Fuzzy Scalar Field Theories: Numerical and Analytical Investigations

TL;DR

This thesis investigates quantum field theories on fuzzy spaces, using numerical simulations and geometric analysis to explore phase structures and the preservation of geometry in a non-perturbative regime.

Contribution

It introduces a new non-perturbative analysis of scalar field theories on fuzzy spaces, revealing phase diagrams and a novel UV-IR mixing phenomenon, and develops a geometric approach to fuzzy S4 via CP3.

Findings

Identification of a third non-uniform phase in the 3D fuzzy scalar field theory.

Evidence of UV-IR mixing in the strong coupling regime.

Geometric interpretation of fuzzy S4 as a Kaluza-Klein reduction from CP3.

Abstract

This thesis is devoted to the study of Quantum Field Theories (QFT) on fuzzy spaces. Fuzzy spaces are approximations to the algebra of functions of a continuous space by a finite matrix algebra. In the limit of infinitely large matrices the formulation is exact. An attractive feature of this approach is that it transparently shows how the geometrical properties of the continuous space are preserved. In the study of the non-perturbative regime of QFT, fuzzy spaces provide a possible alternative to the lattice as a regularisation method. The thesis is divided into two parts. We perform Monte Carlo simulations of a $\lambda \phi^4$ theory on a 3-dimensional Euclidean space. We identify the phase diagram of this model. In addition to the usual disordered and uniform ordered phases we find a third phase of non-uniform ordering. This indicates the existence of the phenomenon called UV-IR mixing in the strong coupling regime. Second we present a geometrical analysis of the scalar field theory on a 4-dimensional fuzzy sphere, S4_F. Nevertheless a fuzzy version of S4 cannot be achieved by quantisation of the classical space. The problem is circumvented by defining a scalar theory on a larger space, CP3 which is 6-dimensional. It includes degrees of freedom related to S^4 plus others beyond S4. Those extra degrees of freedom are dynamically suppressed through a probabilistic method. The analysis of the geometrical structures allows us to interpret this procedure as a Kaluza-Klein reduction of CP3 to S4.