0Introduction

III Quantum Field Theory



0 Introduction
The idea of quantum mechanics is that photons and electrons behave similarly.
We can make a photon interfere with itself in double-slit experiments, and
similarly an electron can interfere with itself. However, as we know, lights are
ripples in an electromagnetic field. So photons should arise from the quantization
of the electromagnetic field. If electrons are like photons, should we then have
an electron field? The answer is yes!
Quantum field theory is a quantization of a classical field. Recall that in
quantum mechanics, we promote degrees of freedom to operators. Basic degrees
of freedom of a quantum field theory are operator-valued functions of spacetime.
Since there are infinitely many points in spacetime, there is an infinite number
of degrees of freedom. This infinity will come back and bite us as we try to
develop quantum field theory.
Quantum field theory describes creation and annihilation of particles. The
interactions are governed by several basic principles locality, symmetry and
renormalization group flow. What the renormalization group flow describes is
the decoupling of low and high energy processes.
Why quantum field theory?
It appears that all particles of the same type are indistinguishable, e.g. all
electrons are the same. It is difficult to justify why this is the case if each particle
is considered individually, but if we view all electrons as excitations of the same
field, this is (almost) automatic.
Secondly, if we want to combine special relativity and quantum mechanics,
then the number of particles is not conserved. Indeed, consider a particle trapped
in a box of size
L
. By the Heisenberg uncertainty principle, we have
p & ~/L
.
We choose a particle with small rest mass so that m E. Then we have
E = ∆p · c &
~c
L
.
When
E &
2
mc
2
, then we can pop a particle-antiparticle pair out of the
vacuum. So when
L .
~
2mc
, we can’t say for sure that there is only one particle.
We say
λ
=
~/
(
mc
) is the “compton wavelength” the minimum distance
at which it makes sense to localize a particle. This is also the scale at which
quantum effects kick in.
This argument is somewhat circular, since we just assumed that if we have
enough energy, then particle-antiparticle pairs would just pop out of existence.
This is in fact something we can prove in quantum field theory.
To reconcile quantum mechanics and special relativity, we can try to write a
relativistic version of Schr¨odinger’s equation for a single particle, but something
goes wrong. Either the energy is unbounded from below, or we end up with some
causality violation. This is bad. These are all fixed by quantum field theory by
the introduction of particle creation and annihilation.
What is quantum field theory good for?
Quantum field theory is used in (non-relativistic) condensed matter systems.
It describes simple phenomena such as phonons, superconductivity, and the
fractional quantum hall effect.
Quantum field theory is also used in high energy physics. The standard model
of particle physics consists of electromagnetism (quantum electrodynamics),
quantum chromodynamics and the weak forces. The standard model is tested
to very high precision by experiments, sometimes up to 1 part in 10
10
. So it is
good. While there are many attempts to go beyond the standard model, e.g.
Grand Unified Theories, they are mostly also quantum field theories.
In cosmology, quantum field theory is used to explain the density pertur-
bations. In quantum gravity, string theory is also primarily a quantum field
theory in some aspects. It is even used in pure mathematics, with applications
in topology and geometry.
History of quantum field theory
In the 1930’s, the basics of quantum field theory were laid down by Jordan,
Pauli, Heisenberg, Dirac, Weisskopf etc. They encountered all sorts of infinities,
which scared them. Back then, these sorts of infinities seemed impossible to
work with.
Fortunately, in the 1940’s, renormalization and quantum electrodynamics
were invented by Tomonaga, Schwinger, Feynman, Dyson, which managed to
deal with the infinities. It was a sloppy process, and there was no understanding
of why we can subtract infinities and get a sensible finite result. Yet, they
managed to make experimental predictions, which were subsequently verified by
actual experiments.
In the 1960’s, quantum field theory fell out of favour as new particles such
as mesons and baryons were discovered. But in the 1970’s, it had a golden
age when the renormalization group was developed by Kadanoff and Wilson,
which was really when the infinities became understood. At the same time, the
standard model was invented, and a connection between quantum field theory
and geometry was developed.
Units and scales
We are going to do a lot of computations in the course, which are reasonably
long. We do not want to have loads of
~
and
c
’s all over the place when we do
the calculations. So we pick convenient units so that they all vanish.
Nature presents us with three fundamental dimensionful constants that are
relevant to us:
(i) The speed of light c with dimensions LT
1
;
(ii) Planck’s constant ~ with dimensions L
2
MT
1
;
(iii) The gravitational constant G with dimensions L
3
M
1
T
2
.
We see that these dimensions are independent. So we define units such that
c
=
~
= 1. So we can express everything in terms of a mass, or an energy, as we
now have
E
=
m
. For example, instead of
λ
=
~/
(
mc
), we just write
λ
=
m
1
.
We will work with electron volts
eV
. To convert back to the conventional SI
units, we must insert the relevant powers of
c
and
~
. For example, for a mass of
m
e
= 10
6
eV, we have λ
e
= 2 × 10
12
m.
After getting rid of all factors of
~
and
c
, if a quantity
X
has a mass dimension
d, we write [X] = d. For example, we have [G] = 2, since we have
G =
~c
M
2
p
=
1
M
2
p
,
where M
p
10
19
GeV is the Planck scale.