(Received January 15, 1950)
The covariant quantum field theory developed by Tomonaga, Schwinger and Feynman has been applied by many authors to the enormous calculations of various processes. However, difficulties which seemed to attributable to the mathematical defects in usual quantum electrodynamics still appear with the various types everywhere in these calculations. Although these causes may be analyzed from the different points of view, one can treat some of them as the problems of ambiguity by distinguishing them from the fatal difficulties of so-called divergence.
A part of these problems is characterized by the aspect that the results of calculations of matrix elements contradict with the formal requirements, such as the gauge-invariance, divergence theorem and equivalence theorem. Although these discrepancies would be removed by formal procedure, there would still remain various ambiguities in the results which could not be tested by any requirement, so that these conclusions are quite unreliable.
Recently, Pauli and Villars proposed the regulator method by which one can automatically remove these difficulties, and the circumstances seemed to be much improved. However, this method has not only the own difficulties that its procedure contradicted with the present concepts of field theory, but also has no definite rule on what conditions one may rationalized the results. Accordingly, we can hardly find the consistent conditions by which one separate physically significant results from non-physical terms throughout in the field theory.
On the other hand, the method of the mixed field theory analyzed by Umezawa and Kawabe, Feldman and Rayski succeeded in the problem of photon self-energy, but not offered the answers for the removal of ambiguity in general forms.
In this paper, we try to examine whether the method satisfying the following three requirements simultaneously exists or not:
1. The matrix elements which have the formal properties, such as the gauge invariancy, divergence theorem and equivalence theorem should also preserve them after the calculations.
2. The ambiguous terms in the matrix elements which have no formally required properties should also be removed.
3. All physically significant terms should be retained after the removal of ambiguous terms.
It sill be concluded that we can hardly find the mathematically consistent theory satisfying above these requirement. In spite of this conclusion, if we utilize the points of the mathematical defects in the opposite sense, we may find a way of the temporal escape, though it seem very paradoxical.
A procedure will be proposed in this paper as an example of such a method. From the results obtained in this way, we will conclude that the regulator method has been raied the unnecessary confusions and more analytical method should be necessitated by shich one can overcome these difficulties throughout.
URL : http://ptp.ipap.jp/link?PTP/5/272/
DOI : 10.1143/PTP.5.272