Introduction to Quantum Electrodynamics, страница 5

                                       ,

  are  eigenvalues of  .

        Spectrum of the Dirac equation for the Coulomb field:

   

    

                positive-energy-continuum states

     ___________________________     

                     ________________________________________   

                                                                       ß bound states 

                     _________________________________________   

        0     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _     

 

                negative-energy-continuum states

  and  are the operators satisfying the following anticommutation relations

 ,

 ,

 

The energy operator

                 .

As usual, the vacuum state is defined by

              .

Then, we have

              ,

              .

Let us consider a state

   .

Its energy

is not restricted from the bottom!  For this reason, let us redefine the vacuum state

              .

For the new vacuum, we have             

    ,  for any  ,

      ,  for any  .

If we denote

 ,

we get  

        ,  for any  .

In what follows, we imply

                                         .

The   operator:

 .

The physical sense of    and   :

   is  a creation operator for electrons,

   is  a creation operator for positrons.

 Therefore,

    a one-electron state,

    a one-positron state.

The energy operator

        .

 .

The last term represents the vacuum energy. It can be removed by using  the normal product:

                 .

The electric charge density for the vacuum state:

                  .

Here   . In the free-field case  ,

the two sums cancel each other. For the Coulomb field of a nucleus, ,  the functions    for

 are pulled closer to the nucleus while   for

 are pushed out.  For the case of an extended heavy nucleus,  it yields the following picture for the vacuum-polarization charge density multiplied with  (after renormalization):

Vacuum-polarization charge density multiplied with  for a heavy extended nucleus :

         QED of  interacting fields

In what follows, we will use the relativistic units:

                               .

The Hamiltonian of the system:

,

where, in the Coulomb gauge,

 

  ,

 

        .

The calculations can be performed by perturbation theory.

However, manifestly covariant expressions, which are required  for the renormalization ,  can be most readily obtained if we use the interaction representation instead of the Schroedinger or the Heisenberg representation. The Hamiltonian of the system:

,

where  .

The Schroedinger representation:

 .

For any operator  :

 .

The average value of   :

 .

The transition to the Heisenberg representation is performed by the substitutions:

,

 .

We have

 ,

 .

The average value of   :

    

                       .

              Interaction representation

By the substitutions

,

 ,

we obtain

 ,

 .

The average value of   :

   

                          .

In what follows, we will consider the interaction representation. In the Feynman gauge

 ,

where

 ,

 ,

                   .

Here   ,  ,

,   ,  ,   .

In the interaction representation, the wave function  obeys the equation

 ,

Let us introduce the evolution  operator  by

                         .

We obtain

             

with the boundary condition

                                .

These equations can be combined into a single integral equation

        .