Abstract

Is it possible that space is not really what we think it is? And what if changing our perspective on this concept could help us understand its quantum nature at the infinitely small scale?

Modern physics has taught us that, at microscopic scales, nature is governed by the laws of quantum mechanics, where phenomena are described in probabilistic terms, whilst, at macroscopic scales, general relativity describes space and time as a dynamic geometric structure, where matter and energy warp spacetime. Bringing these two perspectives together is one of the great challenges of theoretical physics.

This project contributes to this research from a mathematical perspective, developing new tools to study the quantisation of physical theories in non-commutative geometries. Non-commutative geometry starts from the idea of changing our perspective on space: instead of describing it through the points that make it up, it encodes its geometry through algebraic objects and operators that capture its essential properties. It is a radical shift: space is no longer necessarily the starting point, but can emerge from the algebraic structures that encode its geometry.

A central component of the project is the Batalin–Vilkovisky (BV) formalism, a mathematical tool developed to tackle one of the most challenging problems in quantum field theory: how to quantise gauge theories, that is, theories with symmetries. My aim is to extend this type of quantisation to non-commutative geometry, by studying theories such as Yang–Mills and Chern–Simons, first in their formulation as matrix models, and then extending these results to genuinely infinite-dimensional systems.

By analysing which new mathematical structures emerge when BV quantisation and non-commutative geometry intersect, the project aims to contribute to the development of the theoretical tools needed to explore new models of fundamental physics and, more generally, to forge new connections between mathematics and physics.