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## Quantum Field Theory, an Introduction

Walter Pfeifer

2009

Dr. Walter Pfeifer: Stapfenackerweg
9, CH 5034 Suhr, Switzerland

### Preface

Quantum field theory describes the interaction between elementary particles. The relativistic version of the theory underlies high energy physics, particle physics and astrophysics. The nonrelativistic form is applied - for instance – to condensed matter up to quantum Hall fluids.

In this book the quantum field theory is only applied to electrons, positrons and photons. Although their interactions are already treated by quantum electrodynamics – the extremely successful prototype of modern quantum field theories – this lead-in is chosen because it is instructive and opens the access to other phenomena.

This publication is conceived as an introduction. The detailed developments and the numerous references to preceding places make it easier to follow. However, knowledge of the elements of quantum mechanics, relativistic mechanics and electrodynamics is a prerequisite. We use the units of the system SI (MKSA-system), no Einstein convention of summation over repeated indices and no natural units (with ).

As in our previous publication, Pfeifer, W. , 2004, we utilize the following symbols: operators are written in bold letters, three- dimensional vectors are marked with an arrow, two- or four-spinors or -vectors and quantities with more dimensions are underlined and the symbols for matrices are doubly underlined.

**Contents**

1 The lagrangian formulation of classical mechanics 3

1.1 The lagrangian principle 3

1.2 Hamiltons principle 4

1.3 Continuous systems 5

1.4 The energy-momentum tensor 8

1.5 The Hamilton formalism, Poisson brackets 11

2 Canonical quantization 14

2.1 Nonrelativistic quantum fields 14

2.2 Quantization rules for Bose particles 15

2.3 Quantization rules for Fermi particles 19

3 Spin- fields 22

3.1 The Dirac equation 22

3.2 Wave functions of free Dirac particles 24

3.3 Quantum fields of Dirac particles 30

3.4 The Feynman propagator for Dirac fields 36

4 The electromagnetic field 45

4.1 The Maxwell equations 45

4.2 The Lagrangian and the Hamiltonian of the Maxwell field 46

4.3 Coupled Maxwell and Dirac fields 48

4.4 Plane wave expansion of the vector field 49

4.5 Canonical quantization of the photon field 51

4.6 The Hamiltonian of the quantized Maxwell field 54

4.7 The Feynman propagator for photons 57

5 Interacting quantum fields 59

5.1 The interaction picture 59

5.2 The time evolution operator 60

5.3 The scattering matrix 64

5.4 Wick’s theorem 65

5.5 Interaction between quantized Dirac- and Maxwell fields 72

5.6 Electron-electron scattering 77

5.7 Compton scattering 81

6 Scattering cross sections 84

6.1 The scattering cross section of electrons 84

6.2 The cross section for the scattering of photons by free electrons 103

7 Epilogue 119

References 120

Index 121

### 1 The lagrangian formulation of classical mechanics

The lagrangian and the hamiltonian
principle are elegant tools to solve mechanical problems. They often are used in
quantum field theory. A short introduction is given here.

### 1.1 The lagrangian principle

We begin with a system of
mass
points with masses
.The
coordinate of every point has
components
(for the most part
).
Therefore our system is described by
components
. The multiplet which contains all the components is
named

In non-relativistic mechanics we define the
Lagrange function
(Lagrangian) for simple situations

where
is
the kinetic energy of the system and
is
its potential energy with the well-known expressions

where
is
the *j***’**th component of the force on the corresponding particle. The
second Newtonian law reads

Inserting
and
one
obtains

.

Because neither does
depend
on
nor
on
the
following equation is equivalent to

These expressions are the Euler-Lagrange
equations. They are also valid for
non-Canonical variables and - surprisingly - for relativistic mechanics.

For later applications we define the generalized momenta

### 1.2 Hamiltons principle

We start with
the Lagrangian,
,
and restrict our discussion to the case where all the
are
independent (not correlated by given functions). The **action**
of
a system is defined as the following time integral

The
**Hamilton****
principle** states that varying the coordinate
functions
the
action
attains
an extremum as soon as the functions
have the true physical course satisfying the equations of motion. In
order to show its consequences we consider a variation of
to
an arbitrary neighbouring path and demand

I.e

with the condition

With the partial integration

we obtain
.

Taking into account that the functions
are
arbitrary, every summand in
must vanish, i.e.

We have obtained the Euler-Lagrange
equations of motion,
.
This result confirms the

### 1.3 Continuous systems

Now we want to apply

,

where we presuppose
.
The element of the string along
is
enlarged by about
and
its potential energy is increased by
.
The potential energy of the whole string amounts to

Making use of the one-dimensional density
the
whole kinetic energy is written as follows

.

