Novel Approaches to Renormalization Group Transformations in the Continuum and on the Lattice

TL;DR

This thesis introduces new methods for renormalization group transformations in continuum and lattice quantum field theories, emphasizing lattice formulations and novel approaches like Gradient Flow and Stochastic RG, supported by simulation results.

Contribution

It presents a continuous lattice RG approach using Gradient Flow and explores the relationship between FRG and stochastic processes, advancing computational and analytical techniques.

Findings

Gradient Flow effectively implements RG transformations on the lattice.

Simulation results demonstrate the applicability of new RG methods to scalar and gauge theories.

Theoretical connections between FRG and stochastic processes are established.

Abstract

This thesis is about new methods of achieving RG transformations, in both a continuum spacetime background and on a lattice discretization thereof. The subject is explored from the point of view of euclidean quantum field theory. As a thesis grounded on the computational method of lattice simulation, I emphasize the role of lattice formulations throughout the work, especially in the first two chapters. In the first, I describe the essential aspects of lattice theory and its symbiosis with RG. In the second, I present a new, continuous approach to RG on the lattice, based on a numerical tool called Gradient Flow (GF). Simulation results from quartic scalar field theory in 2 and 3 dimensions ($\phi^4_d$) and 4-dimensional 12-flavor SU(3) gauge theory will be presented. In the third and fourth chapters, the focus becomes more analytic. Chapter 3 is an introductory review of Functional Renormalization Group (FRG). In chapter 4, I introduce the concept of Stochastic RG (SRG) by working out the relationship between FRG and stochastic processes.