Causality is all you need: a stochastic perspective on 2D quantum gravity

Thesis

This thesis develops a causality–first perspective on two-dimensional quantum gravity. Working within Causal Dynamical Triangulations (CDT) and closely related solvable models, we combine stochastic methods, matrix models and renormalisation group techniques to investigate how matter couples to causal random geometry.

A central toy model of matter in this thesis is the Ising model and its coupling to CDT. In Chapter 4 we consider the conformal dimensions of the Ising conformal field theory operators on a CDT graph. We construct topological defects and Dehn-twist operators directly on causal triangulations and follow them to the continuum, showing that (for the Ising model and more generally in the settings considered) the critical exponents are those of the flat lattice—i.e. there is no Knizhnik-Polyakov-Zamolodchikov (KPZ) dressing on CDT. The argument relies on the continuity of the causal stochastic process and the fusion category structure of defects.

In Chapter 5, we formulate and analyse a matrix model for Ising spins on CDT using the Functional Renormalisation Group. Within a single-trace truncation we derive beta functions, identify Gaussian and pure-CDT fixed points, and find non-trivial fixed points with three relevant directions—matching the three primary operators of the Ising CFT.

In the final chapter, the stochastic approach once again comes into its own where we use it to show how hyperbolic geometry emerges from CDT in two different ways. The first builds on the previous work in [1], making it rigorous and finding a new stochastic process which incorporates the average hyperbolic profile as well as the fluctuations around it. The second is a consequence of a stochastic time-reparameterisation known as a Lamperti time-change. This change of time transforms the process that describes the spatial length of CDT from a Bessel type process to an exponential Brownian process. The emergent constant negative curvature is then direct.

It is clear that the stochastic approach to analysing continuum CDT is very powerful and has the potential to solve many of the remaining questions in CDT by leveraging the vast machinery of stochastic calculus. However, underlying the ability to describe CDT as a stochastic process in the first place is the central tenet of causality. It appears to tame the two-dimensional gravitational path integral; you might say that, “causality is all you need”.

Authors: Barouki, R

• DOI: 10.5287/ora-z6zyy0x2m
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: stochastic methods; quantum gravity; causal dynamical triangulations; two-dimensional quantum gravity
• Subjects: Quantum Gravity; Theoretical Physics