ABSTRACT
So far the calculations for scattering cross sections have been performed being restricted to the Born approximation. The matrix elements were fairly simple in form involving no integration with respect to the four-dimensional momenta of virtual particles. The corresponding diagrams, including a single internal line, looked just like the trees we have drawn in early childhood. In analogy, Born diagrams are usually called the tree diagrams, whereas the Born approximation is called the tree approximation. In this section we investigate higher orders of the perturbation theory. This is motivated
not only by the necessity to improve the accuracy of the calculation but also by our wish to make sure of the consistency and correctness of the QED. In the process we have to consider two problems: (i) elucidation of the meaning for the parameters m and e involved in the Lagrangian, and (ii) appearance of divergent integrals. In concept, in the original Dirac equations m and e represent a mass and a charge of
a free electron, i.e., the electron completely isolated from the effect of an electromagnetic field. In other words, m and e are the characteristics of a particular hypothetical objectthe bare electron. It is clear that an energy spectrum of the interacting fields is no longer coincident with that of free fields. Because of this, the mass of the bare electron should be different from that of the physical electron, i.e., different from a minimum energy of the singly charged state of the interacting fields. It makes sense to assume that the electron mass is given by the quantity:
mexp ≡ mR = (Ψp, HΨp) (Ψp,Ψp)
|p=0,
where H is a total Hamiltonian of a system, calculated within the required accuracy in higher orders of the perturbation theory. In the case of a physical charge of the electron we proceed in a similar way. To illustrate, a physical charge may be determined in a investigation of low-frequency electromagnetic waves scattered from the electron at rest. As the process under study is purely classical, in the limiting case p → 0 no quantum correction should occur. Then a physical charge of the electron eexp ≡ eR is determined as a factor entering into the Thomson formula for the effective scattering of the photon at
Particles, Fields, and Quantum
p→ 0, σ =
8pi
( e2R
4pimR
)2 .
