Dφ e

−S

1/2

φ,g

(Λ)]

φ(x

) ···φ(x

We can invent a canonically normalized field

ϕ(x) =

φ,

so that the kinetic term looks right. Then defining

hϕ(x

) ···ϕ(x

)i =

≤Λ

Dφ e

−S

[ϕ,g

(Λ)]

ϕ(x

) ···ϕ(x

we find

hφ(x

) ···φ(x

)i = Z

−n/2

hϕ(x

) ···ϕ(x

)i.

Note that we don’t have to worry about the factor of

1/2

coming from the

scaling of the path integral measure, as the partition function

is scaled by the

same amount.

Definition (Γ

(n)

). We write

(n)

({x

}, g

) =

≤Λ

Dφ e

−S

[φ,g

]

φ(x

) ···φ(x

) = hϕ(x

) ···ϕ(x

)i.

Now suppose 0

< s <

1, and that we’ve chosen to insert fields only with

energies

< s

Λ. Then we should equally be able to compute the correlator using

the low energy theory S

sΛ

. We’re then going to find

−n/2

sΛ

(n)

sΛ

, ··· , x

, g

(sΛ)) = Z

−n/2

(n)

sΛ

, ··· , x

, g

(Λ)).

Differentiating this with respect to s, we find

dΛ

(n)

, ··· , x

, g

(Λ)) =



∂

∂Λ

+ β

∂

∂g

+ nγ



(n)

({x

}, g

(Λ)) = 0.

This is the Callan-Symanzik equation for the correlation functions.

There is an alternative way of thinking about what happens when change Λ.

We will assume we work over

, so that it makes sense to scale our universe

(on a general Riemannian manifold, we could achieve the same effect by scaling

the metric). The coordinates change by

x 7→ sx

. How does Γ

(

, . . . , x

, g

)

relate to Γ

(sx

, . . . , sx

, g

We unwrap the definitions

(n)

({sx

}, g

) =

≤Λ

Dφ e

−S

[φ,g

]

φ(sx

) ···φ(sx

)

We make the substitution

(

) =

aφ

(

), with a constant

to be chosen later so

that things work out. Again, we don’t have to worry about how D

transforms.

However, this change of variables does scale the Fourier modes, so the new cutoff

of ϕ is in fact sΛ. How the S

[φ, g

] transform? Using the chain rule, we have

sΛ

[ϕ, g

] =

(∂ϕ)

(sΛ)

d−d

(ϕ, ∂ϕ)

Putting in the definition of ϕ, and substituting y = sx, we have

= s

−d

(∂φ)

(sΛ)

d−d

(aφ, as∂φ)

where all fields are evaluated at

. We want this to be equal to

[

φ, g

]. By

looking at the kinetic term, we know that we need

a = s

(d−2)/2

By a careful analysis, we see that the other terms also work out (or we know

they must be, by dimensional analysis). So we have

(n)

({sx

}, g

) =

≤Λ

Dφ e

−S

[φ,g

]

φ(sx

) ···φ(sx

)

≤sΛ

Dϕ e

−S

sΛ

[ϕ,g

]

(d−2)n/2

ϕ(x

) ···ϕ(x

)

= s

(d−2)n/2

(n)

sΛ

({x

}, g

)

Thus, we can write

, ··· , x

, g

(Λ)) =



sΛ



n/2

sΛ

, ··· , x

, g

(sΛ))



2−d

sΛ



n/2

(sx

, ··· , sx

, g

(sΛ)).

Note that in the second step, we don’t change the values of the

! We are just

changing units for measuring things. We are not integrating out modes.

Equivalently, if y

= sx

, then what we found is that



, ··· ,

, g

(Λ)





d−2

sΛ



n/2

, ··· , y

, g

(sΛ)).

What does this equation say? Here we are cutting off the Γ at the same energy

level. As we reduce

, the right hand side has the positions fixed, while on the

left hand side, the points get further and further apart. So on the left hand side,

s →

0, we are probing the theory at longer and longer distances. Thus, what

we have found is that “zooming out” in our theory is the same as flowing down

the couplings g

(sΛ) to a scale appropriate for the low energy theory.

Infinitesimally, let s = 1 − δs, with 0 < δs  1. Then we have



(1 − δs)

2−d

(1−δs)Λ



1/2

≈ 1 +



d − 2

+ γ



δs,

where γ

is the anomalous dimension of φ we defined before.

Classically, we’d expect this correlation function

hφ

(

)

···φ

(

)

to scale

with s as



d − s



since that’s what dimensional analysis would tell us. But quantum mechanically,

we see that there is a correction given by γ

, and what we really have is

∆



d − 2

+ γ



So the dependence of the correlation on the distance is not just what we expect

from dimensional analysis, but it gains a quantum correction factor. Thus, we

say γ

is the “anomalous dimension” of the field.

4.4 Renormalization group flow

We now study the renormalization group flow. In other words, we want to

understand how the coupling constants actually change as we move to the

infrared, i.e. take Λ

→

0. The actual computations are difficult, so in this section,

we are going to understand the scenario rather qualitatively and geometrically.

We can imagine that there is a configuration space whose points are the

possible combinations of the

, and as we take Λ

→

0, we trace out a trajectory

in this configuration space. We want to understand how these trajectories look

like.

As in most of physics, we start at an equilibrium point.

Definition

(Critical point)

A critical point is a point in the configuration

space, i.e. a choice of couplings g

= g

∗

such that β

∗

) = 0.

One such example of a critical point is the Gaussian theory, with all couplings,

including the mass term, vanishing. Since there are no interactions at all, nothing

happens when we integrate out modes. It is certainly imaginable that there are

other critical points. We might have a theory where the classical dimensions

of all couplings are zero, and also by a miracle, all quantum corrections vanish.

This happens, for example, in some supersymmetric theories. Alternatively, the

classical dimensions are non-zero, but the quantum corrections happen to exactly

compensate the effect of the classical dimension.

In either case, we have some couplings

∗

that are independent of scale, and

thus the anomalous dimension

(

∗

) =

∗

would also be independent of scale.

This has important consequences.

Example.

At a critical point, the renormalization group equation for a two-point

function becomes

0 =



∂

∂Λ

+ β

∗

)

∂

∂g

+ 2γ

∗

)



(2)

(x, y).

But the β-function is zero, and γ

is independent of scale. So

∂

∂Λ

(2)

(x, y) = −2γ

∗

(2)

(x, y).

On the other hand, on dimensional grounds, Γ must be of the form

(2)

(x, y, g

∗

) = f(Λ|x − y|, g

∗

)Λ

d−2

for some function

. Feeding this into the RG equation, we find that Γ must be

of the form

(2)

(x, y, g

∗

) =

d−2

c(g

∗

)

2∆

|x − y|

2∆

∝

c(g

∗

)

|x − y|

2∆

where

(

∗

) are some constants independent of the points. This is an example of

what we were saying before. This scales as

|x − y|

−2∆φ

, instead of

|x − y|

2−d

and the anomalous dimension is the necessary correction.

Now a Gaussian universe is pretty boring. What happens when we start

close to a critical point? As in, say, IA Differential Equations, we can try to

Taylor expand, and look at the second derivatives to understand the behaviour

of the system. This corresponds to Taylor-expanding the

-function, which is by

itself the first derivative.

We set our couplings to be

= g

∗

+ δg

Then we can write

∂g

∂Λ



∗

+δg

= B

({g

})δg

+ O(δg

where

is (sort of) the Hessian matrix, which is an infinite dimensional matrix.

As in IA Differential Equations, we consider the eigenvectors of

. Suppose

we have an “eigencoupling” σ

. Classically, we expect

(Λ) =





−d

(Λ

and so

δg

gives an eigenvector with eigenvalue

−d

. In the fully quantum

case, we will write the eigenvalue as ∆

− d, and we define

= ∆

− d

to be the anomalous dimension of the operator. Since

was an eigenvector, we

find that

∂σ

∂Λ

= (∆

− d)σ

Consequently, we find

(Λ) =





∆

−d

(Λ

)

to this order.

Suppose ∆

> d

. Then as we lower the cutoff from Λ

to 0, we find that

(Λ)

→

0 exponentially. So we flow back to the theory at

∗

as we move to

lower energies. These operators are called irrelevant.

Assuming that quantum corrections do not play a very large role near the

critical point, we know that there must be infinitely many such operators, as we

can always increase

, hence ∆

by adding more derivatives or fields (for

d >

2).

So we know the critical surface is infinite dimensional.

On the other hand, if ∆

< d

, then

(Λ) increases as we go to the infrared.

These operators hence become more significant. These are called relevant

operators. There are only finitely many such relevant operators, at least for

d > 2. Any RG trajectory emanating from g

∗

is called a critical trajectory.

We can draw a picture. The critical surface

consisting of (the span of) all

irrelevant modes, and is typically infinite dimensional with finite codimension:

A generic QFT will start at scale Λ

with both relevant and irrelevant operators

turned on. As we flow along the RG trajectory, we focus towards the critical

trajectory. This focusing is called universality.

This is, in fact, the reason we can do physics! We don’t know about the

detailed microscopic information about the universe. Further, there are infinitely

many coupling constants that can be non-zero. But at low energies, we don’t

need to know them! Most of them are irrelevant, and at low energies, only the

relevant operators matter, and there is only finitely many of them.

One thing we left out in our discussion is marginal operators, i.e. those

with ∆

. To lowest order, these are unchanged under RG flow, but we

have to examine higher order corrections to decide whether these operators are

marginally relevant or marginally irrelevant, or perhaps exactly marginal.

Marginally relevant or marginally irrelevant may stay roughly constant for

long periods of RG evolution. Because of this, these operators are often important

phenomenologically. We’ll see that most of the couplings we see in QED and

QCD, apart from mass terms, are marginal operators, at least to lowest order.

If we use the classical dimension in place of ∆

, it is straightforward to figure

out what the relevant and marginal operators are. We know that in

dimensions,

the mass dimension [

] =

d−2

for a scalar field, and [

∂

] = 1. Focusing on the

even operators only, we find that the only ones are

Dimension d Relevant operators Marginal operators

2 φ

for all k > 0 (∂φ)

, φ

(∂φ)

for all k > 0

3 φ

for k = 1, 2 (∂φ)

, φ

4 φ

(∂φ)

, φ

> 4 φ

(∂φ)

Of course, there are infinitely many irrelevant operators, and we do not attempt

to write them out.

Thus, with the exception of

= 2, we see that there is a short, finite list of

relevant and marginal operators, at least if we just use the classical dimension.

Note that here we ignored all quantum corrections, and that sort-of defeats

the purpose of doing renormalization. In general, the eigen-operators will not be

simple monomials, and can in fact look very complicated!

4.5 Taking the continuum limit

So far, we assumed we started with an effective theory at a high energy scale Λ

and studied the behaviour of the couplings as we flow down to low energies. This

is pretty much what we do in, say, condensed matter physics. We have some

detailed description of the system, and then we want to know what happens

when we zoom out. Since we have a fixed lattice of atoms, there is a natural

energy scale Λ

to cut off at, based on the spacing and phonon modes of the

lattice.

However, in high energy physics, we want to do the opposite. We instead

want to use what we know about the low energy version of the system, and then

project and figure out what the high energy theory is. In other words, we are

trying to take the continuum limit Λ

→ ∞.

What do we actually mean by that? Suppose our theory is defined at a

critical point g

∗

and some cutoff Λ

. Then by definition, in our path integral

Z(Λ

, g

∗

) =

∞

(M)

≤Λ

Dϕ e

−S

[ϕ,g

∗

]

No matter what values of Λ

we pick (while keeping the

∗

fixed), we are going

to get the same path integral, and obtain the same answers, as that is what

“critical point” means. In particular, we are free to take the limit Λ

→ ∞

, and

then we are now integrating over “all paths”.

What if we don’t start at a critical point? Suppose we start somewhere on

the critical surface,

}

. We keep the same constants, but raise the value of Λ

What does the effective theory at a scale Λ look like? As we increase Λ

, the

amount of “energy scale” we have to flow down to get to Λ increases. So as we

raise Λ

, the coupling constants at scale Λ flow towards the critical point. As

we take this continuum limit Λ

→

0, we end up at a critical point, namely a

conformal field theory. This is perhaps a Gaussian, which is not very interesting,

but at least we got something.

However, suppose our theory has some relevant operators turned on. Then

as we take the limit Λ

→ ∞

, the coupling constants of our theory diverges!

This sounds bad.

It might seem a bit weird that we fix the constants and raise the values of Λ

However, sometimes, this is a reasonable thing to do. For example, if we think

in terms of the “probing distances” of the theory, as we previously discussed,

then this is equivalent to taking the same theory but “zooming out” and probing

it at larger and larger distances. It turns out, when we do perturbation theory,

the “naive” thing to do is to do exactly this. Of course, we now know that the

right thing to do is that we should change our couplings as we raise Λ

, so as to

give the same physical predictions at any fixed scale Λ

. In other words, we

are trying to backtrace the renormalization group flow to see where we came

from! This is what we are going to study in the next chapter.

4.6 Calculating RG evolution

We now want to actually compute the RG evolution of a theory. To do so, we of

course need to make some simplifying assumptions, and we will also leave out a

lot of details. We note that (in

d >

2), the only marginal or relevant operators

that involves derivatives is the kinetic term (

∂ϕ

)

. This suggests we can find a

simple truncation of the RG evolution by restricting to actions of the form

S[ϕ] =



(∂ϕ)

+ V (ϕ)



and write the potential as

V (ϕ) =

d−k(d−2)

(2k)!

