Topics in quantum gravity and quantum field theory

Thesis

In this thesis, we analyse three different quantum systems via operator and path integral techniques: 'two-dimensional Causal-Dynamical-Triangulations (CDT) coupled to hard-dimers', 'scalar solitons' and `Quantum Random Walks (QRW) on the Cayley tree'.

In our first project, we extend previous work on CDT coupled to hard dimers -- which is a discretized model of two-dimensional quantum gravity coupled to matter -- and solve this model exactly with all dimer types present subject to a single restriction. We find that, depending on the dimer fugacities, there are, in addition to the usual gravity phase of CDT, two tri-critical and two dense-dimer phases. We establish the properties of these phases, computing their cylinder and disk amplitudes, their scaling limits and their associated continuum Hamiltonians.

In our second project, we investigate scalar solitons in the framework of quantum field theory. We construct general soliton creation operators and compare these with like operators found by Mandelstam in the sine-Gordon model. We find evidence for the fact that the sine-Gordon soliton is the Thirring fermion only at coupling β2 = 4π. We then go on to compute the first quantum corrections to the radius of the kink in the ϕ4 model in two dimensions.

In our third project, we analyse QRWs on the Cayley tree via the generating function method and derive a set of polynomial equations which determine this model exactly. We establish properties of the spectrum of the time-evolution operator U and solve the `perturbed' eigenvalue equation in U.

Authors: Xavier, PD

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: quantum field theory; statistical models; quantum gravity
• Subjects: Quantum field theory; Quantum gravity