This equation can be solved by the iteration procedure:
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
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
that is known as the photon propagator, and calculating the second factor, we obtain
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
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.
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