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

Wojciech Dybalski
January 20, 2009
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.