Zee A. Quantum Field Theory in a Nutshell

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Closing Words

As I confessed in the preface, I started out intending to write a concise introduction to

quantum field theory, but the book grew and grew. The subject is simply too rich. As I

mentioned, after a period of almost being abandoned, quantum field theory came roaring

back. To quote my thesis advisor Sidney Coleman, the triumph of quantum field theory

was veritably “a victory parade” that made “the spectator gasp with awe and laugh with

joy.”

String theory is beautiful and marvellous, but until it is verified, quantum field theory

remains the true theory of everything. All of physics can now be said to be derivable from

field theory. To start with, quantum field theory contains quantum mechanics as a (0 + 1)-

dimensional field theory, and to end (perhaps) with, string theory may be formulated as a

(1 + 1)-dimensional field theory.

Quantum field theory can arguably be regarded as the pinnacle of human thought.

(Hush, you hear the distant howls of the mathematicians, English professors, philoso-

phers, and perhaps even a few stuck-up musicologists?) It is a distillation of basic notions

from the very beginning of the physics: Newton’s realization that energy is the square of

momentum appears in field theory as the two powers of spatial derivative. But yet—you

knew that was coming, didn’t you, with field theory set up as the pinnacle et cetera?—

but yet, field theory in its present form is in my opinion still incomplete and surely some

bright young minds will see how to develop it further.

For one thing, field theory has not progressed much beyond the harmonic paradigm, as

I presaged in the first chapter. The discovery of the soliton and instanton opened up a new

vista, showing in no uncertain terms that Feynman diagrams ain’t everything, contrary

to what some field theorists thought. Duality offers one way of linking perturbative weak

coupling theory to strong coupling, but as yet practically nothing is known of the strong

coupling regime. When speaking of renormalization groups, we bravely speak of flowing

to a strong coupling fixed point, but we merely have the boat ticket: We have little idea of

474 | Closing Words

what the destination looks like. Perhaps in the not too distant future, lattice field theorists

can extract the field configurations that dominate.

Another restriction is to two powers of the derivative, a restriction going back to Newton

as I remarked above. In modern applications of field theory to problems far beyond particle

physics, there is no reason at all to impose this restriction. For example, in studying visual

perception, one encounters field theories much more involved than those we have studied

in this book. (See the appendix for a brief description.) These field theories are Euclidean

in any case and the corresponding functional integral with higher derivatives certainly

makes sense: It is only in Minkowskian theories that we do not know how to handle

higher derivatives. Newton again—certainly economists consider the rate of change of

the acceleration as well as acceleration. Another innovative application is the formulation

of a class of problems in nonequilibrium statistical mechanics as field theories. Typically,

various objects wander around and react when they meet. This class of problems appears

in areas ranging from chemical reactions to population biology.

We can go far beyond the restriction on the number of derivatives in the Lagrangian.

Who said that we can only have integrands of the form “exponential of a spacetime

integral”? Most modifications you can think of might immediately run afoul of some basic

principles (for example,



Dϕe

−



xL(ϕ)−[



L(ϕ)]

would violate locality), but surely

others might not. Another speculative thought I like to entertain goes along the following

line: Classical and quantum physics are formulated in terms of differential equations

and functional integrals, respectively. But how are differential equations contained in

integrals? The answer is that the integrals



Dϕe

−(1/)



xL(ϕ)

contain a parameter  so

that in the limit  going to zero the evaluation of the integrals amounts to solving partial

differential equations. Can we go beyond quantum field theory by finding a mathematical

operation that in the limit of some parameter

k going to zero reduces to doing the integral



Dϕe

−(1/)



xL(ϕ)

The arena of local field theory has always been restricted to the set of d real numbers

. The recent excitement over noncommutative field theory promises to take us beyond.

(I was tempted to discuss noncommutative field theory too, but then the nutshell would

truly burst.)

But perhaps the most unsatisfying feature of field theory is the present formulation of

gauge theories. Gauge “symmetry” does not relate two different physical states, but two

descriptions of the same physical state. We have this strange language full of redundancy

we can’t live without. We start with unneeded baggage that we then gauge-fix away. We even

know how to avoid this redundancy from the start but at the price of discretizing spacetime.

This redundancy of description is particularly glaring in the manufactured gauge theories

now fashionable in condensed matter physics, in which the gauge symmetry is not there

to begin with. Also, surely the way we calculate in nonabelian gauge theories by cutting

the Yang-Mills action up into pieces and doing violence to gauge invariance will be held

By M. Doi, L. Peliti, J. C. Cardy, and others. See for example J. C. Cardy, cond-mat/9607163, “Renormalisation

Group Approach to Reaction-Diffusion Problems,” in: J.-B. Zuber, ed., Mathematical Beauty of Physics, p. 113.