In other words, we leave out higher terms that involve derivatives. This is known

as the local potential approximation (LPA). As before, we split our field into low

and high energy modes,

ϕ = φ + χ,

and we want to compute the effective action for φ:

eff

[φ] = −~ log

∞

(M)

[Λ,Λ

]

Dχ e

−S[φ+χ]

This is still a very complicated path integral to do. To make progress, we assume

we lower the cutoff just infinitesimally, Λ = Λ

− δ

Λ. The action at scale Λ now

becomes

S[φ + χ] = S[φ] +



(∂χ)

(φ) +

000

(φ) + ···



where it can be argued that we can leave out the terms linear in χ.

Since we’re just doing path integral over modes with energies in (Λ

− δ

Λ],

each loop integral takes the form

Λ−δΛ≤|p|≤Λ

p ··· = Λ

d−1

δΛ

d−1

dΩ ··· ,

where dΩ denotes an integral over the unit (

d −

1) sphere. Since each loop

integral comes with a factor of

Λ, to leading order, we need to consider only

1-loop diagrams.

A connected graph with

edges and

loops and

vertices of

-valency

(and arbitrarily many valency in φ) obeys

L − 1 = E −

∞

i=2

Note that by assumption, there are no single-χ vertices.

Also, every edge contributes to two vertices, as there are no

loose ends.

On the other hand, each vertex of type i has i many χ lines. So we have

2E =

Combining these two formulae, we have

L = 1 +

∞

i=2

(i − 2)

The number on the right is non-negative with equality iff

= 0 for all

i ≥

Hence, for 1-loop diagrams, we only need to consider vertices with precisely two

χ-lines attached. Thus, all the contributions look like

, , , . . .

We can thus truncate the action as

S[φ + χ] − S[φ] =



(∂χ)

(φ)



This is still not very feasible to compute. We are only going to do this integral

in a very specific case, where φ is chosen to be constant.

We use the fact that the Fourier modes of

only live between Λ

−δ

< |p| <

Λ.

Then taking the Fourier transform and doing the integral in momentum space,

we have

S[φ + χ] − S[φ] =

Λ−δΛ<|p|≤Λ

2(2π)

˜χ(−p)(p

+ V

(φ))˜χ(p)

d−1

δΛ

2(2π)

[Λ

+ V

(φ)]

d−1

dΩ ˜χ(−Λˆp)˜χ(Λˆp).

The

path integral is finite if we work on a compact space, say

with side

length

, in which case there are only finitely many Fourier modes. Then the

momenta are

2π

, and the path integral is just a product of Gaussians

integrals. Going through the computations, we find

−δ

Dχ e

−(S[φ+χ]−S[φ])

= C



+ V

(φ)



N/2

where

is the number of

modes in our shell of radius Λ and thickness

Λ,

and C is some constant. From our previous formula, we see that it is just

N = vol(S

d−1

)Λ

d−1

δΛ ·



2π



≡ 2aΛ

d−1

δΛ,

where

a =

vol(S

d−1

)

2(2π

)

(4π)

d/2

Γ(d/2)

Consequently, up to field-independent numerical factors, integrating out

leads

to a change in the effective action

eff

δΛ

= a log(Λ

+ V

(φ))Λ

d−1

This diverges as

L → ∞

! This is infrared divergence, and it can be traced to our

simplifying assumption that

is everywhere constant. More generally, it is not

unreasonable to believe that we have

eff

δΛ

= aΛ

d−1

x log(Λ

+ V

(φ)).

This isn’t actually quite the result, but up to the local approximation, it is.

Now we can write down the

-function. As expected, integrating out some

modes has lead to new terms. We have

dΛ

= [k(d − 2) − d]g

− aΛ

k(d−2)

∂

∂φ

log(Λ

+ V

(φ))



φ=0

As before, the first term does not relate to quantum corrections. It is just due to

us rescaling our normalization factors. On the other hand, the 2

th derivative is

just a fancy way to extract the factor of φ

in term.

We can actually compute these things!

Example. Then we find that

dΛ

= −2g −

1 + g

dΛ

= (d − 4)g

−

(1 + g

)

3ag

(1 + g

)

dΛ

= (2d − 6)g

−

(1 + g

)

15ag

(1 + g

)

−

30ag

(1 + g

)

Note that the first term on the right hand side is just the classical behaviour

of the dimensionless couplings. It has nothing to do with the

field. The

remaining terms are quantum corrections (

∼ ~

), and each comes from specific

Feynman diagrams. For example,

1 + g

involves one quartic vertex, and this comes from the diagram

There is one

propagator, and this gives rise to the one 1 +

factor in the

denominator.

The first term in β

comes from

The second term comes from

Note that g

is just the dimensionless mass coupling of χ,

At least perturbatively, we expect this to be a relevant coupling. It increases

as Λ

→

0. Consequently, at scales Λ

 m

, the quantum corrections to these

β-functions are strongly suppressed! This makes sense!

The Gaussian fixed point

From the formulae we derived, we saw that there is only one critical point, with

∗

= 0 for all

k ≥

2. This is free since there are no vertices at all, and hence no

corrections can happen. This is just the free theory.

In a neighbourhood of this critical point, we can expand the

-functions in

lowest order in δg

= g

− g

∗

. We have

= Λ

∂g

∂Λ

= [k(d − 2) − d]g

− ag

2k+2

Writing this linearized β-function as

= B

we see that

is upper triangular, and hence its eigenvalues are just the diagonal

entries, which are

k(d − 2) − d = 2k −4,

in four dimensions.

So vertices

with

k ≥

3 are irrelevant. If we turn them on at some scale,

they become negligible as we fall towards the infrared. The mass term

relevant, as we said before, so even a small mass becomes increasingly significant

in the infrared. Of course, we are making these conclusions based on a rather

perturbative way of computing the

function, and our predictions about what

happens at the far infrared should be taken with a grain of salt. However, we

can go back to check our formulation, and see that our conclusion still holds.

The interesting term is

, which is marginal in

= 4 to lowest order. This

means we have to go to higher order. To next non-trivial order, we have

dΛ

= 3ag

+ O(g

where we neglected

as it is irrelevant. Using the specific value of

= 4,

we find that, to this order,

(Λ)

= C −

16π

log Λ.

Equivalently, we have

(Λ) =

16π



log





−1

for some scale

. If we have no higher order terms, then for the theory to make

sense, we must have g

> 0. This implies that we must pick µ > Λ.

How does this coefficient run? Our coupling

is marginal to leading order,

and consequently it doesn’t run as some power of Λ. It runs only logarithmically

in Λ.

This coupling is irrelevant. In the infrared limit, as we take Λ

→

0, we find

→

0, and so we move towards the Gaussian fixed point. This is rather boring.

On the other hand, if we take Λ

→ ∞

, then eventually

hits 1, and we

divide by zero. So our perturbation theory breaks! Notice that we are not saying

that our theory goes out of control as Λ

→ ∞

. This perturbation theory breaks

at some finite energy scale!

Recall that last term, we were studying

theory. We didn’t really run into

trouble, because we only worked at tree level (and hence wasn’t doing quantum

field theory). But if we actually try to do higher loop integrals, then everything

breaks down. The φ

theory doesn’t actually exist.

The Wilson–Fisher critical point

Last time, we ignored all derivatives terms, and we found, disappointedly, that

the only fixed point we can find is the free theory. This was bad.

Wilson–Fisher, motivated by condensed matter physics rather than funda-

mental physics, found another non-trivial fixed point. What they did was rather

peculiar. They set the dimension to

= 4

−ε

, for some small values of

. While

this might seem rather absurd, because non-integral dimensions do not exist

(unless we want to talk about fractals, but doing physics on fractals is hard),

but we can still do manipulations formally, and see if we get anything sensible.

They proved that there exists a new fixed point with

∗

= −

ε + O(ε

)

∗

+ O(ε

)

∗

∼ O(ε

for k ≥ 3. To study the behaviour near this critical point, we again expand

= g

∗

+ δg

in the

-function we found earlier to lowest non-trivial order, this time expanding

around the Wilson–Fisher fixed point.

If we do this, then in the (g

, g

) subspace, we find that

∂

∂Λ



δg





− 2 −a



1 +



0 ε



δg



The eigenvalues and eigenvectors are

− 2 and ε, with eigenvectors





, σ



−a



3 +



2(3 + ε)



Notice that while the mass term itself is an eigenvector, the quartic coupling is

not! Using the asymptotic expansion



−



∼ −

− γ + O(ε),

where

γ ≈

577 is the Euler–Mascheroni constant, plus fact that Γ(

+ 1) =

xΓ(x), we find that in d = 4 −ε, we have

a =

(4π)

d/2

γ(d/2)



d=4−ε

∼

16π

32π

(1 − γ + log 4π) + O(ε

Since

is small, we know that the eigenvalue of

is negative. This means it is

a relevant operator. On the other hand

, is an irrelevant operator. We thus

have the following picture of the RG flow:

III

We see that theories in region I are massless and free in the deep UV, but flows

to become massive and interacting in the IR. Theories in region II behaves

similarly, but now the mass coefficient is negative. Consequently,

= 0 is a local

maximum of the effective potential. These theories tend to exhibit spontaneous

symmetry breaking.

Finally, theories in III and IV do not have a sensible continuum limit as

both couplings increase without bound. So at least within perturbation theory,

these theories don’t exist. They can only manifest themselves as effective field

theories.

5 Perturbative renormalization

5.1 Cutoff regularization

Our discussion of renormalization has been theoretical so far. Historically, this

was not what people were studying when doing quantum field theory. Instead,

what they did was that they had to evaluate integrals in Feynman diagrams,

and the results happened to be infinite!

For example, consider the scalar

theory in

= 4 dimensions, with action

given by

S[φ] =



(∂φ)



We want to compute the two point function

hφφi

. This has, of course, the tree

level diagram given by just the propagator. There is also a 1-loop diagram given

φ φ

k k

We will ignore the propagators coming from the legs, and just look at the loop

integral. The diagram has a symmetry factor of 2, and thus loop integral is

given by

−

2(2π)

+ m

This integral diverges. Indeed, we can integrate out the angular components,

and this becomes

−

2(2π)

vol(S

)

|p|

d|p|

|p|

+ m

The integrand tends to infinity as we take p → ∞, so this clearly diverges.

Well, this is bad, isn’t it. In light of what we have been discussing so far,

what we should do is to not view

as the Lagrangian in “continuum theory”,

but instead just as a Lagrangian under some cutoff

≤

. Then when doing

the loop integral, instead of integrating over all

, we should integrate over all

such that p

≤ Λ

. And this certainly gives a finite answer.

But we want to take the continuum limit, and so we want to take Λ

→ ∞

Of course the loop integral will diverge if we fix our coupling constants. So

we might think, based on what we learnt previously, that we should tune the

coupling constants as we go.

This is actually very hard, because we have no idea what is the “correct”

way to tune them. Historically, and practically, what people did was just to

introduce some random terms to cancel off the infinity.

The idea is to introduce a counterterm action

[φ, Λ] = ~



δZ

(∂φ)

δm

δλ



where

δz

δm

and

δφ

are some functions of Λ to be determined. We then set the

full action at scale Λ to be

[φ] = S[φ] + S

[φ, Λ].

This action depends on Λ. Then for any physical quantity

hOi

we are interested

in, we take it to be

hOi = lim

Λ→∞



hOi computed with cutoff Λ and action S



Note that in the counterterm action, we included a factor of

in front of

everything. This means in perturbation theory, the tree-level contributions from

would be of the same order as the 1-loop diagrams in S.

For example, in the above 1-loop diagram, we obtain further contributions

to the quadratic terms, given by

φ φ×

δZ

φ φ×

δm

We first evaluate the original loop integral properly. We use the mathematical

fact that vol(S

) = 2π

. Then the integral is

= −

16π

+ m

= −

λm

32π

x dx

1 + x

32π



− m

log



1 +



where we substituted x = p

in the middle.

Including these counter-terms, the 1-loop contribution to hφφi is

32π



− m

log



1 +



+ k

δZ + δm

The objective is, of course, to pick

δz

δm

δφ

so that we always get finite answers

in the limit. There are many ways we can pick these quantities, and of course,

different ways will give different answers. However, what we can do is that

we can fix some prescriptions for how to pick these quantities, and this gives

as a well-defined theory. Any such prescription is known as a renormalization

scheme.

It is important that we describe it this way. Each individual loop integral

still diverges as we take Λ

→ ∞

, as we didn’t change it. Instead, for each fixed

Λ, we have to add up, say, all the 1-loop contributions, and then after adding

up, we take the limit Λ → ∞. Then we do get a finite answer.

On-shell renormalization scheme

We will study one renormalization scheme, known as the on-shell renormalization

scheme. Consider the exact momentum space propagator

x e

ik·x

hφ(x)φ(0)i.

Classically, this is just given by

+ m

where m

is the original mass term in S[φ].

In the on-shell renormalization scheme, we pick our counterterms such that

the exact momentum space propagator satisfies the following two properties:

–

It has a simple pole when

−m

phys

, where

phys

is the physical mass

of the particle; and

– The residue at this pole is 1.

Note that we are viewing k

as a single variable when considering poles.

