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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

Preface 1
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   

  (1.1.1)

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

  (1.1.2) 

  (1.1.3)      

where  is the jth component of the force on the corresponding particle. The second Newtonian law reads

  (1.1.4)           

Inserting  and  one obtains

(1.1.5)  

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

  (1.1.6)

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

  (1.1.7)                 

 

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

                                                          (1.2.1)        

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

                                                                     (1.2.2)     

I.e    (1.2.3)

with the condition

  (1.2.4)     

With the partial integration

  (1.2.5)

we obtain  (1.2.6)

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

  (1.2.7) 

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

 

1.3 Continuous systems

Now we want to apply Hamiltons principle to a continuous system. We start with a flexible string containing the mass  per unit length, stretched by the constant force  between two fixed points at , say, but subject to small transverse displacements in a plane. The function  is the transverse displacement from equilibrium with . If  increases by  along , the corresponding element of the string has the length

(1.3.1)  

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

  (1.3.2)  

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

(1.3.3)           

As in  the Lagrangian reads

  (1.3.4)                  

which we write using the Lagrange density    like this        

(1.3.4a)

From  up to  we obtain

(1.3.5)

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

The action of the system reads analogously to

(1.3.6)     

We perform a variation of with the conditions

(1.3.7)

(1.3.8)

As in  the Hamilton principle  states

(1.3.9)   

Differentiating partially we obtain

 with                                   (1.3.10)             

As in  partial integration produces

                 (1.3.11)

                                       (1.3.12)

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

(1.3.13) 

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

(1.3.14)

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

  (1.3.15)

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

(1.3.16)

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

(1.3.17) 

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

  (1.3.18)           

We insert  into :

  (1.3.19)

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

(1.4.1)

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

(1.4.2)

We take the expression   from , and from  we have

(1.4.3)

which we insert in :

  (1.4.4)

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 Hamilton density.

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 (Hamilton density) is

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

(1.5.1)

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

(1.5.2)

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.

(1.5.3)

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

(1.5.4)

In order to construct a spatial integral we formulate

(1.5.5)

Denoting

  (1.5.6)

we write  like this

(1.5.7)

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

(1.5.8)

The following special case will be interesting

(1.5.9)

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

  (1.5.10)

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

(1.5.11)

Comparing  with  yields

  (1.5.12)

and analogously  (1.5.13)

Moreover,    (1.5.14)

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

(1.5.15)

and because of  we obtain

(1.5.16)

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.

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