An Approach to the Formulation of Quantum Electrodynamics without Use of Indefinite Metric

Abstract

Listen

Translate article

The aim of this paper is to present a new approach to the formulation of. the quantum electrodynamics without using indefinite metric, and the program is examined to the third order in the perturbation expansion. As the wave function of free photons, we adopt the representation in which only the transverse components take part, and the second quantiza­ tion is performed by Fock-Cook-Friedrichs' method. The first order interaction Hamiltonian is so determined as to correspond to that of the transverse photon in the conventional :field theory .. The Coulomb interaction is introduced as a second order additional term which is necessitated by the covariance requirement of the theory. As for the second order diver­ gences which correspond to the self-energy of the photon and to the renormalization of the electron· wave function in the conventional theory, it is shown that the counter terms are necessarily introduced from the requirement of covariance only. The divergence corres­ ponding to the renormalization of the photon wave function does not· appear. To treat the divergence of the self-energy of the electro~, an additional counter term must be added as it is done in the usual theory. As regards the third order vertex part, it is shown that the correct value of the anomalous magnetic moment is obtained, though some new feature appears in the divergent part. §I. Introduction and summary In the formulation of the usual relativistic quantum electrodynamics the four potential plays a fundamental role. According to the general theory of the representation of the inhon1ogeneous Lorentz group, however, only the two component wave function is necessary for massless particles/) and the elimination of the redundant components had been the serious problem of the relativistic quantum electrodynamics until the method of quantization by indefinite metric was developed. 2 l On . the contrary, if the wave function containing only the transv~rse components is adopted as· the photon wave function from the begin­ ning, we need not be worried by such a pathological problem. This is possible, even in the conventional theory, if the use is made of the Coulomb gauge. The Coulomb gauge, however, is not a relativistic concept. In order to formulate the covariant quantum electrodynamics with the photon wave function containing only transverse componts, it seems then necessary to reinterpret the wave func­ tion of the photon not as the electro-magnetic potential but as the state vector of the representation space of the inhomogeneous Lorentz group. Taking the viewpoint mentioned above, the possibility of formulating the