Download The Yukawa Theory of Nuclear Forces in the Light of Present

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.
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