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Progress of Theoretical Physics, Vol•. 0; No. 4, }uly-AlIg\IBt, 1900 The Yukawa Theory of Nuclear Forces in the Light of Present Quantum Theory of Wave ·Fields. w. HEISENBERG Maz Planck-lnstitut fur Pkysik, Gotfing-en. (Received June 0, 1950) The Yukawa theory of nuclear force!llllbas led to many successes and, owing to the present state of quantum theory, to some d'ifficulties. Among the successes one remembers first the existence Of the 7Z'-meson and the possibility of desc.ribing the spin dependency and the saturation of nuclear forces by means of simple vector. fields or pseudo-scalar fields. Among the difficulties we mention the divergence of the interaction at small distances of the nucleons and the impossibility of getting the correct mass defect for heavy nuclei when one takes the cohstants of the Yukawa field from the mass defect of light nuclei.2) FUrthermore, the existence of closed neutron and proton shells in the nucleusS) and the behaviour of the cross section for elastic collisions Of nucleons at very high energies indicate, that the Yukawa potential is not correct at small distances of the nucleons. These difficulties cannot be really solved yet; but the recent progress in quantum theory of wave fields4) shows so clearly the way towards the solution of these problems; that it may be worth while to discuss this way, even if it is still too early to work it out in the mathematical details. In the relativistic quantum theory of wave fields we have learned, that the divergent results arise from the singularities in the commutation function. Therefore the correct theory will have to start with a regular commutation function. This starting point leads to a number of problems, which have been dealt with recently in many papers.5) \Ve mention the most important results: The wave function that obeys a regular commutation rule, corresponds necessarily to several different types of elementary partir.ies, not only to one type. This implies, that nucleons interact, as Bhabha6 ) has suggested, not only by means of 7Z'-mesons but also by other types of particles, in such a way, that the singularity of the force at small distances will disappear. Furthermore, the two coordinated wave functions, that obey the regular commutation rule, cannot be hermitian conjugates in the ordinarysense.5) This leads probably to a change in the hamiltonian foro. malism in the range of the" smallest length" 10 (l0-IO-13 cm), which corresponds to a lack of point-to-point causality, again tending to wash out singularities of the field. w. 524 HEISEN BERG Thereb y already many of the difficulties may have disappeared. The potenti al inside a nucleus will now be rather smooth, certain ly much smooth er than one would expect from potenti als of the type or the corresponding tensor force potenti al. As a result there will be only small forces acting upon a nucleon inside a .nucleus; it is only at the surface of the nucleus that the nucleons will be pulled back into the nucleus by strong forces. This explain s quite na~rall! the ·existence of separat ed neutron and proton orbits and dosed shells ln the nucleus. The order of the closed shells can be understood, according to Haxel, Jensen . Suess~) and Goppert-Mayer,S) from a strong spin-or bit coupling of every nucleon. Gaus7) has. shown that this str.ong spin-Olbit coupling results under certain conditions immed iately from the vector-meson theory of Yukaw a. Theref ore one may at this point conclude· from the experiments, that at larger distanc es of two nucleons the symme trical vector meson theory with the mass of the n-meson will probably give a fairly good approximation, while at smalle r distanc es the higher masses will come into play and the deviations from the hamilto nian formalism will make the definition of a potential rather doubtful, as it was expect ed long ago from the concept of the" smallest length."S) The existen ce of neutral mesons, possibly of the scalar type, may produce forces withou t the ·proper ty of saturation. This would expl:tin natura lly the rather large mass defects of heavy nuclei as compa red to the mass defects of light nuclei. The obsetved saturat ton would then, as Telle£'l> has sugges ted, probab ly be brough t about by the non-linear interac tion terms in the field equation, which. preven t the Yukaw a field to increase above a certain value Anothe r difficulty for the vector-meson theory was the sign of the quadrupolemomen t of the deuteron, which seemed to favour the pseudo-scalar rather than the vector theory. The quadrupole moment of the deutero n is determ ined by the tensor force, which depends strongly on the potenti al at small distanc es i the mass defect and the spin-orbit coupling depend more strong ly on the outer part of the potential function. Therefore the h~gher masses and the deviati on from Hamilt on formalism may be decisive for the quadrupole momen t of the deutero n, while the mass defect and the spin-orbit coupling are mainly produc ed by the vector field of the normal n-mesons. Finally the cross-section for elastic collision of very fast nucleo ns will decrea se more rapidly with increasing energy than one would expect from the x" and the corresponding Yukaw a potenti al tensor potenti al. One may expres s r this mathematical result by stating , that the introduction of the Ie s"mallest length " 10 in the primar y commu tation function leads also to a .. largest force" of the order 10- 2 (or in ordina ry units so that a momen tum transfe r of mqch more than ~ in an elastic collision ~ill be a rather rare event. The collision of very fe-"" .!.e- t), The Yukawa TltI!ory of Nue/ear Forces 525 energetic nucleons will instead as a rule lead to the creation of new particles, first of 1%'-mesons and at still higher energies of' other masses, The quantitative question at which energies the deviations from the simple Yukawa potential appear cannot yet be solved. Refereaees. H. Yukawa, Proc. Phys.-Math. Soc. Jap. 17 (1985), 48. Compare f. i. H. Euler, ZS f. Phys. 105 (1937), 003, or H. Primakotr and T. Holstein, Phys. Rev. 55 (1938), 1218. 3) O. Haxel, J. H. D. Jensen u. H. E. Suess, Naturwiss. 35 (1948), 376; 36 (1949), 153; Phys. Rev. 75 (1949), 1766, and M. Goeppert-Mayer, Phys. Rev. 75 (1949), 11)69. 4) S. Tomonaga, Prog. Theor. Phys. 1 (1946), r1; Phys. Rev. 74 (1948), 224; R. P. Feynman, Phys. Rev. 74 (1948), 939, 1430; J. Schwinger, Phys. Rev. 74 (1948), 1439; 75 (1949) 651; 75 (1945), 790; F. J. Dyson, Phys. Rev. 75 (1949), 486, 1736. 5) Compare W. Heisenberg, Zur Quantentheorie der Elementarteilchen, ZS. f. Naturforschung, to appear shortly, which contains a number of references of the most important papers. 6) H: J. Bhabha, Phys. Rev. 77 (1950), 665. 7) H. Gaus, ZS f. Naturforschung 4a (1949), 721. 8) W. Heisenberg, Ann. d. Phys. 32 (1938), 20. 9) E. Teller, Private communication. 1) 2)