To find the right values of

δm

and

δZ

, we recall that we had the one-particle

irreducible graphs, which we write as

Π(k

) =

1PI

where the dashed line indicates that we do not include the propagator contribu-

tions. For example, this 1PI includes graphs of the form

k k

as well as counterterm contributions

δZ

δm

Then the exact momentum propagator is

∆(k

)

φ φ

1PI

φ φ

1PI 1PI

+ ···

+ m

−

+ m

Π(k

)

+ m

Π(k

)

+ m

Π(k

)

+ m

+ ···

+ m

+ Π(k

)

The negative sign arises because we are working in Euclidean signature with

path integrals weighted by e

−S

Thus, if we choose our original parameter

to be the measured

phys

, then

in the on-shell scheme, we want

Π(−m

phys

) = 0,

and also

∂

∂k

Π(k

)



=−m

phys

= 0.

To 1-loop, the computations at the beginning of the chapter tells us

Π(k

) = δm

+ k

δZ +

32π



− m

log



1 +



We see that no 1-loop contributions involve

, which we can see in our unique

1-loop diagram, because the loop integral doesn’t really involve k in any way.

We see that the second condition forces δZ = 0, and then we must have

δZ = O(λ

)

δm

= −

32π



− m

log



1 +



+ O(λ

Here to 1-loop, we don’t need wavefunction renormalization, but this is merely a

coincidence, not a general phenomenon.

Of course, if we consider higher loop diagrams, then we have further correc-

tions to the counterterms.

5.2 Dimensional regularization

People soon realized this was a terrible way to get rid of the infinities. Doing

integrals from 0 to Λ

is usually much harder than integrals from 0 to

∞

Moreover, in gauge theory, it is (at least naively) incompatible with gauge

invariance. Indeed, say in U(1) gauge theory, a transformation

ψ(x) → e

iλ(x)

ψ(x)

can potentially introduce a lot of high energy modes. So our theory will not be

gauge invariant.

For these reasons, people invented a different way to get rid of infinite

integrals. This is known as dimensional regularization. This method of getting

rid of infinities doesn’t fit into the ideas we’ve previously discussing. It is just

magic. Moreover, this method only works perturbatively — it tells us how to

get rid of infinities in loops. It doesn’t give any definition of a regularized path

integral measure, or describe any full, coherent non-perturbative theory that

predicts the results.

Yet, this method avoids all the problems we mentioned above, and is rather

easy to use. Hence, we will mostly used dimensional regularization in the rest of

the course.

To do dimensional regularization, we will study our theory in an arbitrary

dimension

, and do the integrals of loop calculations. For certain dimensions,

the integral will converge, and give us a sensible answer. For others, it won’t. In

particular, for d = 4, it probably won’t (or else we have nothing to do!).

After obtaining the results for some functions, we attempt to analytically

continue it as a function of

. Of course, the analytic continuation is non-unique

(e.g. we can multiply the result by

sin d

and still get the same result for integer

), but there is often an “obvious” choice. This does not solve our problem yet

— this analytic continuation tends to still have a pole at

= 4. However, after

doing this analytic continuation, it becomes more clear how we are supposed to

get rid of the pole.

Note that we are not in any way suggesting the universe has a non-integer

dimension, or that non-integer dimensions even makes any sense at all. This is

just a mathematical tool to get rid of infinities.

Let’s actually do it. Consider the same theory as before, but in arbitrary

dimensions:

S[φ] =



(∂φ)



In d dimensions, this λ is no longer dimensionless. We have

[φ] =

d − 2

So for the action to be dimensionless, we must have

[λ] = 4 − d.

Thus, we write

λ = µ

4−d

g(µ)

for some arbitrary mass scale

. This

is not a cutoff, since we are not

going to impose one. It is just some arbitrary mass scale, so that

(

) is now

dimensionless.

We can then compute the loop integral

gµ

4−d

(2π)

+ m

gµ

4−d

2(2π)

vol(S

d−1

)

∞

d−1

+ m

We note the mathematical fact that

vol(S

d−1

) =

2π

d/2

Γ(d/2)

While

d−1

does not make sense when

is not an integer, the right-hand

expression does. So replacing the volume with this expression, we can analytically

continue this to all d, and obtain

= µ

4−d

∞

d−1

+ m

4−d

∞

)

d/2−1

+ m





4−d







1 −



The detailed computations are entirely uninteresting, but if one were to do this

manually, it is helpful to note that

s−1

(1 − u)

t−1

du =

Γ(s)Γ(t)

Γ(s + t)

The appearance of Γ-functions is typical in dimensional regularization.

Combining all factors, we find that

2(4π)

d/2





4−d



1 −



This formula makes sense for any value of

in the complex plane. Let’s see what

happens when we try to approach

= 4. We set

= 4

− ε

, and use the Laurent

series

Γ(ε) =

− γ + O(ε)

= 1 +

log x + O(ε

plus the following usual property of the Γ function:

Γ(x + 1) = xΓ(x).

Then, as d → 4, we can asymptotically expand

= −

32π



− γ + log



4πµ



+ O(ε)



Unsurprisingly, this diverges as

ε →

0, as Γ has a (simple) pole at

−

1. The

pole in

reflects the divergence of this loop integral as Λ

→ ∞

in the cutoff

regularization.

We need to obtain a finite limit as

d →

4 by adding counterterms. This time,

the counterterms are not dependent on a cutoff, because there isn’t one. There

are also not (explicitly) dependent on the mass scale

, because

was arbitrary.

Instead, it is now a function of ε.

So again, we introduce a new term

φ φ×

δm

Again, we need to make a choice of this. We need to choose a renormalization

scheme. We can again use the on-shell renormalization. However, we could have

done on-shell renormalization without doing all this weird dimension thing. Once

we have done it, there is a more convenient way of doing this in dimensional

regularization:

(i) Minimal subtraction (MS): we choose

δm

= −

16π

so as to cancel just the pole.

(ii) Modified minimal subtraction (MS): We set

δm

= −

32π



− γ + log 4π



to get rid of some pesky constants, because no one likes the Euler–

Mascheroni constant.

In practice, we are mostly going to use the

scheme, because we really, really,

hate the Euler–Mascheroni constant.

Note that at the end, after subtracting off these counter-terms and taking

ε → 0, there is still an explicit µ dependence! In this case, we are left with

−

32π

log





Of course, the actual physical predictions of the theory must not depend on

, because

was an arbitrary mass scale. This means

must genuinely be a

function of µ, just like when we did renormalization!

What is the physical interpretation of this? We might think that since

is arbitrary, there is no significance. However, it turns out when we do actual

computations with perturbation theory, the quantum corrections tend to look like

log





, where Λ is the relevant “energy scale”. Thus, if we want perturbation

theory to work well (or the quantum corrections to be small), we must pick

be close to the “energy scale” of the scenario we are interested in. Thus, we can

still think of g(µ

) as the coupling constant of the theory “at scale µ

”.

5.3 Renormalization of the φ

coupling

We now try to renormalize the

coupling. At 1-loop, in momentum space, we

receive contributions from

p+k

and also a counter-term:

We first do the first loop integral. It is given by

4−d

(2π)

+ m

(p + k

+ k

)

+ m

This is a complicated beast. Unlike the loop integral we did for the propagator,

this loop integral knows about the external momenta

. We can imagine

ourselves expanding the integrand in

and

, and then the result involves

some factors of

and

. If we invert the Fourier transform to get back to

position space, then multiplication by

becomes differentiation. So these gives

contributions to terms of the form, say, (

∂φ

)

, in addition to the

, which is

what we really care about.

One can check that only the

contribution is divergent in

= 4. This is

reflecting the fact that all these higher operators are all irrelevant.

So we focus on the contribution to

. This is

-independent, and is given

by the leading part of the integral:

4−d

2(2π)

+ m

)

(4π)

d/2





4−d



2 −



How about the other two loop integrals? They give different integrals, but they

differ only in where the

appear in the denominator. So up to leading order,

they are the same thing. So we just multiply our result by 3, and find that the

loop contributions are

− δλ +

2(4π)

d/2





4−d



2 −



∼ −δλ +

32π



− γ + log



4πµ



+ O(ε).

Therefore, in the MS scheme, we choose

δλ =

32π



− γ + log 4π



and so up to O(~), the loop contribution to the φ

coupling is

32π

log





So in

λφ

theory, to subleading order, with an (arbitrary) dimensional regular-

ization scale µ, we have

∼ + + two more +

+ ···

−

32π

log





+ O(~)

Now note that nothing physical (such as this 4-point function) can depend on

our arbitrary scale µ. Consequently, the coupling g(µ) must run so that

∂

∂µ



−

32π

log





+ O(~)



= 0.

This tells us that we must have

β(g) =

32π

Note that this is the same

-function as we had when we did local potential

approximation!

We can solve this equation for the coupling

(

), and find that the couplings

at scales µ and µ

are related by

g(µ)

g(µ

)

16π

log





Thus, if we find that at energy scale

, the coupling takes value

, then at the

scale

= µe

16π

/(3g

)

the coupling g(µ

) diverges. Our theory breaks down in the UV!

This is to be taken with a pinch of salt, because we are just doing perturbation

theory, with a semi-vague interpretation of what the running of

signifies.

So claiming that

(

) diverges only says something about our perturbative

approximation, and not the theory itself. Unfortunately, more sophisticated

non-perturbative analysis of the theory also suggests the theory doesn’t exist.

5.4 Renormalization of QED

That was pretty disappointing. How about the other theory we studied in QFT,

namely QED? Does it exist?

We again try to do dimensional regularization again. This will be slightly

subtle, because in QED, we have the universe and also a spinor space. In genuine

QED, both of these have dimension 4. If we were to do this properly, we would

have to change the dimensions of both of them to

, and then do computations.

In this case, it is okay to to just keep working with 4-dimensional spinors. We

can just think of this as picking as slightly different renormalization scheme than

MS.

In d dimensions, the classical action for QED in Euclidean signature is

S[A, ψ] =



µν

Dψ + m

ψψ



where

Dψ = γ

(∂

+ iA

)ψ.

Note that in the Euclidean signature, we have lost a factor of

, and also we have

{γ

, γ

} = 2δ

µν

To do perturbation theory, we’d like the photon kinetic term to be canonically

normalized. So we introduce

new

old

and then

S[A

new

, ψ] =



µν

ψ(

∂ + ie

A)ψ + m

ψψ



The original photon field necessarily has [

old

] = 1, as it goes together with the

derivative. So in d dimensions, we have

[e] =

4 − d

Thus, we find

new

] = [A

old

] − [e] =

d − 2

From now on, unless otherwise specified, we will use A

new

, and just call it A.

As before, we introduce a dimensionless coupling constant in terms of an

arbitrary scale µ by

= µ

4−d

(µ).

Let’s consider the exact photon propagator in momentum space, i.e.

∆

µν

(q) =

x e

iq·x

(x)A

(0)i

in Lorenz gauge ∂

= 0.

We can expand this perturbatively as

1PI

1PI 1PI

+ ···

The first term is the classical propagator

∆

µν

(q) =



µν

−



and then as before, we can write the whole thing as

∆

µν

(q) = ∆

µν

(q) + ∆

0ρ

(q)Π

(q)∆

σν

(q) + ∆

0ρ

∆

0λ

∆

κν

+ ··· ,

where

−

ρσ

(

) is the photon self-energy, given by the one-particle irreducible

graphs.

We will postpone the computation of the self-energy for the moment, and

just quote that the result is

(q) = q



−



π(q

)

for some scalar function π(q

). This operator



−



is a projection operator onto transverse polarizations. In particular, like any

projection operator, it is idempotent:

= P

This allows us to simply the expression of the exact propagator, and write it as

∆

µν

(q) = ∆

µν

(1 + π(q

) + π

) + ···) =

∆

µν

1 − π(q

)

Just as the classical propagator came from the kinetic term

kin

µν

dx =



µν

−



(−q)

(q) d

so too our exact propagator is what we’d get from an action whose quadratic

term is

quant

(1 − π(q

))q



µν

−



(−q)

(q) d

Expanding

(

) around

= 0, the leading term just corrects the kinetic term

of the photon. So we have

quant

∼

(1 − π(0))

µν

x + higher derivative terms.

One can check that the higher derivative terms will be irrelevant in d = 4.

Computing the photon self-energy

We now actually compute the self energy. This is mostly just doing some horrible

integrals.

To leading order, using the classical action (i.e. not including the countert-

erms), we have

ρσ

1−loop

p − q

= g

4−d

(2π)



(−i

p + m)γ

+ m

−i(

p −

q + m)γ

(p − q)

+ m



where we take the trace to take into account of the fact that we are considering

all possible spins.

To compute this, we need a whole series of tricks. They are just tricks. We

first need the partial fraction identity

B −A



−



((1 − x)A + xB)

Applying this to the two denominators in our propagators gives

+ m

)((p − q)

+ m

)

((p

+ m

)(1 − x) + ((p − q)

+ m

)x)

+ m

+ −2xpq + q

((p − xq)

+ m

+ q

x(1 − x))

Letting p

= p − qx, and then dropping the prime, our loop integral becomes

4−d

(2π)

tr((−i(

p −

qx) + m)γ

(−i(

p −

q(1 − x)) + m)γ

)

+ ∆)

where ∆ = m

+ q

x(1 − x).

We next do the trace over Dirac spinor indices. As mentioned, if we are

working in

dimensions, then we should be honest and work with spinors in

dimensions, but in this case, we’ll get away with just pretending it is four

dimensions.

