Физика \ Электродинамика

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