As in
the
Lagrangian reads

which we write using the Lagrange density
like
this

.

From
up
to
we
obtain

.

Thus, the Lagrange density depends on
and
.
In special cases it depends also on
itself.

The action of the system reads analogously
to

.

We perform a variation of
with the conditions

As in
the

.

Differentiating partially we obtain

with

As in
partial integration produces

Since the virtual displacement function
is
arbitrary the integrand in
must
vanish

.

This is the Euler-Lagrange
equation of the field
.
We verify
by
inserting
of
the string,
,
in
,
which yields

.

This is the well-known wave equation for
small amplitudes
of
a string.

If
exists
in a
-dimensional
space with coordinates
the
equation
has
to be replaced by

If
is
a vectorial field with
components
we
have
Euler-Lagrange
equations of the form

.

The Lagrange function
plays
an important rôle in quantum field theories. In our lead-in example (string) we
have constructed
following
the conventional method for simple mechanical problems. However, general rules
for generating the Lagrange density cannot be given. Effectively, one has to
guess the function
and
to apply the Euler-Lagrange equations to it. The resulting equation must agree
with the known equation of motion of the problem worked on.

For instance, we claim that the quantum
mechanical behaviour of a mass is characterized by the following Lagrange
density

The function
and
the conjugate complex
can
be treated as independent fields. For
and
the Euler-Lagrange
equation
reads

We insert
into
:

which is the well-known Schrödinger
equation. Thus, the choice of
in
is
justified.

### 1.4 The energy-momentum tensor

Here we derive the energy density and the density of
the linear momentum of a field, which we choose as scalar. The conservation of
the total energy and momentum will result.

We consider a uniform, infinitesimal
space-time displacement. Every
is
increased by the constant amount
and analogously
by
the spatially constant amount
.
Consequently the change of
is

.

We investigate the corresponding change of
the Lagrange density
.
Since
does
not depend explicitly on
or
we
have

.

We take the expression
from
,
and from
we
have

,

which we insert in
:

By means of
we
obtain

On the other hand, obviously holds

Comparing
with
and
taking into account that
and
the
are
independent we obtain from the
-terms

With the definitions

it reads

Integrating
over
all the space, using the divergence theorem of Gauss
and considering that the field vanishes at
large distances yields

Therefore the quantity
remains
constant and we expect that it is the energy of the field. Consequently,
,which
is named the 00-element of the energy-momentum tensor, is the energy density
, which is also denoted by
:

It is also named

By means of the Lagrange density
and
by
we
calculate the energy density of the string

as we expect due to
.
This result confirms our interpretation of
as
energy density.

If
is
a vectorial field (see (1.3.16)) the energy density reads

.

Usually the derivative of
given
above is denoted like this

and is named canonically conjugate field of
.
For instance, the canonically conjugate of a quantum
mechanical field
reads
using

Due to
and
the
energy density (

,

We go on equating the expression
with
.
From the
-term
we obtain

.

We define the following elements of the energy-momentum
tensor

and integrate
as
follows

.

With the same argument as for
and
defining
we
obtain

.

Therefore
is
a constant. It must be the
th
component of the linear momentum of the whole field:

.

### 1.5 The Hamilton formalism, Poisson brackets

We will formulate Poisson brackets of the field. They are the starting point for the field quantization.

Generally, as in
and in
,
the Lagrange density
simultaneously depends on the values of the fields
and
of the corresponding complex conjugate quantities. Functions
with
such a dependence on functions are named functionals of the field. It is
customary to denote a functional dependence by square brackets

.

For the moment we take only one field
function
and
define the variation of the functional

.

The variation
depends
on space and time. We look into
in
more detail. We divide up all the space into small cells with central position
vectors
and
volumes
.
The average value of the field in
is
.
We form a special variation of
supposing
that only in one cell (number
)
the value
is
varied by
through
which the function
varies
in the whole space depending on
by
.
I.e.

.

In first order approximation the effect of
all variations
on
the spatial and temporal function
add
up simply to

.

In order to construct a spatial integral we
formulate

.

Denoting

we write
like
this

.

We introduce the Poisson brackets of fields
being based on the field function
and
its canonically conjugate field
,
.
Given two functions
we define the Poisson bracket

.

The following special case will be
interesting

.

For this case we insert the first function
of
in
and
obtain

On the other hand, a variation of
can
be written as

.

Comparing
with
yields

and analogously
.

Moreover,

holds since
and
are
independent functionals. We form the mutual Poisson brackets of the fields
and
using
up
to

and because of
we
obtain

In the next section the fields will be
quantized by replacing the Poisson brackets by commutation relations between
corresponding operators.

This is only an excerpt. The whole publication can be ordered from the category «order for free» in this site.