We have

tr(γ

) = 4δ

ρσ

, tr(γ

) = 4(δ

µν

ρσ

− δ

µν

ρσ

+ δ

µσ

ρν

Then the huge trace expression we have just becomes



− (p + qx)

(p − q(1 − x))

+ (p + qx) ·(p −q(1 − x))δ

ρσ

− (p + qx)

(p − q(1 − x))

+ m

ρσ



Whenever

d ∈ N

, we certainly get zero if any component of

appears an odd

number of times. Consequently, in the numerator, we can replace

7→

ρσ

Similarly, we have

7→

)

d(d + 2)

(δ

µν

ρσ

+ δ

µρ

νσ

+ δ

µσ

ρν

)

The integrals are given in terms of Γ-functions, and we obtain

ρσ

1−loop

(q) =

−4g

4−d

(4π)

d/2



2 −



∆

2−d/2



ρσ

(−m

+ x(1 − x)q

) + δ

ρσ

+ x(1 − x)q

) − 2x(1 − x)q



And, looking carefully, we find that the m

terms cancel, and the result is

ρσ

1−loop

(q) = q



ρσ

−



1−loop

where

1−loop

) =

−8g

(µ)

(4π)

d/2



2 −



dx x(1 − x)



∆



2−d/2

The key point is that this diverges when

= 4, because Γ(0) is a pole, and so

we need to introduce counterterms.

We set d = 4 − ε, and ε → 0

. We introduce a counterterm

[A, ψ, ε] =



δZ

µν

+ δZ

Dψ + δM

ψψ



Note that the counterterms multiply gauge invariant contributions. We do not

have separate counterterms for

∂ψ

and

Aψ

. We can argue that this must be

the case because our theory is gauge invariant, or if we want to do it properly,

we can justify this using the Ward identities. This is important, because if we

did this by imposing cutoffs, then this property doesn’t hold.

For Π

1−loop

µν

, the appropriate counterterm is δZ

. As ε → 0

, we have

1−loop

) ∼ −

(µ)

2π

dx x(1 − x)



− γ + log



4πµ

∆

+ O(ε)



The counterterm

= −

δZ

µν

and must be chosen to remove the

singularity. In the

scheme, we also

remove the −γ + log 4π piece, because we hate them. So what is left is

) = +

(µ)

2π

dx x(1 − x) log



+ q

x(1 − x)



Then this is finite in

= 4. As we previously described, this 1-loop correction

contains the term ∼ log





. So it is small when m

+ q

∼ µ

Notice that the log term has a branch point when

+ q

x(1 − x) = 0.

For x ∈ [0, 1], we have

x(1 − x) ∈





So the branch cut is inaccessible with real Euclidean momenta. But in Lorentzian

signature, we have

= q

− E

so the branch cut occurs when

− q

)x(1 − x) ≥ m

which can be reached whenever

≥

)

. This is exactly the threshold energy

for creating a real e

−

pair.

The QED β-function

To relate this “one-loop” exact photon propagator to the

-function for the

electromagnetic coupling, we rescale back to A

old

= eA

new

, where we have

(2)

eff

old

] =

(1 − π(0))

µν

= −



(µ)

−

2π

dx x(1 − x) log





µν

We call the spacetime parameter

instead of

to avoid confusing it with the

Feynman parameter x.

Since nothing physical can depend on the arbitrary scale µ, we must have

0 = µ

∂

∂µ



(µ)

−

2π

dx x(1 − x) log





Solving this, we find that to lowest order, we have

β(g) =

12π

This integral is easy to do because we are evaluating at

(0), and the then

log

term does not depend on x.

This tells us that the running couplings are given by

(µ)

(µ

)

6π

log





Now suppose µ ∼ m

, where we measure

)

4π

≈

137

the fine structure constant. Then there exists a scale µ

given by

= m

6π

)

∼ 10

286

GeV,

where g

(µ

) diverges! This is known as a Landau pole.

Yet again, the belief is that pure QED does not exist as a continuum quantum

field theory. Of course, what we have shown is that our perturbation theory

breaks down, but it turns out more sophisticated regularizations also break

down.

Physics of vacuum polarization

Despite QED not existing, we can still say something interesting about it.

Classically, when we want to do electromagnetism in a dielectric material, the

electromagnetic field causes the dielectric material to become slightly polarized,

which results in a shift in the effective electromagnetic potential. We shall see

that in QED, vacuum itself will act as such a dielectric material, due to effects

of virtual electron-positron pairs.

Consider the scattering two (distinguishable) Dirac spinors of charges

and

. The S matrix for the process is given by

S(1 2 → 1

) = −

4π

+ p

− p

)¯u

∆

µν

(q)¯u

where q = p

− p

is the momentum of the photon propagator.

Here we are including the exact photon propagator in the diagram. It is given by

∆

µν

(q)

1 − π(q

)

and so we can write

≈

−e

4π

(4)

+ p

− p

)¯u

∆

µν

¯u

(1 + π(q

) + ···).

So the quantum corrections modify the classical one by a factor of (1+

(

···

To evaluate this better, we note that

¯u

∆

µν

¯u

= ¯u

¯u

In the non-relativistic limit, we expect |q

|  |q| and

¯u

≈



giδ



where

, m

are the

(spin) angular momentum quantum numbers. This

tells us it is very likely that the angular momentum is conserved.

Consequently, in this non-relativistic limit, we have

S(1 2 → 1

) ≈

4π|q|

(4)

+ p

− p

)(1 + π(|q|

))δ

This is what we would get using the Born approximation if we had a potential of

V (r) = e

(2π)



1 + π(|q|

)

|q|



iq·r

In particular, if we cover off the

(

|q|

) piece, then this is just the Coulomb

potential.

In the regime |q|

 m

, picking µ = m

, we obtain

π(|q|

) = π(0) +

(µ)

2π

dx x(1 − x) log



1 +

x(1 − x)|q|



≈ π(0) +

(µ)

60π

|q|

Then we find

V (r) ≈ e

(2π)



1 + π(0)

60π

+ ···



iq·r

= e



1 + π(0)

4πr

60π

(r)



So we got a short-range modification to the Coulomb potential. We obtained

-function precisely because we made the assumption

|q|

 m

. If we did

more accurate calculations, then we don’t get a

function, but we still get a

contribution that is exponentially suppressed as we go away form 0.

This modification of the Coulomb force is attributed to “screening”. It leads

to a measured shift in the energy levels of

= 0 bound states of hydrogen. We

can interpret this as saying that the vacuum itself is some sort of dielectric

medium, and this is the effect of loop diagrams that look like

The idea is that the (genuine) charges

and

polarizes the vacuum, which

causes virtual particles to pop in and out to repel or attract other particles.

So far, the conclusions of this course is pretty distressing. We looked at

theory, and we discovered it doesn’t exist. We then looked at QED, and it

doesn’t exist. Now, we are going to look at Yang–Mills theory, which is believed

to exist.

6 Non-abelian gauge theory

6.1 Bundles, connections and curvature

Vector bundles

To do non-abelian gauge theory (or even abelian gauge theory, really), we need

to know some differential geometry. In this section, vector spaces can be either

real or complex — our theory works fine with either.

So far, we had a universe

, and our fields took values in some vector space

. Then a field is just a (smooth/continuous/whatever) function

M → V

However, this requires the field to take values in the same vector space at each

point. It turns out it is a more natural thing to assign a vector space

for each

x ∈ M, and then a field φ would be a function

φ : M →

x∈M

≡ E,

where

means, as usual, the disjoint union of the vector spaces. Of course, the

point of doing so is that we shall require

φ(x) ∈ V

for each x. (∗)

This will open a can of worms, but is needed to do gauge theory properly.

We are not saying that each

x ∈ M

will be assigned a completely random

vector space, e.g. one would get

and another would get

144169

. In fact,

they will be isomorphic to some fixed space

. So what do we actually achieve?

While it is true that

∼

for all

, there is no canonical choice of such an

isomorphism. We will later see that picking such an isomorphism correspond to

picking a gauge of the field.

Now if we just have a bunch of vector spaces

for each

x ∈ M

, then we

lose the ability to talk about whether a field is differentiable or not, let alone

taking the derivative of a field. So we want to “glue” them together in some way.

We can write

E = {(x, v) : x ∈ M, v ∈ V

and we require

to be a manifold as well. We call these vector bundles. There

are some technical conditions we want to impose on

so that it is actually

possible to work on it.

There is a canonical projection map

E → M

that sends (

x, v

) to

. Then

we have

−1

(

{x}

). A natural requirement is that we require this map

be smooth. Otherwise, we might find someone making

into a manifold in a

really stupid way.

We now introduce a convenient terminology.

Definition

(Section)

Let

E → M

be a map between manifolds. A section

is a smooth map s : M → E such that p ◦s = id

Now we can rewrite the condition (

∗

) as saying

is a section of the map

E → M

. In general, our fields will usually be sections of some vector bundle.

Example.

Let

be any manifold, and

be any vector space. Then

M ×V

a manifold in a natural way, with a natural projection

M × V → M

. This

is called the trivial bundle.

Now trivial bundles are very easy to work with, since a section of the trivial

bundle is in a very natural correspondence with maps

M → V

. So we know

exactly how to work with them.

The final condition we want to impose on our vector bundles is not that

they are trivial, or else we have achieved absolutely nothing. What we want to

require is that near every point, the vector bundle looks trivial.

Definition

(Vector bundle)

Let

be a manifold, and

a vector space. A

vector bundle over

with typical fiber

is a manifold

with a map

E → M

such that for all

x ∈ M

, the fiber

∼

−1

(

{x}

) is a vector space that is

isomorphic to V .

Moreover, we require that for each

x ∈ M

, there exists an open neighbourhood

, and a diffeomorphism Φ :

U × V → π

−1

(

) such that

(Φ(

y, v

)) =

for

all y, and Φ(y, ·) : {y}× V → E

is a linear isomorphism of vector spaces.

Such a Φ is called a local trivialization of

, and

is called a trivializing

neighbourhood.

By definition, each point

is contained in some trivializing neighbourhood.

Thus, we can find a trivializing cover

}

with a trivialization on each

such

that

= M.

There are some philosophical remarks we can make here. On

, every

bundle is isomorphic to a trivial bundle. If we only care about Euclidean (or

Minkowski) space, then it seems like we are not actually achieving much. But

morally, requiring that a bundle is just locally trivial, and not globally trivial

in some sense tells us everything we do is “local”. Indeed, we are mere mortals

who can at best observe the observable universe. We cannot “see” anything

outside of the observable universe, and in particular, it is impossible to know

whether bundles remain trivial if we go out of the observable universe. Even

if we observe that everything we find resembles a trivial bundle, we are only

morally allowed to conclude that we have a locally trivial bundle, and are not

allowed to conclude anything about the global geometry of the universe.

Another reason for thinking about bundles instead of just trivial ones is that

while every bundle over

is globally trivial, the choice of trivialization is not

canonical, and there is a lot of choice to be made. Usually, when we have a

vector space

and want to identify it with

, then we just pick a basis for

it. But now if we have a vector bundle, then we have to pick a basis at each

point in space. This is a lot of arbitrary choices to be made, and it is often more

natural to study the universe without making such choices. Working on a vector

bundle in general also prevents us from saying things that depends on the way

we trivialize our bundle, and so we “force” ourselves to say sensible things only.

Recall that for a trivial bundle, a section of the bundle is just a map

M → V

Thus, for a general vector bundle, if we have a local trivialization Φ on

, then

under the identification given by Φ, a section defined on

can be alternatively

be written as a map

U → V

, which we may write, in coordinates, as

(

for

= 1

, ··· , dim V

. Note that this is only valid in the neighbourhood

, and

also depends on the Φ we pick.

Example. Let M be any manifold. Then the tangent bundle

T M =

x∈M

M → M

is a vector bundle. Similarly, the cotangent bundle

∗

M =

x∈M

∗

M → M

is a vector bundle.

Recall that given vector spaces

and

, we can form new vector spaces by

taking the direct sum

V ⊕W

and the tensor product

V ⊗W

. There is a natural

way to extend this to vector bundles, so that if

E, F → M

are vector bundles,

then

E ⊕ F

and

E ⊗ F

are vector bundles with fibers (

E ⊕ F

)

⊕ F

and

(

E ⊗ F

)

⊗ F

. It is an exercise for the reader to actually construct these

bundles. We can also similarly extend then notion of exterior product

vector bundles.

In particular, applying these to

T M

and

∗

gives us tensor product bundles

of the form (T M )

⊗n

⊗ (T

∗

⊗m

, whose sections are tensor fields.

In more familiar notation, (in a local trivialization) we write sections of

tangent bundles as

(

), and sections of the cotangent bundle as

(

Sections of the tensor product are written as X

,...,µ

,...,ν

Example.

There is exactly one non-trivial vector bundle we can visualize.

Consider the circle S

Let’s consider line bundles on

, i.e. vector bundles with fiber

∼

. There is of

course the trivial bundle, and it looks like this:

However, we can also introduce a “twist” into this bundle, and obtain the M¨obius

band:

This is an example of a non-trivial line bundle. How do we know it is non-trivial?

The trivial bundle obviously has a nowhere-vanishing section. However, if we

stare at the M¨obius band hard enough, we see that any section of the M¨obius

band must vanish somewhere. Thus, this cannot be the trivial line bundle.

In fact, it is a general theorem that a line bundle has a nowhere-vanishing

section if and only if it is trivial.

We introduce a bit more notation which we will use later.

Notation.

Let

E → M

be a vector bundle. Then we write Ω

(

) for the

vector space of sections of

E → M

. Locally, we can write an element of this as

, for a = 1, ··· , dim E

More generally, we write Ω

(

) for sections of

E ⊗

∗

M → M

, where

∗

is the bundle of

-forms on

. Elements can locally be written as

...µ

If V is a vector space, then Ω

(V ) is a shorthand for Ω

(V × M).

