We devised in Chapter 4 a method by means of which a
transition from Q.F.T, to a potential theory for a system of
two particles can be effected without assuming much beyond the
general postulates of Axiomatic Q.F.T. This is the decisive
advantage of our method over the ones outlined in Chapter 1.
There, the first major step was the establishment of the
"first" Lippmann -Schwinger equation (4.10a) which is exact
if the energy of the scattering state is below the pion production threshold. Jauch and Rohrlich ( 27), and Heisenberg( 28) claimed to have proved the same equation in Q.F.T. for
energies, but throughout their proofs, they assumed the
possibility of using the eigenstates of the free Hamiltonian
as a basis of representation. Moreover, their formalism then
led to the "second" Lippmann -Schwinger equation (2.9a) in
which the matrix elements of the potential were replaced by
those of the interaction Hamiltonian between two free states.
Since for trilinear interactions the latter matrix elements
vanish due to the violation of conservation of energy -momentum,
their equivalent of (2.9a) means that the reaction matrix
From (4.10a) we obtained the Low equation (4.11a), exact
for the same energy range. This latter should be compared with
the one obtained by Low(29) whose method only works in the
peculiar case of pion- nucleon scattering. His equation is
approximate even in the low energy region. Examination of
(4.11a) showed that we have much freedom to assign values to
the scattering matrix elements involving high energy scattering
states. In particular we can demand the D- matrix elements
between two -particle states to have the same functional form
as those of them which have both energies below the threshold,
which course of action we adopted in Chapter 5, and in Chapter
3 albeit only on the energy-shell. Under this circumstance the
T-matrix elements on the energy-shell defined in both Chapters
for a potential theory should be equal in the case of boson - boson scattering, and hence if elements of the potential on
the energy -shell obtained in these Chapters differ, it must
be due to the fact that the T- matrix elements are split into
two parts, the potential and the rest, differently.
At present, our potential theory suffers from two defects.
The first one is that the wavefunctions of the theory may not
be orthonormal. The second is that we have no reliable method
of calculating the potential defined in it. To a great extent
these defects are due to the fact that in Q.F.T. a satisfactory way of estimating, never mind calculating,
most expressions arising in it, is still waiting.
We have not gone very far with the possible analytic
properties of our T- matrix elements in Q.F.T. We look forward
to the time when that is done so that we can test our definition
of the potential between two particles.