Closing Words | 475

up to ridicule a hundred years from now. I would not be surprised if a brilliant reader of

this book finds a more elegant formulation of what we now call gauge theories.

Look at the development of the very first field theory, namely Maxwell’s theory of

electromagnetism. By the end of the nineteenth century it had been thoroughly studied and

the overwhelming consensus was that at least the mathematical structure was completely

understood. Yet the big news of the early twentieth century was that the theory, surprise

surprise, contains two hidden symmetries, Lorentz invariance and gauge invariance: two

symmetries that, as we now know, literally hold the key to the secrets of the universe. Might

not our present day theory also contain some unknown hidden symmetries, symmetries

even more lovely than Lorentz and gauge invariance? I think that most physicists would

say that the nineteenth-century greats missed these two crucial symmetries because of

their lousy notation

and tendency to use equations of motion instead of the action. Some

of these same people would doubt that we could significantly improve our notation and

formalism, but the dotted-undotted notation looks clunky to me and I have a nagging

feeling

that a more powerful formalism will one day replace the path integral formalism.

Since the point of good pedagogy is to make things look easy, students sometimes do

not fully appreciate that symmetries do not literally leap out at you. If someone had written

a supersymmetric Yang-Mills theory in the mid-1950s, it would certainly have been a long

time before people realized that it contained a hidden symmetry. So it is entirely possible

that an insightful reader could find a hitherto unknown symmetry hidden in our well-

studied field theories.

It is not just a matter of clearer notation and formalism that caused the nineteenth-

century greats to miss two important symmetries; it is also that they did not possess

the mind set for symmetry. The old paradigm “experiments → action → symmetry” had

to be replaced

in fundamental physics by the new paradigm “symmetry → action →

experiments,” the new paradigm being typified by grand unified theory and later by string

theory. Surely, some future physicists will remark archly that we of the early twenty-first

century did not possess the right mind set.

In physics textbooks, many subjects have a finished completed feel to them, but not

quantum field theory. Some people say to me, what else is there to say about field theory?

I would like to remind those people that a large portion of the material in this book was

unknown 30 years ago. Of course, while I feel that further developments are possible, I

have no idea what—otherwise I would have published it—so I can’t tell you what. But let me

mention two recent developments that I find extremely intriguing. (1) Some field theories

may be dual to string theories. (2) In dimensional deconstruction a d-dimensional field

theory may look (d +1)-dimensional in some range of the energy scale: the field theory

can literally generate a spatial dimension. These developments suggest that quantum field

It is said, and I agree, that one of Einstein’s great contributions is the repeated indices summation convention.

Try to read Maxwell’s treatises and you will appreciate the importance of good notation.

I once asked Feynman how he would solve the finite square well using the path integral.

A. Zee, Fearful Symmetry, chap. 6.

476 | Closing Words

theories contain considerable hidden structures waiting to be uncovered. Perhaps another

golden age is in store for quantum field theory.

So boys and girls, the parade is over, and now it’s up to you to get another parade going.

Appendix

An image presented to the visual system can be described as a 2-dimensional Euclidean field ϕ

(x), with ϕ

representing the gray scale from black (ϕ

=−∞) to white (ϕ

=+∞). [You can see that color might be included

by going to a field ϕ transforming under some internal SO(2) group for example.] The image actually perceived,

ϕ(x), is the actual image ϕ

(x) distorted to ϕ

[y(x)] plus some noise η(x). Distortion is described by a map

x → y(x) of the 2-dimensional Euclidean plane. Your brain’s task is to decide whether the actual image is ϕ

(x)

or some other ϕ

(x). Your ability to discriminate between images depends on the functional integral

Z =



Dy(x)



Dη(x)e

−W [y(x)]−(1/2C)



xη(x)

δ{ϕ

[y(x)] +η(x) − ϕ(x)} (1)



Dy(x)e

−W [y(x)]−(1/2C)



x{ϕ(x)−ϕ

[y(x)]}

where for simplicity I have taken the noise, measured by the parameter C , to be Gaussian and white. The

weighting function W [y(x)] is presumably hard wired by evolution into our visual system, telling us that certain

distortions (translations, rotations, and dilations) are much more likely than others. Writing y(x) = x +A(x) we

note that Z defines a field theory of the 2-component field A

(x), which can always be written as A

=∂

η +ε

∂

χ .

