Introduction to Quantum Electrodynamics, страница 6

This equation can be solved by the iteration procedure:

This series can be rewritten in the following form

,

where   is the time-ordered product operator.

This expansion provides a basis for QED calculations by perturbation theory in .

S-matrix and Green’s functions

We define the S-matrix by

.

It connects the state  and the state    by

.

The perturbation expansion for  :

,

where we have used

.

This expression for  can be used for calculations of scattering amplitudes for free particles .

For processes involving bound states, one should use the corresponding expressions for so-called Green’s functions or other methods.

In particular, for a one-electron atom, the Green function can be defined by

.

The Green function    contains the total information  about the energy levels of the atom and can be calculated by perturbation theory according to its definition. There are various mathematical methods to extract the energy levels from

Feynman diagrams

The calculations of  and   are performed using the expressions for the field operators  , , and

in terms of the creation and annihilation operators and the corresponding commutation relations.

For instance, if we want to calculate the electron-electron scattering to the lowest order of the perturbation theory, we have to evaluate

.

We have

.

Denoting

,

that is known as the photon propagator, and calculating the second factor, we obtain

where  .

This expression can be represented by two  Feynman diagrams

The diagram representation can be employed for every term of the perturbation expansion, if we formulate the following correspondence rules (Feynman’s rules):

1)  Internal photon line

2)  Internal electron line

3)  Vertex

4)  Incoming and outgoing electron lines

Outgoing positron with  incoming electron with   .

Incoming positron with  outgoing electron with   .

5)  Symmetry factor , where  is the parity of the permutation of the outgoing particles with  respect to the incoming ones.

6) Factor    for every closed electron loop.

Similar Feynman’s rules can be formulated for Green’s functions.

It turns out that contributions of some diagrams are divergent at high internal momenta (small distances).

It can be shown that a one-loop diagram can be divergent only if

,

where  is the number of external electron lines,

is the number of external photon lines.

Example: