Chapter 9
Quantum Field Theory
9.1
Introduction
We have studied the properties of photons primarily as single particles.
It was Einstein’s great discovery to realize that particulate basis of light.
Granted these are particles that are very different than those that we are
used to.
At the same time we realize that we have to develop a picture
based on photons that adequately describes the many wavelike properties
that are associated with light, the field properties.
The theory that does
that is quantum field theory.
We want to make a quantum theory of the
electromagnetic field.
This is a rather complex field; it is a combination
of two vector fields with a rather complex dynamics.
For this reason, we
will first discuss a simpler field, the stretched string. We will construct it by
realizing that the phenomena that we identify with the field nature of light is
characterized by energies that are large compared to
ω
and therefore states
with many photons. Also our study of the quantum oscillator indicated that
to recover classical motion, we needed states composed of several stationary
states.
Our first problem will be to describe the many photon state.
This is
actually a subtle construction and will lead us in to a new definition of the
identity of particles. We will cover also one of the great theorems of modern
physics, the spin statistics theorem.
9.2
The Many Particle State
These are the things that locally transfer discrete amounts of energy and
momentum and other things. The example that we have been dealing with
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CHAPTER 9.
QUANTUM FIELD THEORY
is the photon. It has a definite energy and momentum. The energy is related
to the time evolution of the state and thus there is a frequency identified,
ω
=
. This frequency is related to the classical frequency. I remind you
though that in the definite energy state of a quantum system nothing is
moving back and forth.
From the classical relationships we know that there is a relationship
between the energy and momentum,
=

p

c
.
The polarization of the light was known from the classical case to be
related to the angular momentum of the light. The photon is said to have
an intrinsic angular momentum L =
. In fact we can do experiments that
measure the angular momentum transferred by the absorption of photons.
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 Spring '07
 Anderson
 Photon, Quantum Field Theory, Light, intrinsic angular momentum, deﬁnite energy

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