(Received July 8, 1946)
In studying the interactions of mesotrons with nuclear particles the perturbation theory has been first applied without being earnestly doubted whether this process will give convergent results in these problems. In the quantum electrodynamics which deals with the interaction between electromagnetic field and electrons this process has given the results which agree very well with experimental evidences so that one can scarcely doubt the applicability of this process in this case. Still, there are several reasons for believing that the perturbation theory would not be applicable to the case of mesotron theory.
The interaction between mesotron field and nuclear particles differs in many respects from that between electromagnetic field and electrons. In the first place, the former interaction is much intensiver than the latter: the coupling constant g specifying the former interaction is, at least, about ten times so large as the constant e which specifies the latter. In the second place, the potential of the mesotronic self-field of a nuclear particle has a singularity of the form 1/r3 or δ(r) which is a stronger singularity than that of the electrostatic self-field of an electron where the potential is of the form of 1/r. This fact means that the interaction between nuclear particle with its mesotronic self-field is much more singular than that between electron and its electric self-field. It is, thus, rather doubtfull whether the perturbation theory does converge in case of our mesotron theory, since this procedure presupposes the weakness of the coupling.
Under such circumstances several authors have attempted to solve the problem involving the interaction of the nuclear particle with the mesotron field by some other ways without using the rather problematic perturbation theory. The aim of the present paper is to give a review of treatments of this kind and to make clear the underlying ideas and discuss the results obtained by these treatments.
That the perturbation theory can not be applicable to our problem is implied by a simple example. One calculates the scattering cross-section of a mesotron on colliding with a nuclear particle. Supposing thereby that the nuclear particle is infinitely heavy so that in this collision its recoil can be neglected, one finds, according to the ordinary perturbation theory, that the cross-section increases indefinitely with increasing energy of the colliding mesotron. It is, however, unreasonable that in such a problem the cross-section exceeds the square of the wave length of the incident particle.
Physically means the application of the perturbation theory the neglect of the field reaction. The fact that the scattering cross-section exceeds the square of the wave length of the colliding particle means that in such a case this reaction is by no means small. What prevents the cross-section from increasing so unreasonablly must be just this field reaction.
The reaction of a field on a particle can be classified into two distinct kinds, each of which has physically as well as mathematically different properties. The first one of these reactions is caused by the following phenomena: Around the particle there exist the so-called self-field. But the field has inertia against its change. When, now, the state of motion of the particle changes, also, its self-field must change so that it exhibits the inertia and an inertial reaction acts on the particle. Since this reaction is, in the first approximation, proportional to the acceleration of the particle, this reaction results in an appearent change of its mass. The other one of the field reactions is the effect of damping which is caused in the following manner: When the particle is in a state of non-uniform motion, it disturbs the surrounding field. Then waves propagate from this center, so that energy will be brought away by these waves, because under the natural boundary condition only outgoing waves can exist at infinity. This phenomena results thus in the dissipation of energy of the particle and, consequently, an irreversible damping force acts on the particle.
As is well known, the inertial reaction of the field is infinitely large in the existing theory, so that one is forced to touch on the fundamental defect of the quantum field theory whenever one wil deal with this reaction. Thus the existing theory is quite powerless to give account of such phenomena in shich the first term of the perturbation theory does not suffice and the field reaction takes an essential part. One can get rid of this difficulty only by a very unsatisfactory procedure of “cut-off”.
Although there exists at present no unumbiguous way of introducing the cut-off process in general cases, it seems fairly certain that this process corresponds in some way to giving a finite size to the particle interacting with the field. In the simplified problem in which the particle is cosidered as infinitely heavy, this process of giving the finite size is introduced in a comparatively unumbiguous way. So we shall in this paper deal with this simplified problem, and ask in what way the nuclear particle with infinite mass and finite size will interact with the mesotron field.
When the problem is simplified in this way, it can be solved also when the coupling is not weak. Thus one can see in what manner the field will react on the particle. When the coupling of the field and the particle is very strong, this can be done by solving the problem classically, since we may expect that in this case the field generated by the particle is so large that the quantum-theoretical fluctuation can be neglected throughout. Indeed, it is seen by the quantum-theoretical treatment of Wentzel, which is a powerfull method applicable to the case of strong coupling, that the classical theory gives a satisfactory approximation in case of strong coupling. Wentzel has nemely treated the problem quantum-theoretically expanding the solution in inverse powers of the coupling constant, whereas in the ordinary perturbation theory one finds the solution in powers of the coupling constant.
If the interaction is not strong enough and neither classical nor Wentzel's treatment is allowed, the problem can be solved approximately by Ritz's procedure. This method gives a reasonable results which connect the results in the exterme cases of strong and weak couplings: Wentzel's results on one hand and the results of the perturbation theory on the other hand.
In the present paper we shall deal with the problem successively by classical, by Wentzel's and by Ritz's methods. We shall show first that the results of the classical teratment do really agree with the quantum-theoretical results of Wentzel when the coupling is strong. We shall then treat the problem by Ritz's procedure. These treatments give reise to several results which differ substantially from those obtained by the perturbation theory. These differences represent just the effects of the field reaction. We shall discuss in the last paragraph the physical consequences of these treatments.
URL : http://ptp.ipap.jp/link?PTP/1/83/
DOI : 10.1143/PTP.1.83