The physical electron charge can be defined in the following way:
Let us consider two classical charged particles being at the rest at a large distance from each other :
The interaction of the charges is described by the Coulomb low
where is the physical charge.
From the other side, the calculation within QED shows that, to the two lowest orders in , the interaction is described by the diagrams
Their calculation yields
where is the regularization parameter for the one-photon-loop diagram. Comparing both formulas, we obtain
Therefore, in our equations we should put
that is known as the charge renormalization.
In particular, the charge renormalization makes the vacuum polarization (VP) contribution to the energy shift of an atomic level to be finite:
the Coulomb field
of the nucleus
For low-Z H-like atoms, the VP energy shift is
For the 2s state in hydrogen:
 J.J. Sakurai, Advanced Quantum Mechanics, 1967.
 V.G. Serbo and I.B. Khriplovich, Lectures on Quantum Mechanics, (Novosibirsk State University, 1999).
 J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, 1964.
 V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii,
Relativistic Quantum Theory, 1971.
 W. Greiner, Quantum Mechanics. Special Chapters,
 W. Greiner and J. Reinhardt, Quantum Electrodynamics, 1992.
We consider aspace of functions
with the boundary conditions
In this space, the momentum operator
is Hermitian and has eigenfunctions
where . The corresponding eigenvalues are . For any from the space under consideration we have
If is real, we obtain
This expansion corresponds to the Fourier expansion in terms of the “cos” and “sin” functions:
For an even function, and, for an odd function, .
The boundary condition does not mean that is an even function.
Let us summarize the mass renormalization procedure
in the non-relativistic case. We start with the Schroedinger equation:
where is the momentum operator and is the bare electron mass. Calculating the energy shift of the level “a” due to interaction with the quantized electromagnetic field, we get
where . From the other side, the interaction
with the quantized electromagnetic field shifts the electron mass by
where is the observable electron mass. Because we know from experiment only , we must express all physical quantities in terms of but not . With this in mind we should put in the Schroedinger equation
and consider the second term (which is ) by perturbation theory.
The observable energy shift is
Where in we also can use instead of , because this replacement does not affect the energy shift
in order .
Чтобы распечатать файл, скачайте его (в формате Word).
Ссылка на скачивание - внизу страницы.