Perturbative renormalization of field theories breaking translation invariance

Overview & Synopsis

The aim of this thesis is to establish a rigorous proof of perturbative renormalization of massive scalar field theories that break translation invariance. First, we study an implicit breaking of translation invariance, by studying the scalar field model in R4 regularized by a lattice. This theory is invariant under the Euclidean symmetries, which is not the case of its lattice regularized version. We establish that the continuum limit (i.e. when the lattice parameter is sent to 0) is invariant under translations and rotations. The central part of this thesis concerns the perturbative analysis of the renormalizability of boundary scalar field theories. We study a toy model of such systems, which is the semi-infinite scalar field model. A field theoretical approach was performed extensively in the litterature up to two loop orders, but a systematic analysis to all loop orders remained open. In this thesis, we give a rigorous proof of renormalizability of this model to all loop orders. Then, we prove that for a particular choice of renormalization conditions the counter-terms are independent of the position in the space. Furthermore, we prove that the counter-terms are of the same structure of the terms appearing in the original Hamiltonian. For Dirichlet boundary conditions, we find that the counter-terms are the same as those that renormalize the translationally invariant theory (i.e. bulk counter-terms). However, the Robin and Neumann boundary conditions require in addition to the usual bulk counter-terms some additional surface counter-terms which are supported on the surface.