Construction and Analysis of a Hierarchical Massless Quantum Field Theory

This dissertation focuses on critical phenomena in statistical mechanics and Quantum Field Theory. This involves the analysis of systems with infinitely many degrees of freedom across different length scales coupled together via interactions which can be easy to describe locally but give rise to a rich class of emergent phenomena. We adopt the framework of mathematical physics and probability where these systems are represented as measures on certain infinite dimensional spaces. The primary approach used in this dissertation is the \emph{Renormalization Group}, a powerful and elegant framework that reveals how the collective influence of degrees of freedom manifest at different length scales within these systems. Pioneered by the physicist Kenneth Wilson, the philosophy of the RG approach is to reduce the analysis of these complex systems to the study of a ``tractable'' infinite dimensional dynamical system of effective potentials. The first project develops a Renormalization Group for spatially inhomogenous systems that allows one to establish a rigorous correspondence between orbits in Wilson's dynamical system and the measures one expects them to represent - this is done in the setting of a hierarchical approximation to Wilson's $4 - \epsilon$ expansion. This culminates in the construction of a translation invariant, rotation invariant, and partially scale invariant generalized random field corresponding to the Wilson-Fisher fixed point. The second project leverages methods from statistical mechanics to strengthen this result and show that this generalized random field is in fact fully scale invariant.