Let’s return to the definition of a vector bundle. Suppose we had two trivial-

izing neighbourhoods

and

, and that they have a non-trivial intersection

∩ U

. We can then compare the two trivializations on U

and U

∩ U

) × V π

−1

× U

) (U

∩ U

) × V.

Composing the maps gives us a map

αβ

: Φ

−1

◦ Φ

: (U

∩ U

) × V → (U

∩ U

) × V

that restricts to a linear isomorphism on each fiber. Thus, this is equivalently a

map U

∩ U

→ GL(V ). This is called the transition function.

These transition functions satisfy some compatibility conditions. Whenever

x ∈ U

∩ U

, we have

αβ

(x) · t

βγ

(x) = t

αγ

(x).

Note that on the left, what we have is the (group) multiplication of

(

These also satisfy the boring condition

αα

. These are collectively known

as the cocycle conditions.

Exercise.

Convince yourself that it is possible to reconstruct the whole vector

bundle just from the knowledge of the transition functions. Moreover, given any

cover

}

and functions

αβ

∩ U

→ GL

(

) satisfying the cocycle

conditions, we can construct a vector bundle with these transition functions.

This exercise is crucial. It is left as an exercise, instead of being spelt out

explicitly, because it is much easier to imagine what is going on geometrically

in your head than writing it down in words. The idea is that the transition

functions tell us how we can glue different local trivializations together to get a

vector bundle.

Now we want to do better than this. For example, suppose we have

, which comes with the Euclidean inner product. Then we want the local

trivializations to respect this inner product, i.e. they are given by orthogonal

maps, rather than just linear isomorphisms. It turns out this is equivalent to

requiring that the transition functions

αβ

actually land in O(

) instead of just

GL(n, R). More generally, we can have the following definition:

Definition

(

-bundle)

Let

be a vector space, and

G ≤ GL

(

) be a Lie

subgroup. Then a

-bundle over

is a vector bundle over

with fiber

equipped with a trivializing cover such that the transition functions take value

in G.

Note that it is possible to turn a vector bundle into a

-bundle into many

different ways. So the trivializing cover is part of the data needed to specify the

G-bundle.

We can further generalize this a bit. Instead of picking a subgroup

G ≤

(

), we pick an arbitrary Lie group

with a representation on

. The

difference is that now certain elements of G are allowed to act trivially on V .

Definition

(

-bundle)

Let

be a representation,

a Lie group, and

G →

GL(V ) a representation. Then a G-bundle consists of the following data:

(i) A vector bundle E → M.

(ii) A trivializing cover {U

} with transition functions t

αβ

(iii)

A collection of maps

αβ

∩ U

→ G

satisfying the cocycle conditions

such that t

αβ

= ρ ◦ ϕ

αβ

Note that to specify a

-bundle, we require specifying an element

αβ

(

)

∈ G

for each

x ∈ M

, instead of just the induced action

(

αβ

(

))

∈ GL

(

). This

is crucial for our story. We are requiring more information than just how the

elements in

transform. Of course, this makes no difference if the representation

is faithful (i.e. injective), but makes a huge difference when

is the trivial

representation.

We previously noted that it is possible to recover the whole vector bundle just

from the transition functions. Consequently, the information in (i) and (ii) are

actually redundant, as we can recover

αβ

from

αβ

by composing with

. Thus,

-bundle is equivalently a cover

}

, and maps

αβ

∩ U

→ G

satisfying the cocycle condition.

Note that this equivalent formulation does not mention ρ or V at all!

Example.

Every

-dimensional vector bundle is naturally a

(

) bundle —

we take ρ to be the identity map, andϕ

αβ

= t

αβ

Principal G-bundles

We are halfway through our voyage into differential geometry. I promise this

really has something to do with physics.

We just saw that a

-bundle can be purely specified by the transition

functions, without mentioning the representation or fibers at all. In some sense,

these transition functions encode the “pure twisting” part of the bundle. Given

this “pure twisting” information, and any object

with a representation of

, we can construct a bundle with fiber

, twisted according to this prescription.

This is what we are going to do with gauge theory. The gauge group is the

group

, and the gauge business is encoded in these “twisting” information.

Traditionally, a field is a function

M → V

for some vector space

. To do gauge

coupling, we pick a representation of

. Then the twisting information

allows us to construct a vector bundle over

with fiber

. Then gauge-

coupled fields now correspond to sections of this bundle. Picking different local

trivializations of the vector bundle corresponds to picking different gauges, and

the transition functions are the gauge transformations!

But really, we prefer to work with some geometric object, instead of some

collection of transition functions. We want to find the most “natural” object

for

to act on. It turns out the most natural object with a

-action is not a

vector space. It is just G itself!

Definition

(Principal

-bundle)

Let

be a Lie group, and

a manifold.

A principal

-bundle is a map

P → M

such that

−1

(

{x}

)

∼

for each

x ∈ M

. Moreover,

P → M

is locally trivial, i.e. it locally looks like

U × G

and transition functions are given by left-multiplication by an element of G.

More precisely, we are given an open cover

}

and diffeomorphisms

: U

× G → π

−1

)

satisfying π(Φ

(x, g)) = x, such that the transition functions

−1

◦ Φ

: (U

∩ U

) × G → (U

∩ U

) × G

is of the form

(x, g) 7→ (x, t

αβ

(x) · g)

for some t

αβ

: U

∩ U

→ G.

Theorem.

Given a principal

-bundle

P → M

and a representation

G → GL

(

), there is a canonical way of producing a

-bundle

E → M

with

fiber V . This is called the associated bundle.

Conversely, given a

-bundle

E → M

with fiber

, there is a canonical way

of producing a principal

-bundle out of it, and these procedures are mutual

inverses.

Moreover, this gives a correspondence between local trivializations of

P → M

and local trivializations of E → M.

Note that since each fiber of

P → M

is a group, and trivializations are

required to respect this group structure, a local trivialization is in fact equivalent

to a local section of the bundle, where we set the section to be the identity.

Proof. If the expression

P ×

V → M

makes any sense to you, then this proves the first part directly. Otherwise,

just note that both a principal

-bundle and a

-bundle with fiber

can be

specified just by the transition functions, which do not make any reference to

what the fibers look like.

The proof is actually slightly less trivial than this, because the same vector

bundle can have be many choices of trivializing covers, which gives us different

transition functions. While these different transition functions patch to give the

same vector bundle, by assumption, it is not immediate that they must give the

same principal G-bundle as well, or vice versa.

The way to fix this problem is to figure out explicitly when two collection of

transition functions give the same vector bundle or principal bundle, and the

answer is that this is true if they are cohomologous. Thus, the precise statement

needed to prove this is that both principal

-bundle and

-bundles with fiber

biject naturally with the first

Cech cohomology group of

with coefficients

in G.

We now get to the physics part of the story. To specify a gauge theory

with gauge group

, we supplement our universe

with a principal

-bundle

P → M

. In QED, the gauge group is U(1), and in QCD, the gauge group is

SU(3). In the standard model, for some unknown reason, the gauge group is

G = SU(3) × SU(2) × U(1).

Normally, a field with values in a vector space

is is given by a smooth map

M → V

. To do gauge coupling, we pick a representation

G → V

, and

then form the associated bundle to

P → M

. Then a field is now a section of

this associated bundle.

Example.

In Maxwell theory, we have

= U(1). A complex scalar field is a

function

G → C

. The vector space

has a very natural action of U(1) given

by left-multiplication.

We pick our universe to be

, and then the bundle is trivial. However,

we can consider two different trivializations defined on the whole of

. Then

we have a transition function

M →

U(1), say

(

) =

iθ(x)

. Then under this

change of trivialization, the field would transform as

φ(x) 7→ e

iθ(x)

φ(x).

This is just the usual gauge transformation we’ve seen all the time!

Example.

Recall that a vector bundle

E → M

with fiber

is naturally a

(

)-bundle. In this case, there is a rather concrete description of the principal

GL(n)-bundle that gives rise to E.

At each point

x ∈ M

, we let

(

) be the set of all ordered bases of

We can biject this set with

(

) as follows — we first fix a basis

}

Then given any other basis

}

, there is a unique matrix in

(

) that sends

}

. This gives us a map

(

)

→ GL

(

), which is easily seen to be a

bijection. This gives a topology on Fr(E

The map constructed above obviously depends on the basis

chosen. Indeed,

changing the

corresponds to multiplying

on the right by some element

(

). However, we see that at least the induced smooth structure on

(

) is well-defined, since right-multiplication by an element of

(

) is a

diffeomorphism.

We can now consider the disjoint union of all such

(

). To make this

into a principal

(

)-bundle, we need to construct local trivializations. Given

any trivialization of

on some neighbourhood

, we have a smooth choice

of basis on each fiber, since we have bijected the fiber with

, and

has

a standard basis. Thus, performing the above procedure, we have a choice of

bijection between

(

) between

(

). If we pick a different trivialization,

then the choice of bijection differs by some right-multiplication.

This is almost a principal

(

)-bundle, but it isn’t quite so — to obtain

a principal

(

)-bundle, we want the transition functions to be given by left

multiplication. To solve this problem, when we identified

(

) with

(

)

back then, we should have sent

}

to the inverse of the matrix that sends

}

to {f

In fact, we should have expected this. Recall from linear algebra that under

a change of basis, if the coordinates of elements transform by

, then the basis

themselves transform by

−1

. So if we want our principal

(

)-bundle to have

the same transition functions as

, we need this inverse. One can now readily

check that this has the same transition functions as

. This bundle is known as

the frame bundle, and is denoted Fr(E).

Note that specifying trivializations already gives a smooth structure on

(

)

→ M

. Indeed, on each local trivialization on

, we have a bijection

between

−1

(

) and

U × GL

(

), and this gives a chart on

−1

(

). The fact

that transition functions are given by smooth maps

U → GL

(

) ensures the

different charts are compatible.

Recall that we previously said there is a bijection between a section of a

principal

-bundle and a trivialization of the associated bundle. This is very

clearly true in this case — a section is literally a choice of basis on each fiber!

Connection

Let’s go back to the general picture of vector bundles, and forget about the

structure group

for the moment. Consider a general vector bundle

E → M

and a section

M → E

. We would like to be able to talk about derivatives of

this section. However, the “obvious” attempt looking like

s(x + ε) − s(x)

|ε|

doesn’t really make sense, because

(

) and

(

) live in different vector

spaces, namely E

x+ε

and E

We have encountered this problem in General Relativity already, where we

realized the “obvious” derivative of, say, a vector field on the universe doesn’t

make sense. We figured that what we needed was a connection, and it turns

out the metric on

gives us a canonical choice of the connection, namely the

Levi-Civita connection.

We can still formulate the notion of a connection for a general vector bundle,

but this time, there isn’t a canonical choice of connection. It is some additional

data we have to supply.

Before we meet the full, abstract definition, we first look at an example of a

connection.

Example.

Consider a trivial bundle

M × V → M

. Then the space of sections

Ω

(

) is canonically isomorphic to the space of maps

M → V

. This we know

how to differentiate. There is a canonical map d : Ω

(

)

→

Ω

(

) sending

f 7→ df , where for any vector X ∈ T

M, we have

df(X) =

∂f

∂X

∈ V.

This is a one-form with values in

(or

M × V

) because it takes in a vector

and returns an element of V .

In coordinates, we can write this as

df =

∂f

∂x

We can now define a connection:

Definition

(Connection)

A connection is a linear map

∇

: Ω

(

)

→

Ω

(

)

satisfying

(i) Linearity:

∇(α

+ α

) = α

(∇s

) + α

(∇s

)

for all s

, s

∈ Ω

(E) and α

, α

constants.

(ii) Leibnitz property:

∇(fs) = (df)s + f(∇S)

for all

s ∈

Ω

(

) and

f ∈ C

∞

(

), where, d

is the usual exterior

derivative of a function, given in local coordinates by

df =

∂f

∂x

Given a vector field

, the covariant derivative of a section in the direction

of V is the map

∇

: Ω

(E) → Ω

(E)

defined by

∇

s = V y∇s = V

∇

In more physics settings, the connection is usually written as D

Consider any two connections

∇, ∇

. Their difference is not another connec-

tion. Instead, for any f ∈ C

∞

(M) and s ∈ Ω

(E), we have

(∇

− ∇)(f s) = f(∇

− ∇)(s).

So in fact the difference is a map Ω

(

)

→

Ω

(

) that is linear over functions

∞

(

). Equivalently, it is some element of Ω

(

End

(

)), i.e. some matrix-

valued 1-form A

(x) ∈ End(E

In particular, consider any open set

U ⊆ M

equipped with a local trivializa-

tion. Then after picking the trivialization, we can treat the bundle on

as a

trivial one, and our previous example showed that we had a “trivial” connection

given by d. Then any other connection ∇ can be expressed as

∇s = ds + As

for some

A ∈

Ω

(

End

(

)), where the particular

depends on our trivialization.

This is called the connection 1-form, or the gauge field . In the case where

the tangent bundle, this is also known as the Christoffel symbols.

This was all on a general vector bundle. But the case we are really interested

in is a

-bundle. Of course, we can still say the same words as above, as any

-bundle is also a vector bundle. But can we tell a bit more about how the

connection looks like? We recall that specifying a

-bundle with fiber

equivalent to specifying a principal

-bundle. What we would like is to have

some notion of “connection on a principal G-bundle”.

Theorem.

There exists a notion of a connection on a principal

-bundle.

