Degenerate perturbation theory for models of quantum field theory with symmetries

TL;DR

This paper proves that eigenvalues of certain quantum field theory models with symmetries persist under coupling and depend analytically on parameters, using operator renormalization and representation theory.

Contribution

It extends degenerate perturbation theory to models with symmetry-protected degeneracies in quantum field theory using operator theoretic renormalization.

Findings

Eigenvalues persist under nonzero coupling.

Eigenvalues depend analytically on external parameters.

Results apply to ground and resonance states.

Abstract

We consider Hamiltonians of models describing non-relativistic quantum mechanical matter coupled to a relativistic field of bosons. If the free Hamiltonian has an eigenvalue, we show that this eigenvalue persists also for nonzero coupling. The eigenvalue of the free Hamiltonian may be degenerate provided there exists a symmetry group acting irreducibly on the eigenspace. Furthermore, if the Hamiltonian depends analytically on external parameters then so does the eigenvalue and eigenvector. Our result applies to the ground state as well as resonance states. For our results we assume a mild infrared condition. The proof is based on operator theoretic renormalization. It generalizes the method used in [15] to non-degenerate situations, where the degeneracy is protected by a symmetry group, and utilizes Schur's lemma from representation theory.