Note that the field A

(x) appears “inside” an “external” field ϕ

. From symmetry considerations we might argue

that

W =−





η∂

η +

χ∂



with two coupling constants f and g. I can give here only the briefest of sketches and refer the interested reader

to the literature.

Clearly, one can think of other examples. This particular example serves only to show that there

are many more field theories than those described in standard texts.

As the Beatles said, quantum fields forever!

W. Bialek and A. Zee, Statistical mechanics and invariant perception, Phys. Rev. Lett. 58: 741, 1987; Under-

standing the efficiency of human perception, Phys. Rev. Lett. 61: 1512, 1988.

Part N

While quantum field theory was discovered and developed in the twentieth century, I will

introduce in this part, added in the second edition of this text, some topics that have been

worked out in the twenty-first century. At the rate these topics are rapidly evolving, I may

be quite foolish to include them here. But I am taking the plunge as I think that I would

serve my readers better by letting them have a taste of the twenty-first century rather

than expanding on the twentieth. More likely than not, by the time this second edition

is published, there will be better ways of treating the material contained here. You should

read part N in this spirit and regard what is given here as an entry key to a fast growing

research literature.

Indeed, by the time the manuscript was copyedited (April 2009) it had been discovered that the amplitudes

discussed in chapters N.2–4 could be written even more simply using a twistor and dual twistor formalism. See

p. 494 and N. Arkani-Hamed, P. Cachazo, C. Cheung, and J. Kaplan, arXiv:09032110.

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N.1 Gravitational Waves and Effective Field Theory

An unfinished symphony

One astounding prediction of Einstein gravity is the existence of ripples crisscrossing the

fabric of spacetime, what one writer refers to as Einstein’s unfinished symphony.

Massive

detectors have been built, with more to come, in a human “curious George” effort to tune

in to the song of the cosmos.

Consider a black hole of size r

=2Gm (its Schwarzschild radius—see chapter I.10, with

m its mass) a distance r

from another black hole, moving with velocity v. As the black

holes spiral into each other they emit gravitational waves, with a characteristic wavelength

determined by the orbital period λ = 2πr

/v. Thus the physics contains three distance

scales: r

, r

, and λ. We will stay within the simple post Newtonian regime r

 r

 λ.

Toward the end, as r

r

and v  1, relativistic effects rear their nasty heads, from which

we will prudently stay away.

In the Closing Words to the first edition of this text, I mention that one intriguing

development over the last few decades has been the use of effective field theory to describe

situations involving more than one energy scale (or equivalently, length and time scales).

The physics at the high energy scale M is then represented in the low energy effective

Lagrangian by higher dimensional terms, suppressed by powers of M but constrained by

the symmetries we know. Examples abound in this book, from the quantum Hall effect

to surface growth to proton decay. The latter provides a classic example: while we profess

ignorance of the physics responsible for proton decay, we can nevertheless make useful

predictions by adding 4-fermion interactions invariant under the low energy gauge group

SU(3) ⊗ SU(2) ⊗ U(1), as shown in chapter VIII.3.

An interesting recent development is an elegant description of the emission of gravi-

tational waves by inspiraling black holes using effective field theory. Here, in contrast to

proton decay, we actually know the short distance physics involved. Effective field theory

M. Bartusiak, Einstein’s Unfinished Symphony.

480 | Part N

nevertheless offers an efficient and sensible way to organize and compartmentalize physics

on the various distance scales. We will merely touch upon one aspect of this approach.

Finite size objects in general relativity

Since r

r

, the leading approximation would be to treat the black hole as a point particle

using the action (I.11.12) S

=−m



dτ =−m





μν

=−m



dτ



μν

where

=dX

/dτ . Let us now include the corrections due to the finite size of the black

hole. As you will see, the following discussion actually applies not only to black holes but

to any finite sized object, including you.

In the spirit of effective field theory, we add to S

higher dimensional terms to be formed

out of the point particle degree of freedom

and the ambient g

μν

the particle moves in,

subject to local coordinate invariance, of course. The invariant tensors we can form out of

μν

are, to leading order, the scalar curvature R, the Ricci curvature R

μν

, and the Riemann

curvature tensor R

μλνρ

. You might start with the scalar curvature and the Ricci curvature,

and add to S

the terms S

drop



dτ(c

R(X) + c

μν

(X)

). The curvatures R(X)

and R

μν

(X) are evaluated on the worldline X

(τ ) of the particle of course.

