The understanding of the two- nucleon interaction is
the central problem in the theory of nuclear forces.
Whether specifically many -body forces turn out to be
important or not, the investigation of the two -body
interaction is bound to give us some insight into the
mechanism responsible for the strongest binding known
So far, the only attempts, in theory, to go beyond
pure guesswork (or phenomenology, as it is politely
called) have been along two essentially different
directions. The first type of computations, which were
initiated by Yukawa and have since been assiduously
followed up by many physicists (e.g. Taketani et al.)
may be collectively classed as "meson- theoretical calculations". The main idea behind these efforts if the
belief (perhaps quite true) that nuclear forces are
mediated by a meson cloud and, naturally, the closer
together the nucleons are, the larger the number of pions taking part. This rather loose inverse relationahip between the range of a force and the corresponding mass of
the intermediate state is now widely accepted among
elementary particle physicists. However, the quantitative
realisation of the Taketani programme takes the form of
computing one perturbation graph after another in the
framework of field theory. The one pion exchange graph,
of course, yielded the famous Yukawa potential which, as expected, was very successful in explaining the long range
effect of the nucleon force. Higher order effects have
not had the same success in the "medium" range, with the
consequent flourishing of phenomenology. Apart from the
increasing technical difficulty of calculating higher
order graphs, it is very probable that the perturbation
series is divergent in which case the whole project is
essentially illusory, and any occasional agreements with
experiments must be regarded as fortuitous. There have
been many variations from straight expansion in the
coupling strength or the number of intermediate particles.
A comprehensive survey is given in the review of
Y oravcsik & Noyes(16).
A more hopeful approach has appeared in recent years,
following Fandelstam's proposal of double dispersion
relations for the scattering amplitude in a two-particles-in, two-particles-out process. It was initiated by Champ
& Fubini(4), who indicated how one can obtain a useful
"potential ", starting from the Mandelstam representation.
But he could do it reliably only for the case of scalar
particles, mainly because no one had derived a Kandelstam
representation, or even a single variable dispersion
relation, for two particles of spin 0.5 interacting through
a general spin and velocity- dependent potential. Up to
now, there have appeared only two papers on this subject
and both have touched the problem only partially.
Hamilton(9) has proved dispersion relations which are
those (12) similar to those that Khuri obtained for central potentials,
considering only an additional tensor term. The other
attempt has been by Buslayev(3) who considers a particle
scattered by a spin-orbit potential.
The present work is intended to fill this gap; we
obtain dispersion relations (in energy, for fixed momentum
transfer) for two spin 2 particles interacting through a complete potential subject only to reasonable physical
requirements. The relations obtained turn out to be
substantially different in form from those derived or
postulated as yet; and the methods used have the virtue
of being generalisable to the scattering between systems
of arbitrary spin, as indicated in the last chapter.
The derivation of these dispersion relations is also a step forward in the fulfilment of the Charap-Fubini
programme of deducing a realistic two-nucleon potential.
We will now give a brief outline of this manuscript.
In Chapter II, we discuss the form of the potential
for a two-nucleon system, and then go on to derive some
properties of the resolvent of the Hamiltonian. We also
have a brief look at the spectral decomposition of the
total Hamiltonian. Most of the matter contained in
this chapter is of a preliminary character, and no originality is claimed.
Chapter III forms the bulk of the thesis in which we
obtain the analytic properties and asymptotic behaviour
of the Hamiltonian Green's function in the complex energy
plane. We start by deriving an integral equation for the
Green's function incorporating the outgoing boundary
condition. This equation does not have a bounded or
square integrable kernel so that the usual methods of
solution do not apply. However, it is seen that the kernel
is only "weakly" singular (to use the terminology of
Yikhlin(15)), and Fredholm's theorems can still be applied.
This is the central point of the argument, which permits
us to infer that the Green's function is analytic except
on the spectrum of the Hamiltonian. We then proceed to
investigate the spectrum of the total Hamiltonian, and
the approach is in part borrowed from Povzner(19). The
chapter concludes with a detailed investigation of the
Green's function at high energy.
The analyticity and asymptotics obtained in Chapter III
help us in writing down an explicit form for the scattering
amplitude in Chapter IV. The amplitude is treated in
detail, and split into five parts with different spin-invariants, as e.g. in Goldberger, Nambu & Oehme(8) . Each
coefficient separately obeys a dispersion relation which
is then written down. We close the chapter with a few
physical remarks of interest.
Chapter V contains a generalisation of the above
results to systems with arbitrarily high spins. It
gives only a sketch of the arguments without going into
much cumbersome detail.