Theory of Mesons and Nuclear Forces

C. Møller and L. Rosenfeld

Editor’s Note

Mesons were so-called because they are intermediate in mass between the electron and the proton. In 1935, in a Japanese journal, Hideki Yukawa and colleagues proposed that mesons could account for the strong forces between nucleons (protons and neutrons) much as photons account for the electrical forces between charged particles of all kinds. This article by Rosenfeld and Møller, both protegés of Niels Bohr at Copenhagen, suggests how physical considerations require particular forms of the mathematical expressions (called wave functions) that agree better with experiments on the decay of light elements emitting β-particles. This prediction proved correct.　　中文

AS was first pointed out by Yukawa, it is in principle possible to account for the short-range forces between nuclear particles by the assumption of virtual emission and absorption processes involving intermediary particles of integral spin, the so-called mesons1, the mass of which is determined by the range of the forces. As has been shown by Kemmer2, the simplest wave-equations for the mesons which satisfy, besides the claim of relativistic invariance, the condition of giving a positive definite expression for the energy, reduce to four types, characterized by different co-variance properties of the wave-functions, and each allowing the existence of neutral as well as positively and negatively charged mesons. Starting from such wave-equations, including the interaction of the meson field with the heavy nuclear constituents, the estimation of the resulting expressions for the nuclear forces has hitherto been carried out by using the ordinary perturbation method of quantum theory, and taking into consideration only the first non-vanishing approximation, in spite of the well-known lack of convergence of the method. It would thus seem desirable to discuss more closely the reliability of such results, and for this purpose a possible method of attack is suggested by an analogous situation in quantum electrodynamics, where a suitable canonical transformation allows us to separate, from the expression of the total energy of a system consisting of electrons and an electromagnetic field, a term depending only on the coordinates of the electrons and representing the Coulomb potential energy.　　中文

A similar method3 is, actually, applicable to a system consisting of nuclear particles and a meson field. For such a system it is, in fact, possible to find canonical transformations effecting the separation of a “static” interaction between the nuclear particles, defined as the part of the interaction which is obtained when one neglects the time-variations of the variables characterizing the positions, spins and proton or neutron states of the heavy particles. This static interaction is in all cases exactly the same as that obtained as a first approximation in the perturbation method, and there exists a lower limit, smaller but unfortunately not much smaller, than the range of the nuclear forces, to the mutual distances between two heavy particles for which the static interaction is more important than the additional non-static contributions arising from the terms, in the transformed Hamiltonian, which describe the remaining interactions between the heavy particles and the meson field.　　中文

Although no improvement can, of course, be obtained in this way as regards the self-energy difficulties, it would seem that consistent results can be derived from the transformed Hamiltonian by considering only the last-mentioned interactions as a perturbation and applying a method of treatment analogous to the correspondence methods used in electrodynamics. It is especially to be noted that if, following Yukawa, we also introduce an interaction between the meson field and electrons and neutrinos, the transformed Hamiltonian contains a term which represents a direct interaction between the heavy particles and the electrons and neutrinos, and which, when treated as a small perturbation, immediately gives the probabilities of β-disintegration processes. It is perhaps to be regarded as a satisfactory feature of the point of view just outlined that, contrary to previous treatments, where the nuclear forces came out in the same stage of the perturbation method as the probabilities of β-decay, account is here taken at the outset of at least the static part of the nuclear forces.　　中文

As regards the form of these static interactions, it is well known that the type of potential resulting from the four-vector meson field generally considered hitherto has the defect of including a term of dipole interaction energy which is so strongly singular for infinitely small mutual distances of the nuclear particles that it would not in general allow the existence of stationary states for a system of such particles. In order to remedy this defect, it seems necessary2 to introduce besides the four-vector wave-function a further pseudoscalar wave-function for the meson field which has the property of giving rise to a static interaction of a form just capable of cancelling the dipole interaction term without affecting the others. The consideration of such a pseudoscalar meson field would also seem to be useful from the point of view of the theory of β-decay. While, for example, the four-vector theory yields4 exactly the same form of the β-spectrum as Fermi’s original theory, the introduction of a pseudoscalar wave-function in addition to the four-vector one gives rise to a modification of the energy distribution of the β-rays which seems to open a new possibility of a better adaptation to the experimental results.　　中文

A detailed account of our work will appear shortly in the Proceedings of the Copenhagen Academy.　　中文

(143, 241-242; 1939)

C. Møller and L. Rosenfeld: Institute for Theoretical Physics, Copenhagen, Jan. 6.

References:

1. See Bhabha, Nature (in the Press).

2. Kemmer, Proc. Roy. Soc., A, 166, 127 (1938); Proc. Camb. Phil. Soc., 34, 354 (1938).

3. Independently of our work, essentially the same method has been proposed by Stückelberg (Phys.Rev., 54, 889; 1938), to whom we are very thankful for the kind communication of his manuscript.

4. Yukawa, Sakata, Kobayasi, Taketani, Proc. Phys. Math. Soc. Japan, 20, 720 (1938).