Einstein’s equation of motion R

μν

−

μν

R = 0 implies R

μν

(X) = 0 and thus also

R(X) = 0. Following the discussion in chapter VIII.3, we now show that, as we might

intuitively feel, we are allowed to drop S

drop

. For our problem we have the total action

S = S

+ S

drop

with the Einstein-Hilbert action (VIII.1.1) S



√

−gM

Under a field redefinition g

μν

→ g

μν

+ δg

μν

δS



√

−gM

μν

−

μν

R)δg

μν

(1)

Note that was how we would have derived the equation of motion for gravity, by varying g

μν

Here we are not making an arbitrary variation, but rather our goal is to choose a specific

δg

μν

so that the resulting δS

negates S

drop

. Since S

drop

consists of an integral over the

worldline of the particle while δS

is given by an integral over spacetime, we need a delta

function in δg

μν

to switch from one kind of integral to another. The choice

δg

μν

(x) =



−g(x)M



dτδ

(x − X(τ))[ag

μν

(X) + b

] (2)

gives

δS =



dτ(−aR(X) + b



μν

(X) −

μν

R(X)



) (3)

So, with some appropriate values of a and b, we can indeed cancel off

drop

, thus

vindicating our intuition that the particle does not feel the Ricci and scalar curvatures

for the obvious reason that they vanish.

A technicality: field redefinition also induces a contact interaction of the form



dτ(1/

√

−g)δ

(τ ) −

(τ )) between the two massive objects. Going from field theory to the point particle description represents a

conceptual step backward, so that we should expect delta function effects at the location of the point particles.

N.1. Gravitational Waves | 481

How about terms we can construct out of the Riemann curvature tensor R

μλνρ

? For your

convenience, I list its symmetry properties

here: R

τρμν

=−R

τρνμ

=−R

ρτμν

, R

τρμν

μντρ

, and R

τρμν

τμνρ

τνρμ

=0. Thus, due to the antisymmetry, we are not able to

contract all four indices of R

μλνρ

with

. We can contract at most two indices, to form

the two objects E

μν

(X) ≡ R

μλνρ

(X)

and B

μν

(X) ≡

μλνρ

(X)

, where

μλνρ

(x) ≡

√

−g

μλσ η

ση

νρ

(x)

denotes the dual of the curvature tensor. These are 2-indexed tensors and we need to square

them to form scalars to put into the action. Hence, to next order the particle action becomes



dτ(−m + c

μν

+ c

μν

...

) (4)

Note that the unknown constants c

and c

have dimension of inverse mass cubed.

Before we explore the physical content of this effective action, let us understand the

meaning of E and B by retreating to the more familiar case of a point particle mov-

ing in an electromagnetic field F

μν

(in flat space). Form E

≡ F

μν

and B

≡

μν

where

μν

(x) ≡

μνσ η

ση

. Going to the rest frame of the particle, where

= 1 and

= 0, we see that, as the notation suggests, this is just the familiar decomposition of

electromagnetism into electricity and magnetism. Similarly, E

μν

and B

μν

represent the

decomposition of curvature into its “electric” and “magnetic” components.

In chapter I.11 we varied the first term in (4) to obtain the standard geodesic equation

that is at the heart of Einstein’s theory. Here we obtain

dτ

+ 

μν

(X(τ ))

dτ

= f

(X(τ )) (5)

where f

(X(τ )) comes from varying the E and B terms in (4). A finite sized body

experiences a tidal force f

due to the varying gravitational force acting on it. It no longer

follows a geodesic.

The fact that we had to square the electric and magnetic components of the curvature

to form the effective action (4) means that the effects of these correction terms are highly

suppressed. Since Riemann curvature contains two derivatives, the correction terms in-

volve four derivatives. To estimate the magnitude of c

and c

we exploit a rather cute

argument as follows.

Consider the scattering of a graviton off this point particle (which, remember, is a

black hole in the problem we are studying) generated by the couplings in (4): iM ∼

...

+ ic

E,B

...

where ω denotes the energy of the graviton. The powers of ω

follows from the four derivatives just mentioned. (If you don’t understand the powers

of M

you need to read chapter VIII.1 again.) Here c

E,B

denotes the two unknown

couplings c

∼ c

generically. Imagine calculating the total scattering cross section for

a graviton on a black hole. Squaring the amplitude M etc., we would end up with σ(ω)∼

...

+ c

E,B

...

S. Weinberg, Gravitation and Cosmology, p. 141.