The Use of the Proper Time in Quantum Electrodynamics I

The space-time approach to quantum electrodynamics, as has been developed by Feynman, seems to offer a very attractive and useful idea to this domain of physics. His ingenious method is indeed attractive, not only because of its intuitive procedure which enables one to picture to oneself the complicated interactions of elementary particles, its ease and relativistic correctness with which one can calculate the necessary matrix elements or transition probabilities, but also because of its way of thinking which seems somewhat strange at first look and resists our minds that are accustomed to causal laws. According to the new standpoint, one looks upon the world in its four-dimensional entirety. A phenomenon that will come into play in this theatre is now laid out beforehand in full detail from immemorial past to ultimate future and one investigates the whole of it at glance. The time itself loses sense as the indicator of the development of phenomena; there are particles which flow down as well as up the stream of time; the eventual creation and annihilation of pairs that may occur now and then, is no creation nor annihilation, but only a change of directions of moving particles, from past to future, or from future to past; a virtual pair, which, according to the ordinary view, is foredoomed to exist only for a limited interval of time, may also be regarded as a single particle that is circulating round a closed orbit in the four-dimensional theatre; a real particle is then a particle whose orbit is not closed but reaches to infinity. In such a view, a state with prescribed number of particles including real as well as virtual does not exactly correspond to a four-dimensional state in the ordinary sense, that is, a state represented by the wave function satisfying the time dependent Schroedinger equation. But the former is rather a part of the latter in which any number of virtual particles may be allowed to occur. To obtain an idea of the actual state we shall have to sum over all possibilities as to the number of virtual particles. The interpretation of the four-dimensional state in the present sense becomes also somewhat different from the conventional one as giving the transition probability or amplitude from a given state A to a final state F in the three-dimensional space. We can rather ask for the relative probability that a four-dimensional state A-I-F with prescribed real as well as virtual (intermediate) particles be realized in nature. This will be zero unless the arbitrary chosen A-I-F is not such as is an actually possible transition under the Schroedinger equation. The above-mentioned view of the entire space-time behavior of nature sub specie aeternitatis, however, might not appeal to a reason which is liable to think in the language of differential equations and pursue the development of things along a certain parameter. In fact we find it hard to regard the world line of a particle as a mere status of that particle, but are unconciously following the motion of an imaginary mass point along the world line. Thus, in Feynman's theory where the ordinary time loses its rôle as the indicator of the development of the world, it would still be convenient to introduce some parameter with which the four-dimensional world is going to shape itself. How this is possible to a certain extent we shall see in what follows.