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UBC Theses and Dissertations

Relativistic few-body quantum mechanics Monahan, Adam Hugh


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.

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