#### Degree

Doctor of Philosophy

#### Program

Applied Mathematics

#### Supervisor

Miranskyy, Volodya A.

#### 2nd Supervisor

McKeon, Dennis G. C.

Co-Supervisor

#### Abstract

My thesis covers several topics in the quantization and renormalization of gauge fields, ranging from the application of Dirac constraint procedure on the light front, to the manipulation of Faddeev-Popov method to enable use of the transverse-traceless gauge in first order gravity. Last, I study renormalization group ambiguities and carry out a new characterization method for models with one, two and five couplings.

In chapter 2 we apply the Dirac constraint procedure to the quantization of gauge theories on the light front. The light cone gauge is used in conjunction with the first class constraints that arise and the resulting Dirac brackets are found. These gauge conditions are not used to eliminate degrees of freedom from the action prior to applying the Dirac constraint procedure. This approach is illustrated by considering Yang-Mills theory and the superparticle in a $2 + 1$ dimensional target space.

We consider the first order form of the Einstein-Hilbert action and quantize it using the path integral in chapter 3. Two gauge fixing conditions are imposed so that the graviton propagator is both traceless and transverse. It is shown that these two gauge conditions result in two complex Fermionic vector ghost fields and one real Bosonic vector ghost field. All Feynman diagrams to any order in perturbation theory can be constructed from two real Bosonic fields, two Fermionic ghost fields and one real Bosonic ghost field that propagate. These five fields interact through just five three point vertices and one four point vertex.

Finally in chapter 4 we study the ambiguities inherent in renormalization when using mass independent renormalization in massless theories that involve two coupling constants. We review how unlike models in which there is just one coupling constant there is no renormalization scheme in which the $\beta$-functions can be chosen to vanish beyond a certain order in perturbation theory, and also the $\beta$-functions always contain ambiguities beyond first order. We examine how the coupling constants depend on the coefficients of the $\beta$-functions beyond one loop order. A way of characterizing renormalization schemes that doesn't use coefficients of the $\beta$-function is considered for models with one, two and five couplings. The renormalization scheme ambiguities of physical quantities computed to finite order in perturbation theory are also examined. The renormalization group equation makes it possible to sum the logarithms that have explicit dependence on the renormalization scale parameter $\mu$ in a physical quantity R and this leads to a cancellation with the implicit dependence of R on $\mu$ through the running couplings, thereby removing the ambiguity associated with the renormalization scale parameter $\mu$. It is also shown that there exists a renormalization scheme in which all radiative contributions beyond lowest order to R are incorporated into the behavior of the running couplings and the perturbative expansion for R is a finite series.

COinS