# Spectral and Scattering Theory for Translation Invariant Models in Quantum Field Theory

By Morten Grud Rasmussen
PhD Dissertations
June 2010
Abstract:

This thesis is concerned with a large class of massive translation invariant models in quantum field theory, including the Nelson model and the Fröhlich polaron. The models in the class describe a matter particle, e.g. a nucleon or an electron, linearly coupled to a second quantised massive scalar field e.g. describing mesons or phonons.

The models are given by three inputs: $\Omega$ - the dispersion relation for the matter particle, $\omega$ - the dispersion relation for the field particle, and $v$ - the (UV cut-off) coupling function. The assumptions imposed on $\Omega$, $\omega$ and $v$ are rather weak and are satisfied by the physically relevant choices.

The translation invariance implies that the Hamiltonian may be decomposed into a direct integral over the space of total momentum where the fixed momentum fiber Hamiltonians are given by $H(\xi)=\mathrm{d}\Gamma(\omega)+\Omega(\xi-\mathrm{d}\Gamma(k))+\Phi(v)$, where $\xi$ denotes total momentum and $\Phi(v)$ is the Segal field operator. The fiber Hamiltonians are known to satisfy an HVZ theorem, implying that the essential spectrum equals a translation of the positive half-axis.

For the full model, the results include a Mourre estimate and absence of singular continuous spectrum in the region of the energy-momentum spectrum lying between the bottom of the essential energy-momentum spectrum and either the two-body threshold, if there are no exited isolated mass shells, or the one-body threshold pertaining to the first exited isolated mass shell, if it exists.

For the model restricted to the vacuum and one-particle sectors, the absence of singular continuous spectrum is proven to hold globally and scattering theory of the model is studied using time-dependent methods, of which the main result is asymptotic completeness.