This thesis develops relativistic quantum mechanical models with a finite number of
degrees of freedom and the scattering theories associated with these models.
Starting from a consideration of the Poincare Group and its irreducible unitary representations,
we develop such representations on Hilbert Spaces of physical states of one,
two, and three particles. In the two- and three- particle cases, we consider systems in
which the particles are non-interacting and in which the particles experience mutual interactions.
We are also careful to ensure that for the three-body system, the formalism
predicts that subsystems separated by infinite spatial distances behave independently.
We next develop the Faddeev equations, which simplify the solution of multi-channel
scattering equations. These are specialised to the three-body system introduced earlier
and a series solution of the Faddeev Equations is obtained. A simple mechanical model
is introduced to provide a heuristic understanding of this solution. The series solution is
also expressed in a diagrammatic form complementary to this mechanical model.
A system in which particle production and annihilation are allowed is then introduced
by working on an Hilbert Space which is the direct sum of the two- and three-body Hilbert
Spaces considered earlier. It is found that in this 2-3 system, as the mass operator and
the number operators do not commute, it is not possible for a system to simultaneously
have a sharply defined mass and number of particles. The Faddeev Equations for this
system are then considered, and a series solution of these equations is developed and
discussed. It is also shown that the particle production and annihilation potential has a
non-trivial effect on pure two-body and three-body scattering.
In the last chapter we consider an attempt to derive from a more elementary field theory, using the dressing transformation, a form for the potential coupling the two- and
three-body sectors of the Hilbert Space in the 2-3 system. It is found that this method
is inherently ambiguous and is not, therefore, able to provide such information.Science, Faculty ofPhysics and Astronomy, Department ofGraduat