(Received September 23, 1948)
In the first paper of the same title Tati and one of the present
authors have proposed a subtraction procedure to be used in the
treatment of quantum-theoretical problems involving infinite field
reactions. This method consists in generalizing the method of
canonical transformation used first by Bloch and Nordsieck and then by
Pauli and Fierz in the treatment of similar problems in a
non-relativistic approximation, and by means of this transformation
the electron and the radiation fields are separated into two parts,
one bound with each other and the other unbound and propagating freely
in the absence of an external field. Then infinite energies, one
related to the infinite self-energy of the free electron and the other
to that giving rise to the vacuum polarization of the radiation field,
are separated as the interaction energies between bound parts of the
fields. Then these infinite terms in the energy are dropped off
considering that only the remaining finite terms have physical
Now when an external field is present, the unbound fields do no longer propagate freely, and, by virtue of the external field, transitions take place. The electron wave is able to change its state of propagation not only elastically affected by the extenrnal field but also emitting or absorbing unbound photons; thus an interaction appears between electron and radiation which were free from interaction with each other in the absence of the external field. This interaction causes now a radiation reaction upon the electron so that the motion of the electron will be modified. We shall give in this paper an example how one can calculate this reaction and the result obtained is really finite as the consequence of our subtraction procedure introduced in I. Thus, for example, we are able to obtain a finite radiative level-shift of a bound electron in the external field and a finite e2-correction to the scattering cross section, the latter problem having been discussed recently by Koba and one of us.
As will be shown below we can find finite effective interaction energies which describe the interaction between electron and radiation as mentioned above. One of these interaction energies is first order in e and is essentially of the form eψ†OA[German Symbol A]°ψ, [German Symbol A]° being the potential (multiplied by e) of the external field and O some operator containing Dirac matrices and operators upon ψ†, A, [German Symbol A]° and ψ. This energy is responsible for the Brems-strahlung of an electron in the field [German Symbol A]° or allied phenomena. Another interaction energy is of the second order in e and has the form e2ψ†O[German Symbol A]°ψ. This energy as well as trivial zero order term ψ†γ[German Symbol A]°ψ give rise to elastic scattering of the electron in the field [German Symbol A]°, the second order term giving rise to the e2-correction to the ordinary elastic scattering casused by the zero order term. The famous energy-level shift of a bound electron observed by Lamb and Retherford can be calculated as a combined effect of these two terms, namely as the first order modification of the energy-level due to the second order term combined with the second order effect due to the first order term.
Although we have thus successfully obtained finite answers for these field reaction problems and they are of the magnitude agreeing with experimental results, we must nevertheless confess that the calculation carried out in this paper is still unsatisfactory because we had to make a non-relativistic treatment in the evaluation of the effective energies, which have the form ∞-∞. In the calculation with such improper expressions, from which we wish to draw a finite conclusion, the results are often affected by the way of calculation so that it will possibly occur that the approximate treatment would miss the essential point. But we think we have been able to confirm, at least, that the result converges by virtue of our subtraction procedure. A method more satisfactory from the relativistic point of view is now being investigated.
URL : http://ptp.ipap.jp/link?PTP/4/47/
DOI : 10.1143/PTP.4.47