Locally on a trivializing neighbourhood

, the connection 1-form is an element

(x) ∈ Ω

(g), where g is the Lie algebra of G.

Every connection on a principal

-bundle induces a connection on any asso-

ciated vector bundle. On local trivializations, the connection on the associated

vector bundle has the “same” connection 1-form

(

), where

(

) is regarded

as an element of End(V ) by the action of G on the vector space.

Readers interested in the precise construction of a connection on a principal

-bundle should consult a textbook on differential geometry. Our previous work

doesn’t apply because G is not a vector space.

It is useful to know how the connection transforms under a change of local

trivialization. For simplicity of notation, suppose we are working on two trivial-

izations on the same open set

, with the transition map given by

U → G

We write

and

for the connection 1-forms on the two trivializations. Then

for a section s expressed in the first coordinates, we have

g · (ds + As) = (d + A

)(g · s).

So we find that

= gAg

−1

− gd(g

−1

This expression makes sense if

is a matrix Lie group, and so we can canonically

identify both

and

as subsets of

(

n, R

) for some

. Then we know what

it means to multiply them together. For a more general Lie group, we have to

replace the first term by the adjoint representation of

, and the second by

the Maurer–Cartan form.

Example.

In the U(1) case,our transition functions

αβ

are just multiplication

by complex numbers. So if g = e

−iλ

, then we have

= −gdg

−1

+ gA

−1

= −gdg

−1

+ A

= i(dλ − iA

Note that since U(1) is abelian, the conjugation by

has no effect on

. This is

one of the reasons why non-abelian gauge theory is simple.

Minimal coupling

So how do we do gauge coupling? Suppose we had an “ordinary” scalar field

ψ : M → C on our manifold, and we have the usual action

S[ψ] =

|∂ψ|

+ ··· .

We previously said that to do gauge theory, we pick a representation of our

gauge group

= U(1) on

, which we can take to be the “obvious” action by

multiplication. Then given a principal U(1)-bundle

P → M

, we can form the

associated vector bundle, and now our field is a section of this bundle.

But how about the action? As mentioned, the

∂ψ

term no longer makes

sense. But in the presence of a connection, we can just replace

∂

with

∇

! Now,

the action is given by

S[ψ] =

|∇ψ|

+ ··· .

This is known as minimal coupling.

At this point, one should note that the machinery of principal

-bundles

was necessary. We only ever have one principal

-bundle

P → M

, and a single

connection on it. If we have multiple fields, then we use the same connection on

all of them, via the mechanism of associated bundles. Physically, this is important

— this means different charged particles couple to the same electromagnetic field!

This wouldn’t be possible if we only worked with vector bundles; we wouldn’t

be able to compare the connections on different vector bundles.

Curvature

Given a connection

∇

, we can extend it to a map Ω

(

)

→

Ω

p+1

(

) by requiring

it to satisfy the conditions

∇(α

+ α

) = α

(∇s

) + α

(∇s

∇(ω ∧ s) = (dω) ∧ s + (−1)

deg ω

ω ∧ ∇s.

whenever ω ∈ Ω

(M) and s ∈ Ω

p−q

(E).

We can think of

∇

as a “covariant generalization” of the de Rham operator

d. However, where the ordinary d is nilpotent, i.e. d

= 0, here this is not

necessarily the case.

What kind of object is ∇

, then? We can compute

∇

(ω ∧ s) = ∇(dω ∧ s + (−1)

ω ∧ ∇s)

= d

ω ∧ s + (−1)

q+1

dω ∧ ∇s + (−1)

ω ∧ ∇

= ω ∧ ∇

So we find that

∇

is in fact a map Ω

(

)

→

Ω

q+2

(

) that is linear over any

forms! Specializing to the case of q = 0 only, we can write ∇

∇

(s) = F

∇

for some

∇

∈

Ω

(

End

(

)). It is an easy exercise to check that the same

formula works for all q. In local coordinates, we can write

∇

µν

(x))

∧ dx

Since we have

∇s = ds + As,

we find that

∇

s = ∇(ds + As) = d

s + d(As) + A(ds + As) = (dA + A ∧ A)s.

Note that by

A ∧ A

, we mean, locally in coordinates,

∧ A

, which is still a

form with values in End(V ).

Thus, locally, we have

F = dA + A ∧ A

= (∂

+ A

) dx

∧ dx

(∂

− ∂

+ A

− A

) dx

∧ dx

(∂

− ∂

+ [A

, A

]) dx

∧ dx

Of course, when in the case of U(1) theory, the bracket vanishes, and this is

just the usual field strength tensor. Unsurprisingly, this is what will go into the

Lagrangian for a general gauge theory.

Crucially, in the non-abelian theory, the bracket term is there, and is non-zero.

This is important. Our

is no longer linear in

. When we do Yang–Mills later,

the action will contain a

term, and then expanding this out will give

and

terms. This causes interaction of the gauge field with itself, even without

the presence of matter!

We end by noting a seemingly-innocent identity. Note that we can compute

∇

s = ∇(∇

s) = ∇(F s) = (∇F ) ∧ s + F ∧ (∇s).

On the other hand, we also have

∇

s = ∇

(∇s) = F ∧ ∇s.

These two ways of thinking about

∇

must be consistent. This implies we have

∇(F

∇

) ≡ 0.

This is known as the Bianchi identity.

6.2 Yang–Mills theory

At the classical level, Yang–Mills is an example of non-abelian gauge theory

defined by the action

S[∇] =

Y M

µν

, F

µν

)

√

where (

·, ·

) denotes the Killing form on the Lie algebra

of the gauge group,

and

Y M

is the coupling constant. For flat space, we have

√

= 1, and we will

drop that term.

For example, if G = SU(n), we usually choose a basis such that

, t

) =

and on a local U ⊆ M , we have

S[∇] =

Y M

µν

b,µν

with

µν

= ∂

− ∂

+ f

Thus, Yang–Mills theory is the natural generalization of Maxwell theory to the

non-Abelian case.

Note that the action is treated as a function of the connection, and not the

curvature, just like in Maxwell’s theory. Schematically, we have

∼ (dA + A

)

∼ (dA)

+ A

dA + A

So as mentioned, there are non-trivial interactions even in the absence of charged

matter. This self-interaction is proportional to the structure constants

. So

in the abelian case, these do not appear.

At the level of the classical field equations, if we vary our connection by

∇ 7→ ∇+ δa, where δa is a matrix-valued 1-form, then

δF

∇

= F

∇+δa

− F

∇

= ∇

[µ

δa

ν]

In other words,

δF

µν

= ∂

[µ

δa

ν]

+ [A

, δa

The Yang–Mills equation we obtain from extremizing with respect to these

variations is

0 = δS[∇] =

Y M

(δF

µν

, F

µν

) d

x =

Y M

(∇

δa

, F

µν

) d

x = 0.

So we get the Yang–Mills equation

∇

µν

= ∂

µν

+ [A

, F

µν

] = 0.

This is just like Gauss’ equation. Recall we also had the Bianchi identity

∇ ∧ F = 0,

which gives

∇

νλ

+ ∇

λµ

+ ∇

µν

= 0,

similar to Maxwell’s equations.

But unlike Maxwell’s equations, these are non-linear PDE’s for

. We no

longer have the principle of superposition. This is much more similar to general

relativity. In general relativity, we had some non-linear PDE’s we had to solve

for the metric or the connection.

We all know some solutions to Einstein’s field equations, say black holes and

the Schwarzschild metric. We also know many solutions to Maxwell’s equations.

But most people can’t write down a non-trivial solution to Yang–Mills equations.

This is not because Yang–Mills is harder to solve. If you ask anyone who

does numerics, Yang–Mills is much more pleasant to work with. The real reason

is that electromagnetism and general relativity were around for quite a long time,

and we had a lot of time to understand the solutions. Moreover, these solutions

have very direct relations to things we can observe at everyday energy scales.

However, this is not true for Yang–Mills. It doesn’t really describe everyday

phenomena, and thus less people care.

Note that the action contains no mass terms for A, i.e. there is no A

term.

should describe a massless particle, and this gives rise to long-range force,

just like Coulomb or gravitational forces. When Yang–Mills first introduced this

theory, Pauli objected to this, because we are introducing some new long-range

force, but we don’t see any!

To sort-of explain this, the coupling constant

Y M

plays no role in the

classical (pure) theory. Of course, it will play a role if we couple it to matter.

However, in the quantum theory,

Y M

appears together with

Y M

. So

the classical theory is only a reasonable approximation to the physics only if

Y M

∼

0. Skipping ahead of the story, we will see that

Y M

is marginally

relevant. So at low energies, the classical theory is not a good approximation for

the actual quantum theory.

So let’s look at quantum Yang–Mills theory.

6.3 Quantum Yang–Mills theory

Our first thought to construct a path integral for Yang–Mills may be to compute

the partition function

naive

DA e

−S

Y M

[A]

where

is the “space of all connections” on some fixed principal

-bundle

P → M

. If we were more sophisticated, we might try to sum over all possible

principal G-bundles.

But this is not really correct. We claim that no matter how hard we try to

make sense of path integrals, this integral must diverge.

We first think about what this path integral can be. For any two connections

∇, ∇

, it is straightforward to check that

∇

= t∇ + (1 − t)∇

is a connection of

. Consequently, we can find a path between any two

connections on

. Furthermore, we saw before that the difference

∇

− ∇ ∈

Ω

(

). This says that

is an (infinite-dimensional) affine space modelled on

Ω

(

)

∼

∇

. This is like a vector space, but there is no preferred origin 0.

There is even a flat metric on A, given by

(δA

, δA

) d

i.e. given any two tangent vectors

, a

∈

Ω

(

)

∼

∇

, we have an inner

product

, a

∇

1µ

, a

) d

Importantly, this is independent of the choice of ∇.

This all is trying to say that this

is nice and simple. Heuristically, we

imagine the path integral measure is the natural “

” measure on

as an affine

space. Of course, this measure doesn’t exist, because it is infinite dimensional.

But the idea is that this is just the same as working with a scalar field.

Despite this niceness, S

Y M

[∇] is degenerate along gauge orbits. The action

is invariant under gauge transformations (i.e. automorphisms of the principal

-bundle), and so we are counting each connection infinitely many times. In

fact, the group

of all gauge transformations is an infinite dimensional space

that is (locally)

Maps

(

M, G

), and even for compact

and

, the volume of

this space diverges.

Instead, we should take the integral over all connections modulo gauge

transformations:

Y M

A/G

dµ; e

−S

Y M

[∇]/~

where

A/G

is the space of all connections modulo gauge transformation, and d

is some sort of measure. Note that this means there is no such thing as “gauge

symmetry” in nature. We have quotiented out by the gauge transformations

in the path integral. Rather, gauge transformations are a redundancy in our

description.

But we now have a more complicated problem. We have no idea what the

space

A/G

looks like. Thus, even formally, we don’t understand what d

on this

space should be.

In electromagnetism, we handled this problem by “picking a gauge”. We are

going to do exactly the same here, but the non-linearities of our non-abelian

theory means this is more involved. We need to summon ghosts.

6.4 Faddeev–Popov ghosts

To understand how to do this, we will first consider a particular finite-dimensional

example. Suppose we have a field (

x, y

) :

{pt} → R

on a zero-dimensional

universe and an action

[

x, y

]. For simplicity, we will often ignore the existence

of the origin in R

The partition function we are interested in is

dx dy e

−S[x,y]

Suppose the action is rotationally invariant. Then we can write the integral as

dx dy e

−S[x,y]

2π

dθ

∞

r dr e

−S[r]

= 2π

∞

r dr e

−S[r]

We can try to formulate this result in more abstract terms. Our space

fields has an action of the group

(2) by rotation. The quotient/orbit space of

this action is

\ {0}

SO(2)

∼

Then what we have done is that we have replaced the integral over the whole of

, namely the (

x, y

) integral, with an integral over the orbit space

, namely

the r integral. There are two particularly important things to notice:

–

The measure on

is not the “obvious” measure d

. In general, we have

to do some work to figure out what the correct measure is.

–

We had a factor of 2

sticking out at the front, which corresponds to the

“volume” of the group SO(2).

In general, how do we figure out what the quotient space

is, and how do we

find the right measure to integrate against? The idea, as we have always done,

is to “pick a gauge”.

We do so via a gauge fixing function. We specify a function

→ R

, and

then our gauge condition will be

(

) = 0. In other words, the “space of gauge

orbits” will be

C = {x ∈ R

: f(x) = 0}

f(x) = 0

For this to work out well, we need the following two conditions:

(i) For each x ∈ R

, there exists some R ∈ SO(2) such that f(Rx) = 0.

(ii) f

is non-degenerate. Technically, we require that for any

such that

f(x) = 0, we have

∆

(x) =

∂

∂θ

f(R

(x))



θ=0

6= 0,

where R

is rotation by θ.

The first condition is an obvious requirement — our function

does pick out a

representative for each gauge orbit. The second condition is technical, but also

crucial. We want the curve to pick out a unique point in each gauge orbit. This

prevents a gauge orbit from looking like

where the dashed circle intersects the curved line three times. For any curve that

looks like this, we would have a vanishing ∆

(

) at the turning points of the

curve. The non-degeneracy condition forces the curve to always move radially

outwards, so that we pick out a good gauge representative.

