Singular perturbations of a free quantum field Hamiltonian

TL;DR

This paper investigates solutions to the eigenstate equation of a free quantum field Hamiltonian, demonstrating the existence of finite-norm, negative-energy eigenstates for scalar and vector fields, with implications for renormalized states in quantum field theory.

Contribution

It introduces a novel approach to construct admissible eigenstates of free quantum fields using singular perturbation theory, applicable to models with asymptotic freedom.

Findings

Existence of finite-norm, negative-energy eigenstates for scalar and vector fields.

Functional eigenstates are analogous to infinite-dimensional eigenfunctions of differential operators.

Potential application in constructing renormalized states in quantum field models.

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.