(Received September 12, 1949)
Recent researches developed by Tomonaga and his collaborators, Schwinger and Feynman have brought the theory of quantum electrodynamics to an advanced stage. These authors have extended the field theories initiated by Heisenberg-Pauli's work fully relativistically, and have given them beautiful appearances. On the other hand, the development of ingenious and minute experiment which has brilliant representatives in the determination of energy levels of hydrogen atoms or of the value of the anomalous magnetic moment of an electron, makes it possible to estimate the effects of the interaction between the radiation field and charged particles. These situations force us to harmonize the frequently said divergence difficulties of quantum field theory with the real experimental facts. Bethe, Schwinger, Weisskopf, Tomonaga and others restored the idea of electromagnetic mass and constructed a formally closed theory by the use of "ronormalization" method and obtained satisfactory agreements with experiments though this method has an unsatisfactoriness of handling the quantities which are divergent in practice.
In spite of many successes, it was pointed out by many authors that these theories contain ambiguous features concerning with the photon self-energy problems and are being investigated in many other cases.
This defects apparently come from the fact that they are treating infinite quantities in separate way in each cases.
We believe that these divergence difficulties are closely connected with the infinitesimal character of the commutators of field variables and more essentially with the point character of the interaction between fields which was the corner stone of the present quantum field theories. Without altering this fundaments it would be impossible to get a satisfactory theory and the future theory would be the one in which no divergence quantities appear explicitly. Of course it may be probable that the present difficulties come from the defects of the mathematical devices. But in our opinion the improvement of such mathematical tools would necessarily cause the change of the present physical concepts.
Recently, Pauli and Villars proposed a remedy for these difficulties. They have “regularized” the infinite quantities and removed ambiguities concerning them. Although this method contains some ambiguities in the case of γ-instability of neutral Bose particles, we think that this is an good approximation to the correct theory in the sense that it gives satisfactory results.
It was shown that Pauli's method is equivalent to the mixed fields. We have brought these fields together in a closed form by extending the present theory to a five dimensional formulation, and got results in agreement with Pauli's one. In our formalism the added coordinate is independent from the usual space-time coordinates and thus has a conventional character. It is, so to speak, an “unphysical variable”. We are reconciled ourselves, however, to this circumstance, though it is very unsatisfactory, in view of the transitional character of the present formalism. We think that the new freedoms, if it were necessary to add them, would not have the equivalent character to the space-time coordinates, and we hope that our formalism would become an approximation of some 5-dimensional covariant form.
URL : http://ptp.ipap.jp/link?PTP/5/14/
DOI : 10.1143/PTP.5.14