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

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.

Keywords:

*none*