Spectral Theory of Automorphism Groups and Particle Structures in Quantum Field Theory

TL;DR

This thesis develops a refined spectral theory for automorphism groups in quantum field theory, introducing a new decomposition of local observables that clarifies the existence and structure of particles and infra-particles.

Contribution

It introduces a novel spectral decomposition including a 'point-continuous' subspace and establishes conditions for particle existence based on this framework.

Findings

A new spectral decomposition with a 'point-continuous' subspace.

The 'point-continuous' subspace is trivial under regularity conditions.

Existence of particles linked to stress-energy tensor presence.

Abstract

This Thesis presents some physically motivated criteria for the existence of particles and infra-particles in a given quantum field theory. It is based on a refined spectral theory of automorphism groups describing the energy-momentum transfer of local observables. In particular, a novel decomposition of the algebra of local observables into spectral subspaces is constructed. Apart from the counterparts of the pure-point and absolutely continuous subspaces, familiar from the spectral theory of operators, there appears a new 'point-continuous' subspace. It belongs to the singular-continuous part of the decomposition, but is finite-dimensional in a large class of models; its dimension carries information about the infrared structure of a theory. It is shown that this point-continuous subspace is trivial in all theories complying with a regularity condition proposed in this work. Moreover, this condition entails the existence of particles if the theory admits a stress-energy tensor. The uniqueness of the decomposition of the algebra of observables into the pure-point and continuous subspace is established by proving an ergodic theorem for automorphism groups. The proof is based on physically motivated phase space conditions which have a number of interesting consequences pertaining to the vacuum structure such as the convergence of physical states to a unique vacuum under large timelike translations.