Lecture Notes in Loop Quantum Gravity. LN2: Cauchy problems and pre-quantum states

TL;DR

This paper analyzes the structure of covariant equations in relativistic models, focusing on the principal symbol of PDE systems, and introduces concepts like pre-quantum configurations and Cauchy bubbles to establish well-posed evolution problems in general relativity.

Contribution

It provides a general analysis of the principal symbol in quasi-linear PDE systems and introduces pre-quantum configurations and Cauchy bubbles for setting up evolution problems in relativistic models.

Findings

Analysis of covariant equations and their principal symbols.

Introduction of pre-quantum configurations and Cauchy bubbles.

Application to standard General Relativity.

Abstract

We discuss the structure of covariant equations, relating analytical properties of solutions to algebraic properties of the corresponding differential operator, specifically of its principal symbol. The principal symbol and its globality is discussed for a general quasi-linear PDE system, regardless the algebraic structure the configuration space can have. We also discuss how the typical relativistic model can be under-determined and over-determined at the same time as well as how one can define out of it a well-posed Cauchy problem. This issue leads us to pre-quantum configurations and Cauchy bubbles as the way to set up evolution problems in a compact region of spacetime, taking into account that relativistic models are defined on bare manifolds. The typical application we shall sketch is standard GR.