#### Document Type

Dissertation

#### Date of Degree

Fall 2009

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Applied Mathematical and Computational Sciences

#### First Advisor

William H. Klink

#### Abstract

This dissertation investigates part of the strong nuclear force in point form QCD. The quark sector is neglected to focus on gluons and their self-interactions. The structure of gluons is investigated by building up a field theory for massless particles. Single gluon states are defined, and a condition is implemented to make the wave function inner product positive definite. The transformation between gluon and classical gluon fields generates a differentiation inner product, and the kernels allow for transition between momentum and position space. Then, multiparticle gluon states are introduced as symmetric tensor products of gluon Hilbert spaces generated by creation and annihilation operators. In order to assure that the resulting Fock space inner product is positive definite, an annihilator condition is needed and gauge transformations are introduced. The four momentum operator consists of the stress-energy tensor integrated over the forward hyperboloid. The free gluon four momentum operator introduced via the Lagrangian and stress-energy tensor is shown to be equivalent to that generated by gluon irreducible representations when acting on the physical Fock space.

Next the vacuum problem is discussed, where the vacuum state is the state that is annihilated by the the four momentum operator and is invariant under Lorentz and color transformations. To find such a state, the vacuum problem is simplified by considering a one degree of freedom model. The Hamiltonian for such a model, the one dimensional energy operator, is solved under a variety of different ansatzes. It is shown that the Hamiltonian has a continuous eigenvalue spectrum, and that the vacuum can be constructed in a way that eliminates the interaction term of the Hamiltonian. This one dimensional vacuum model is adapted to the full problem where it is shown that such a result cannot be replicated.

#### Pages

vi, 71

#### Bibliography

71

#### Copyright

Copyright 2009 Kevin Christoher Murphy