It is important to note that the non-degeneracy condition in general does not

guarantee that each gauge orbit has a unique representative. In fact, it forces

each gauge orbit to have two representatives instead. Indeed, if we consider a

simple gauge fixing function

(

x, y

) =

, then this is non-degenerate, but the

zero set looks like

It is an easy exercise with the intermediate value theorem to show that there

must be at least two representatives in each gauge orbit (one will need to use

the non-degeneracy condition).

This is not a particularly huge problem, since we are just double counting

each gauge orbit, and we know how to divide by 2. Let’s stick with this and

move on.

To integrate over the gauge orbit, it is natural to try the integral.

dx dy δ(f(x))e

−S(x,y)

Then the δ-function restricts us to the curve C, known as the gauge slice.

However, this has a problem. We would want our result to not depend on

how we choose our gauge. But not only does this integral depend on

. It in

fact depends on f as well!

To see this, we can simply replace f by cf for some constant c ∈ R. Then

δ(f(x)) 7→ δ(cf (x)) =

|c|

δ(f(x)).

So our integral changes. It turns out the trick is to include the factor of ∆

previously defined. Consider the new integral

dx dy δ(f(x))|∆

(x)|e

−S(x)

. (∗)

To analyze how this depends on

, we pretend that the zero set

actually

looks like this:

rather than

Of course, we know the former is impossible, and the zero set must look like the

latter. However, the value of the integral (

∗

) depends only on how

behaves

locally near the zero set, and so we may analyze each “branch” separately,

pretending it looks like the former. This will make it much easier to say what

we want to say.

Theorem. The integral (∗) is independent of the choice of f and C.

Proof.

We first note that if we replace

(

) by

(

)

(

) for some

c >

0, then we

have

δ(cf) =

|c|

δ(f), |∆

(x)| = c(r)|∆

and so the integral doesn’t change.

Next, suppose we replace

with some

, but they have the same zero set.

Now notice that

(

) and

∆

depend only on the first-order behaviour of

. In particular, it depends only on

∂f

∂θ

. So for all practical purposes,

changing

is equivalent to multiplying

by the ratio of their derivatives.

So changing the function

while keeping

fixed doesn’t affect the value of (

∗

Finally, suppose we have two arbitrary

and

, with potentially different

zero sets. Now for each value of r, we pick a rotation R

θ(r)

∈ SO(2) such that

f(x) ∝ f(R

θ(r)

x).

By the previous part, we can rescale

, and assume we in fact have equality.

We let

θ(r)

. Now since the action only depends on the radius, it

in particular is invariant under the action of

θ(r)

. The measure d

is also

invariant, which is particularly clear if we write it as d

θ r

instead. Then we

have

dx dy δ(f(x))|∆

(x)|e

−S(x)

δ(f(x

))|∆

)|e

−S(x

)

δ(

f(x))|∆

(x)|e

−S(x

)

dx dy δ(

f(x))|∆

(x)|e

−S(x)

Example.

We choose

to be the

-axis with

(

) =

. Then under a rotation

f(x) = y 7→ y sin θ −x sin θ,

we have

∆

(x) = −x.

So we have

dx dy δ(f)∆

(x)e

−S(x,y)

dx dy δ(y)|x|e

−S(x,y)

∞

−∞

dx |x|e

−S(x,0)

= 2

∞

d|x| |x|e

−S(|x|)

= 2

∞

r dr e

−S(r)

So this gives us back the original integral

∞

r dr e

−S(r)

of along the gauge orbit we wanted, except for the factor of 2. As we mentioned

previously, this is because our gauge fixing condition actually specifies two points

on each gauge orbit, instead of one. This is known as the Gribov ambiguity.

When we do perturbation theory later on, we will not be sensitive to this

global factor, because in perturbation theory, we only try to understand the

curve locally, and the choice of gauge is locally unique.

The advantage of looking at the integral

dx dy δ(f)∆

−S(x,y)

is that it only refers to functions and measures on the full space

, which we

understand well.

More generally, suppose we have a (well-understood) space

with a measure

. We then have a Lie group

acting on

. Suppose locally (near the identity),

we can parametrize elements of

by parameters

for

= 1

, ··· , dim G

. We

write

for the corresponding element of

(technically, we are passing on to

the Lie algebra level).

To do gauge fixing, we now need many gauge fixing functions, say

, again

with a = 1, ··· , dim G. We then let

∆

= det



∂f

∂θ



θ=0



This is known as the Fadeev–Popov determinant.

Then if we have a function

−S[x]

that is invariant under the action of

then to integrate over the gauge orbits, we integrate

dµ |∆

dim G

a=1

δ(f

(x))e

−S[x]

Now in Yang–Mills, our spaces and groups are no longer finite-dimensional, and

nothing makes sense. Well, we can manipulate expressions formally. Suppose we

have some gauge fixing condition f. Then the expression we want is

Z =

A/G

Dµ e

−S

Y M

DA δ[f]|∆

(A)|e

−S

Y M

[A]

Suppose the gauge group is

, with Lie algebra

. We will assume the gauge

fixing condition is pointwise, i.e. we have functions

g → g

, and the gauge

fixing condition is

f(A(x)) = 0 for all x ∈ M.

Then writing n = dim g, we can write

δ[f] =

x∈M

(n)

(f(A(x))).

We don’t really know what to do with these formal expressions. Our strategy is

to write this integral in terms of more “usual” path integrals, and then we can

use our usual tools of perturbation theory to evaluate the integral.

We first take care of the

[

] term. We introduce a new bosonic field

h ∈ Ω

(g), and then we can write

δ[f] =

Dh exp



−

(x)f

(A(x)) d



This

acts as a “Lagrange multiplier” that enforces the condition

(

)) = 0,

and we can justify this by comparing to the familiar result that

ip·x

dp = δ(x).

To take care of the determinant term, we wanted to have

∆

= det

δf

(x)]

δλ

(y)

where λ

(y) are our gauge parameters, and A

is the gauge-transformed field.

Now recall that for a finite-dimensional n × n matrix M , we have

det(M) =

c d

¯c e

¯cMc

where

c, ¯c

are

-dimensional fermion fields. Thus, in our case, we can write the

Fadeev—Popov determinant as the path integral

∆

Dc D¯c exp



M×M

x d

y ¯c

(x)

δf

(x))

δλ

(y)



where

c, ¯c

are fermionic scalars, again valued in

under the adjoint action. Since

we assumed that

is local, i.e.

(

)(

) is a function of

(

) and its derivative

at x only, we can simply write this as

∆

Dc d¯c exp



x ¯c

(x)

δf

)

δλ

(x)c

(x)



The fermionic fields

and

¯c

are known as ghosts and anti-ghosts respectively.

We might find these a bit strange, since they are spin 0 fermionic fields, which

violates the spin statistic theorem. So, if we do canonical quantization with this,

then we find that we don’t get a Hilbert space, as we get states with negative

norm! Fortunately, there is a subspace of gauge invariant states that do not

involve the ghosts, and the inner product is positive definite in this subspace.

When we focus on these states, and on operators that do not involve ghosts,

then we are left with a good, unitary theory. These

and

¯c

aren’t “genuine”

fields, and they are just there to get rid of the extra fields we don’t want. The

“physically meaningful” theory is supposed to happen in A/G, where no ghosts

exist.

Will all factors included, the full action is given by

S[A, ¯c, c, h] =



Y M

µν

a,µν

+ ih

(A) − ¯c

δf

)

δλ



and the path integral is given by

Z =

DA Dc D¯c Dh exp



−S[A, ¯c, c, h]



Example.

We often pick Lorenz gauge

(

) =

∂

. Under a gauge transfor-

mation, we have A 7→ A

= A + ∇λ. More explicitly, we have

)

= A

+ ∂

+ f

So the matrix appearing in the Fadeev–Popov determinant is

δf

)

δλ

= ∂

∇

Thus, the full Yang–Mills action is given by

S[A, ¯c, c, h] =



Y M

µν

a,µν

∂

− ¯c

∂

∇



Why do have to do this? Why didn’t we have to bother when we did

electrodynamics?

If we did this for an abelian gauge theory, then the structure constants were

zero. Consequently, all

terms do not exist. So the ghost kinetic operator

does not involve the gauge field. So the path integral over the ghosts would be

completely independent of the gauge field, and so as long as we worked in a

gauge, then we can ignore the ghosts. However, in a non-abelian gauge theory,

this is not true. We cannot just impose a gauge and get away with it. We need

to put back the Jacobian to take into account the change of variables, and this

is where the ghosts come in.

The benefit of doing all this work is that it now looks very familiar. It seems

like something we can tackle using Feynman rules with perturbation theory.

6.5 BRST symmetry and cohomology

In the Faddeev–Popov construction, we have introduced some gauge-fixing terms,

and so naturally, the Lagrangian is no longer gauge invariant. However, we would

like to be able to use gauge symmetries to understand our theory. For example,

we would expect gauge symmetry to restrict the possible terms generated by the

renormalization group flow, but we now can’t do that.

It turns out that our action still has a less obvious symmetry arising form

gauge invariance, known as BRST symmetry. This was discovered by Becchi,

Rouet and Stora, and also independently by Tyruin.

To describe this symmetry, we are going to construct an BRST operator.

Since we want to prove things about this, we have to be more precise about

what space this is operating on.

We let

be the (complex) space of all polynomial functions in the fields and

their derivatives. More precisely, it is defined recursively as follows:

–

Let Ψ be any of

, c

, ¯c

, h

}

, and

∂

be any differential operator (e.g.

∂

). Then ∂

Ψ ∈ B.

– Any complex C

∞

function on M is in B.

– If a, b ∈ B, then a + b, ab ∈ B.

We impose the obvious commutativity relations based on fermionic and bosonic

statistics. Note that by construction, this space only contains polynomial

functions in the fields. This is what we are going to assume when we try to prove

things, and makes things much more convenient because we can now induct on

the degree of the polynomial.

However, for a general gauge-fixing function, we cannot guarantee that the

Yang–Mills Lagrangian lives in

(even though for the Lorenz gauge, it does).

What we do can be extended to allow for more general functions of the fields,

but this is difficult to make precise and we will not do that.

This

is in fact a

-graded algebra, or a superalgebra, i.e. we can

decompose it as

B = B

⊕ B

where

, B

are vector subspaces of

. Here

contains the “purely bosonic”

terms, while B

contains the purely fermionic terms. These satisfy

⊆ B

s+t

where the subscripts are taken modulo 2. Moreover, if

y ∈ B

and

x ∈ B

, then

we have the (graded-)commutativity relation

yx = (−1)

xy.

belongs to one of

, we say it has definite statistics, and we write

|x| = s if x ∈ B

Definition (BRST operator). The BRST operator Q is defined by

= ∇

c Q¯c = ih

Qc = −

[c, c] Qh = 0.

This extends to an operator on

by sending all constants to 0, and for

f, g ∈ B

of definite statistics, we set

Q(fg) = (−1)

|f|

fQg + (Qf)g, Q(∂

f) = ∂

Qf.

In other words, Q is a graded derivation.

There are a couple of things to note:

–

Even though we like to think of

¯c

as the “complex conjugate” of

, formally

speaking, they are unrelated variables, and so we have no obligation to

make Q¯c related to Qc.

– The expression [c, c] is defined by

[c, c]

= f

where

are the structure constants. This is non-zero, even though the

Lie bracket is anti-commutative, because the ghosts are fermions.

–

The operator

exchanges fermionic variables with bosonic variables. Thus,

we can think of this as a “fermionic operator”.

– It is an exercise to see that Q is well-defined.

We will soon see that this gives rise to a symmetry of the Yang–Mills action.

To do so, we first need the following fact:

Theorem. We have Q

= 0.

Proof. We first check that for any field Ψ, we have Q

Ψ = 0.

– This is trivial for h.

– We have

¯c = Qih = 0.

– Note that for fermionic a, b, we have [a, b] = [b, a]. So

c = −

Q[c, c] = −

([Qc, c] + [c, Qc]) = −[Qc, c] =

[[c, c], c].

It is an exercise to carefully go through the anti-commutativity and see

that the Jacobi identity implies this vanishes.

– Noting that G acts on the ghosts fields by the adjoint action, we have

= Q∇

= ∇

(Qc) + [QA

, c]

= −

∇

[c, c] + [∇

c, c]

= −

([∇

c, c] + [c, ∇

c]) + [∇

c, c]

= 0.

To conclude the proof, it suffices to show that if

a, b ∈ B

are elements of definite

statistics such that

= 0, then

= 0. Then we are done by

induction and linearity. Using the fact that |Qa| = |a| + 1 (mod 2), we have

(ab) = (Q

a)b + aQ

b + (−1)

|a|

(Qa)(Qb) + (−1)

|a|+1

(Qa)(Qb) = 0.

We can now introduce some terminology.

Definition

(BRST exact)

We say

a ∈ B

is BRST exact if

for some

b ∈ B.

Definition (BRST closed). We say a ∈ B is BRST closed if Qa = 0.

By the previous theorem, we know that all BRST exact elements are BRST

closed, but the converse need not be true.

In the canonical picture, as we saw in Michaelmas QFT, the “Hilbert space”

of this quantum theory is not actually a Hilbert space — some states have

negative norm, and some non-zero states have zero norm. In order for things

to work, we need to first restrict to a subspace of non-negative norm, and then

quotient out by states of zero norm, and this gives us the space of “physical”

states.

The BRST operator gives rise to an analogous operator in the canonical

picture, which we shall denote

. It turns out that the space of non-negative

norm states is exactly the states

|ψi

such that

Q|ψi

= 0. We can think of this

as saying the physical states must be BRST-invariant. Moreover, the zero norm

states are exactly those that are of the form Q|φi = 0. So we can write

phys

BRST closed states

BRST exact states

This is known as the cohomology of the operator

. We will not justify this or

further discuss this, as the canonical picture is not really the main focus of this

course or section.

