arXiv subjects

Gabriel Schmid

Publications and source records attributed to Gabriel Schmid.

8 recordsLinked to original sources

arXiv2026

On the Quantisation of Linear Gauge Theories on Lorentzian Manifolds: Maxwell's Theory via Complete Gauge Fixing

This thesis is devoted to the study of hyperbolic differential operators on globally hyperbolic manifolds, linear gauge theories and their quantisation. In the first part, we treat the Cauchy problem for symmetric hyperbolic systems and normally hyperbolic operators on globally hyperbolic manifolds from first principles. Although hyperbolic equations are usually studied with local interactions, there are strong motivations from several areas of mathematical physics to consider also nonlocal interactions. As an intermezzo, we therefore take a small deviation from the classical local theory and prove well-posedness of the Cauchy problem for symmetric hyperbolic systems coupled to a broad class of nonlocal potentials. The second part presents a detailed exposition of linear gauge theories in globally hyperbolic spacetimes. Linear gauge theories are yet another deviation from the concept of hyperbolicity: the corresponding equations of motion are generically non-hyperbolic; however, can always be reduced to a constrained hyperbolic dynamics once an appropriate gauge fixing procedure has been applied. We give a thorough analysis of their Cauchy problem and classical phase space, complemented by a detailed discussion of many examples of physical interest, and discuss their quantisation following the algebraic approach to quantum field theory. The final chapter is devoted to the quantisation of Maxwell's theory on globally hyperbolic spacetimes, with the goal of proving the existence of Hadamard states. The novelty of our approach lies in a new gauge-fixing procedure at the level of initial data, which allows us to suppress the unphysical degrees of freedom. This gauge is achieved by means of a new Hodge decomposition for differential k-forms in Sobolev spaces on complete Riemannian manifolds, while states are constructed using tools from pseudodifferential calculus.

math-ph↗

arXiv2026

The Semiclassical Einstein-Klein-Gordon System: Asymptotic Analysis of Minkowski Spacetime

We establish the linear instability of the semiclassical Einstein-Klein-Gordon system linearised about the Minkowski vacuum spacetime. The proof relies on formulating a forcing problem for both metric and state perturbations within the space of past-compact sections. This geometric framework admits a unique tensor decomposition which, in conjunction with the quantum M{\o}ller operator, enables the decoupling of the linearised system into two distinct Cauchy problems. Consequently, the metric perturbations are shown to be governed by a higher-order, nonlocal hyperbolic partial differential equation. By relegating the nonlocal contributions to subleading order, we establish the well-posedness of this forcing problem. Furthermore, we provide a rigorous asymptotic analysis for physically admissible choices of the renormalisation constants. We prove that the system exhibits a late-time linear instability: the metric perturbations grow exponentially, bounded strictly by a universal scale H, thereby indicating a quantum backreaction-driven transition toward a de Sitter cosmological spacetime. Provided the parameters governing the system are restricted to a physically relevant regime, this universal scale is compatible with the measured expansion of our universe.

math-ph↗

arXiv2025

Pseudodifferential Weyl calculus on vector bundles

We develop a geometric framework for Weyl quantization on pseudo-Riemannian manifolds, in which pseudodifferential operators act on sections of vector bundles equipped with pseudo-Hermitian metrics and compatible connections. We construct the associated star product and compute its semiclassical expansion up to third order in the semiclassical parameter. A central feature of our approach is a correspondence, modulo smoothing remainders, between formally self-adjoint symbols and formally self-adjoint operators, extending known results from flat space to curved geometries. In addition, we analyze the Moyal equation satisfied by the Wigner function in this setting and provide explicit computations of Weyl symbols for several physically significant operators, including the Dirac, Maxwell, linearized Yang-Mills, and linearized Einstein operators. Our results lay the foundation for future developments in quantum field theory on curved spacetimes, semiclassical analysis, and chiral kinetic theory.

math-ph↗

arXiv2025

The Cauchy Problem for Symmetric Hyperbolic Systems with Nonlocal Potentials

In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly bounded in time, we establish well-posedness of the Cauchy problem on a time strip, proving existence, uniqueness, and regularity of solutions. If the potential is not retarded but has only short time range, we show that strong solutions still exist, under the additional assumptions that the uniform bound in time is sufficiently small compared to the range in time and that its kernel decays sufficiently fast in time with respect to the zero-order terms of the system. Furthermore, we present a counterexample demonstrating that when the uniform bound is too large compared to the time range, solutions may fail to exist. As an application, we discuss Maxwell's equations in linear dispersive media on ultrastatic spacetimes, as well as the Dirac equation with nonlocal potential naturally arising in the theory of causal fermion systems. Our paper aims to represent the starting point for a rigorous study for the Cauchy problem for the semiclassical Einstein equations.

math.AP↗

arXiv2024

On boundary conditions for linearised Einstein's equations

We investigate the properties of a fairly large class of boundary conditions for the linearised Einstein equations in the Riemannian setting, ones which generalise the linearised counterpart of boundary conditions proposed by Anderson. Through the prism of the quest to quantise gravitational waves in curved spacetimes, we study their properties from the point of view of ellipticity, gauge invariance, and the existence of a spectral gap.

math.AP↗

arXiv2024

The Quantization of Maxwell Theory in the Cauchy Radiation Gauge: Hodge Decomposition and Hadamard States

The aim of this paper is to prove the existence of Hadamard states for the Maxwell equations on any globally hyperbolic spacetime. This will be achieved by introducing a new gauge fixing condition, the Cauchy radiation gauge, that will allow to suppress all the unphysical degrees of freedom. The key ingredient for achieving this gauge is a new Hodge decomposition for differential k-forms in Sobolev spaces on complete (possibly non-compact) Riemannian manifolds.

math.AP↗

arXiv2022

Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model

We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We analyse the transition amplitudes for the simplest boundary topology, the 2-sphere, and prove that they factorize into a sum entirely given by the combinatorics of the boundary spin network state and that the leading order is given by graphs representing the closed 3-ball in the large N limit. This is the first step towards a more detailed study of the holographic nature of coloured Boulatov-type GFT models for topological field theories and quantum gravity.

hep-th↗

arXiv2022

On 3-Dimensional Quantum Gravity and Quasi-Local Holography in Spin Foam Models and Group Field Theory

This thesis is devoted to the study of 3-dimensional quantum gravity as a spin foam model and group field theory. In the first part of this thesis, we review some general physical and mathematical aspects of 3-dimensional gravity, focusing on its topological nature. Afterwards, we review some important aspects of the Ponzano-Regge spin foam model for 3-dimensional Riemannian quantum gravity and explain in some details how it is related to the discretized path integral of general relativity in its first-order formulation. Furthermore, we discuss briefly some related spin foam models and review the notion of spin network states in order to properly define transition amplitudes of these models. The main results of this thesis are contained in the second part. We start by reviewing the Boulatov group field theory and explain how it is related to the Ponzano-Regge model and some advantages of introducing colouring. Afterwards, we give a very detailed review of the topology of coloured graphs with non-empty boundary and review techniques, which are devolved in crystallization theory, a branch of geometric topology. In the last part of this chapter, we apply these techniques in order to define suitable boundary observables and transition amplitudes of this model and in order to set up a formalism for dealing with transition amplitudes in the coloured Boulatov model in a more systematic way by writing them as topological expansions. We also apply these techniques to the simplest possible boundary state representing a 2-sphere. Last but not least, we review some results regarding quasi-local holography in the Ponzano-Regge model, construct some explicit examples of coloured graphs representing manifolds with torus boundary and discuss the transition amplitude of some fixed boundary graph representing a 2-torus.

gr-qc↗