# 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

.