Let’s return to our main focus, which was to find a symmetry of the Yang–

Mills action.

Theorem.

The Yang–Mills Lagrangian is BRST closed. In other words,

= 0.

This theorem is often phrased in terms of the operator

εQ

, where

is a

new Grassmannian variable. Since

= 0, we know (

δa

)(

δb

) = 0 for any

a, b

. So

the function that sends any

a 7→ a

δa

is well-defined (or rather, respects

multiplication), because

ab + δ(ab) = (a + δa)(b + δb).

So this

behaves like an infinitesimal transformation with

“small”. This is

not true for Q itself.

Then the theorem says this infinitesimal transformation is in fact a symmetry

of the action.

Proof.

We first look at the (

µν

, F

µν

) piece of

. We notice that for the purposes

of this term, since

δA

is bosonic, the BRST transformation for

looks just like

a gauge transformation

7→ A

∇

. So the usual (explicit) proof that this

is invariant under gauge transformations shows that this is also invariant under

BRST transformations.

We now look at the remaining terms. We claim that it is not just BRST

closed, but in fact BRST exact. Indeed, it is just

Q(¯c

[A]) = ih

[A] − ¯c

δf

δλ

So we have found a symmetry of the action, and thus if we regularize the path

integral measure appropriately so that it is invariant under BRST symmetries

(e.g. in dimensional regularization), then BRST symmetry will in fact be a

symmetry of the full quantum theory.

Now what does this actually tell us? We first note the following general fact:

Lemma.

Suppose

is some operator such that Φ

7→

Φ +

Φ is a symmetry.

Then for all O, we have

hδOi = 0.

Proof.

hO(Φ)i = hO(Φ

)i = hO(Φ) + δOi.

Corollary.

Adding BRST exact terms to the Lagrangian does not affect the

expectation of BRST invariant functions.

Proof.

Suppose we add

to the Lagrangian, and

is BRST invariant. Then

the change in hOi is



Qg(x) d

x O



hQ(gO)i d

x = 0.

If we want to argue about this more carefully, we should use

εQ

instead of

This has some pretty important consequences. For example, any gauge

invariant term that involves only

is BRST invariant. Also, changing the

gauge-fixing function just corresponds to changing the Lagrangian by a BRST

exact term,

(

¯c

[

]). So this implies that as long as we only care about gauge-

invariant quantities, then all correlation functions are completely independent of

the choice of f .

6.6 Feynman rules for Yang–Mills

In general, we cannot get rid of the ghost fields. However, we can use the

previous corollary to get rid of the h field. We add the BRST exact term

−i

Q(¯c

) =

to the Lagrangian, where

is some arbitrary constant. Then we can complete

the square, and obtain

[A] +



[A]



2ξ

[A]f

[A].

Moreover, in the path integral, for each fixed A, we should have

Dh exp

−



[A]



Dh exp



−

)



since we are just shifting all

by a constant. Thus, if we are not interested in

correlation functions involving

, then we can simply factor out and integrate

out

, and it no longer exists. Then the complete gauge-fixed Yang–Mills action

in Lorenz gauge with coupling to fermions ψ is then

S[A, ¯c, c,

ψ, ψ]



µν

µν,a

2ξ

(∂

)(∂

) − ¯c

∂

∇

ψ(

∇ + m)ψ



where we absorbed the factor of g

Y M

into A.

Using this new action, we can now write down our Feynman rules. We have

three propagators

= D

µν

(p) = −



µν

− (1 − ξ)



= S(p) =

p + m

¯c

= C

(p) =

We also have interaction vertices given by

= g

Y M

abc

((k − p)

µν

+ (p − q)

νλ

+ (q −k)δ

µλ

)

−g

Y M

abe

cde

(δ

µλ

νσ

− δ

µσ

νλ

)

−g

Y M

ace

bde

(δ

µν

σλ

− δ

µσ

νλ

)

−g

Y M

ade

bce

(δ

µν

σλ

− δ

µλ

νσ

)

¯c

(p)

= g

Y M

abc

ψ ψ

(p)

= −g

Y M

What is the point of writing all this out? The point is not to use them.

The point is to realize these are horrible! It is a complete pain to work with

these Feynman rules. For example, a

gg → ggg

scattering process involves

∼

10000 terms when we expand it in terms of Feynman diagrams, at tree level!

Perturbation theory is not going to right way to think about Yang–Mills. And

GR is only worse.

Perhaps this is a sign that our theory is wrong. Surely a “correct” theory

must look nice. But when we try to do actual computations, despite these

horrific expressions, the end results tends to be very nice. So it’s not really

Yang–Mills’ fault. It’s just perturbation theory that is bad.

Indeed, there is no reason to expect perturbation theory to look good.

We formulated Yang–Mills in terms of a very nice geometric picture, about

principal

-bundles. From this perspective, everything is very natural and

simple. However, to do perturbation theory, we needed to pick a trivialization,

and then worked with this object

. The curvature

µν

was a natural geometric

object to study, but breaking it up into d

A ∧A

is not. The individual terms

that give rise to the interaction vertices have no geometric meaning — only

∇

and

do. The brutal butchering of the connection and curvature into these

non-gauge-invariant terms is bound to make our theory look messy.

Then what is a good way to do Yang–Mills? We don’t know. One rather

successful approach is to use lattice regularization, and use a computer to

actually compute partition functions and correlations directly. But this is very

computationally intensive, and it’s difficult to do complicated computations with

this. Some other people try to understand Yang–Mills in terms of string theory,

or twistor theory. But we don’t really know.

The only thing we can do now is to work with this mess.

6.7 Renormalization of Yang–Mills theory

To conclude the course, we compute the

-function of Yang–Mills theory. We

are mostly interested in the case of

(

). While a lot of the derivations (or

lack of) we do are general, every now and then, we use the assumption that

G = SU(N), and that the fermions live in the fundamental representation.

Usually, to do perturbation theory, we expand around a vacuum, where all

fields take value 0. However, this statement doesn’t make much sense for gauge

theory, because

is only defined up to gauge transformations, and it doesn’t

make sense to “set it to zero”. Instead, what we do is that we fix any classical

solution

to the Yang–Mills equation, and take this as the “vacuum” for

This is known as the background field . We write

= A

+ a

Whenever we write ∇

, we will be using the background field, so that

∇

= ∂

+ A

We can compute

µν

= ∂

− ∂

+ [A

, A

]

= F

µν

+ ∂

− ∂

+ [a

, a

] + [A

, a

] + [a

, A

]

= F

µν

+ ∇

− ∇

+ [a

, a

Thus, if we compute the partition function, we would expect to obtain

something of the form

Z ≡ e

−S

eff

[A]

= exp



−

Y M

µν

, F

0,µν

) d



(something).

A priori, the “something” will be a function of

, and also the energy scale

Then since the result shouldn’t actually depend on

, this allows us to compute

the β-function.

We will work in Feynman gauge, where we pick ξ = 1. So we have

S[a, ¯c, c,

ψ, ψ] =



µν

+ ∇

[µ

ν]

+ [a

, a

])

−

(∂

+ ∂

)

− ¯c∂

∇

c − ¯c∂

c +

ψ(

∇ + m)ψ +

aψ



This allows us to rewrite the original action in terms of the new field

. We

will only compute the β-function up to 1-loop, and it turns out this implies we

don’t need to know about the whole action. We claim that we only need to know

about quadratic terms.

Let

be the number of loops,

the number of vertices, and

the number

of propagators. Then, restricting to connected diagrams, Euler’s theorem says

P − V = L − 1.

By restricting to 1-loop diagrams, this means we only care about diagrams with

P = V .

For each node

, we let

q,i

be the number of “quantum” legs coming out of

the vertex, i.e. we do not count background fields. Then since each propagator

connects two vertices, we must have

2P =

vertices i

q,i

Also, almost by definition, we have

V =

vertices i

So this implies we only care about fields with

0 =

q,i

− 2).

It can be argued that since

satisfies the Yang–Mills equations, we can ignore

all linear terms. So this means it suffices to restrict to the quadratic terms.

Restricting to the quadratic terms, for each of

c, ψ, a

, we have a term that

looks like, say

Dc D¯c e

x ¯c∆c

for some operator ∆. Then the path integral will give

det

∆. If the field is a

boson, then we obtain

det

instead, but ultimately, the goal is to figure out what

this ∆ is.

Note that each particle comes with a representation of

(

) (or rather, the

spin group) and the gauge group

. For our purposes, all we need to know about

the representations of

(

) is the spin, which may be 0 (trivial),

(spinor) or

1 (vector). We will refer to the representation of

as “

”, and the spin as

We then define the operator

∆

R,j

= ∇

+ 2



µν

(j)



where

}

are the images of the generators of

in the representation, and

µν

(j)

are the generators of so(d) in the spin j representation. In particular,

µν

(0)

= 0, J

µν

(

)

= S

µν

[γ

, γ

], J

µν

(1)

= i(δ

− δ

For simplicity, we will assume the fermion masses

= 0. Then we claim that

(up to a constant factor), the ∆ of the

and

fields are just ∆

adj,0

∆

and ∆

adj,1

respectively. This is just a computation, which we shall omit.

Thus, if there are n many fermions, then we find that we have

Z = exp



−

Y M

µν

, F

0,µν

) d



(det ∆

adj,0

)(det ∆

)

n/2

(det ∆

adj,1

)

1/2

We are going to view these extra terms as being quantum corrections to the

effective action of

, and we will look at the corrections to the coupling

Y M

they induce. Thus, we are ultimately interested in the logarithm of these extra

terms.

To proceed, write write out ∆ explicitly:

∆

R,j

= −∂

+ (∂

+ A

∂

(R)

| {z }

∆

(1)

+ A

µ,a

| {z }

∆

(2)

+ 2



µν

(j)



(R)

| {z }

∆

(J)

Then we can write the logarithm as

log det ∆

R,j

= log det(−∂

+ ∆

(1)

+ ∆

(2)

+ ∆

(J)

)

= log det(−∂

) + tr log(1 − (∂

)

−1

(∆

(1)

+ ∆

(2)

+ ∆

(J)

)).

Again, the first term is a constant, and we will ignore it. We want to know the

correction to the coupling

Y M

. Since everything is covariant with respect to

the background field

, it is enough to just compute the quadratic terms in

, because all the other terms are required to behave accordingly.

We now see what quadratic terms we obtain in the series expansion of

log

In the case of

(

), We have

tr t

tr J

µν

= 0. So certain cross terms

vanish, and the only quadratic terms are

log det ∆

R,j

∼ tr(−∂

−2

∆

(2)

)

| {z }

(a)

tr(∂

−2

∆

(1)

∂

−2

∆

(1)

)

| {z }

(b)

tr(∂

−2

∆

(J)

∂

−2

∆

(J)

)

| {z }

(c)

(a) =

(2π)

(k)A

(−k)

(2π)

)

µν

d(j)

(b) =

(2π)

(k + 2p)

tr(t

(k)A

(k)

(p + k)

(2π)

(k)A

(−k)(k

µν

− k

)

(p + k)

C(j) tr(t

where

(

) is the number of spin components of the field and

(

) is some

constant depending on j. Explicitly, they are given by

scalar Dirac 4-vector

d(j) 1 4 4

C(j) 0 1 2

In terms of Feynman diagrams, we can interpret (

) as being given by the

loop diagram

while (b) and (c) are given by diagrams that look like

We now define the quantity C(R) by

) = C(R)δ

where the trace is to be taken in the representation

. For

(

), we have

C(adj) = N, C(fund) =

Then, evaluating all those integrals, we find that, in dimensional regularization,

we have

log ∆

R,j

Γ(2 −

)

x µ

4−d

(−∂

)

d−4

µν

a,µν

C(R)

(4π)



d(j)

− 4C(j)



Thus, combining all the pieces, we obtain

eff

[A] =

Y M

x F

µν

a,µν

−



2 −



x µ

4−d

(−∂

)

d−4

µν



adj,1

− C

adj,0

−

R,1/2



where

R,j

C(R)

(4π)



d(j)

− 4C(j)



Explicitly, we have

R,j

C(R)

(4π)











scalars

−

Dirac

−

vectors

As always, the Γ function diverges as we take

d →

4, and we need to remove

the divergence using counterterms. In the

scheme with scale

(with

Y M

4−d

(

)), we are left with logarithmic dependence on

. The independence

of µ gives the condition

dµ



(µ)

+ log



something



adj,1

− C

adj,0

−



= 0.

−

(µ)

β(g) + 2



adj,1

− C

adj,0

−

R,1



= 0.

In other words, we find

β(g) = g

(µ)



adj,1

− C

adj,0

−

R,1



= −

(µ)

(4π)



C(adj) −

C(R)



= −

(4π)



N −



Thus, for

sufficiently small, the

-function is negative! Hence, at least for

small values of

, where we can trust perturbation theory, this coupling now

increases as Λ → 0, and decreases as Λ → ∞.

The first consequence is that this theory has a sensible continuum limit! The

coupling is large at low energies, and after a long story, this is supposed to lead

to the confinement of quarks.

In fact, in 1973, Gross–Coleman showed that the only non-trivial QFT’s in

= 4 with a continuum limit are non-abelian gauge theories. The proof was by

exhaustion! They just considered the most general kind of QFT we can have,

and then computed the

-functions etc., and figured the only ones that existed

were non-abelian gauge theories.