Classical theory of mesons

Abstract

The quantum theory of the meson (Kemmer 1938; Fröhlich, Heitler and Kemmer 1938; Bhabha 1938; Yukawa, Sakata and Taketani 1938; Stueckelberg 1938), in spite of its great similarity to the quantum theory of radiation, differs from it in certain important respects. The total Hamiltonian for the system of protons or neutrons and mesons contains terms in the interaction energy of mesons with the heavy particles which increase with increasing energy of the mesons. This has brought physicists to the prevalent view that for high energies this theory leads to large probabilities for multiple processes, and to explosions of the type first investigated by Heisenberg (1936). Indeed, the possibility of such processes has recently caused Heisenberg (1938) to take the position that they set a limit to the applicability of quantum mechanics. Connected with the same behaviour of the interaction is the fact that many effects, such for example as the perturbation of the self-energy of a proton calculated to the second order in the interaction, diverge more acutely than in ordinary radiation theory, and this has led Heitler (1938) and others to doubt the correctness of the fundamental equations even for mesons of energy comparable with their rest mass. All the above views are in essence based on the results of second-order perturbation calculations. There is, however, another approach to the problem. Using the commutation rules for the observables and the well-known equations of motion of quantum mechanics, we can derive exactly from the same Hamiltonian on the one hand the Dirac equation for the proton or neutron under the influence of a given meson field, and on the other hand the equations of the meson field influenced by the presence of neutrons. (For brevity we shall henceforth only speak of neutrons, whereas our remarks will apply equally well to protons, since the two are on the same footing as far as this theory is concerned.) Treating the Hamiltonian “classically”, that is, treating all the observables occurring in it as commuting variables, we can as usual again derive the same two sets of equations. In this paper, as a first step in the problem, we shall deal with classical equations, since it is possible either to solve them exactly, or at least to give approximate solutions the errors of which can be strictly estimated.