The ancient Greeks believed that everything in the Universe should
be describable in terms of geometry. This thesis takes several
steps towards realising this goal by introducing geometric
descriptions of systems such as quantum gravity, fermionic
particles and the origins of the Universe itself. The author
extends the applicability of previous work by Vilkovisky, DeWitt
and others to include theories with spin 1/2 and spin 2 degrees of
freedom. In addition, he introduces a geometric description of the
potential term in a quantum field theory through a process known as
the Eisenhart lift. Finally, the methods are applied to the theory
of inflation, where they show how geometry can help answer a
long-standing question about the initial conditions of the
Universe. This publication is aimed at graduate and advanced
undergraduate students and provides a pedagogical introduction to
the exciting topic of field space covariance and the complete
geometrization of quantum field theory.
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