Singular perturbations of a free quantum field Hamiltonian

Abstract

We study solutions of the functional eigenstate equation of a free quantum field Hamiltonian. Admissible solutions are to have a finite norm and a finite eigenvalue w.r.t. the norm and eigenvalue of the ground state of the free theory. We show that in the simple cases of a scalar field and of a vector field in the Coulomb gauge the admissible eigenstates exist and possess negative energy. The functionals can be treated as infinite-dimensional counterparts of the eigenfunctions of the theory of singular perturbations of differential operators, and can be deployed for construction of the renormalized states of models with asymptotic freedom.