Spectral analysis of a massless charged scalar field with spacial cut-off

TL;DR

This paper analyzes a massless charged scalar quantum field with a spatial cut-off, proving key spectral properties and the existence of a ground state under certain conditions.

Contribution

It introduces a spatial cut-off to define the Hamiltonian as a self-adjoint operator and establishes its spectral properties and charge decomposition.

Findings

Hamiltonian strongly commutes with total charge operator

Decomposition of state space into fixed charge sectors

Existence of a ground state for arbitrary coupling constants

Abstract

The quantum system of a massless charged scalar field with a self-interaction is investigated. By introducing a spacial cut-off function, the Hamiltonian of the system is realized as a linear operator on a boson Fock space. It is proven that the Hamiltonian strongly commutes with the total charge operator. This fact implies that the state space of the charged scalar field is decomposed into the infinite direct sum of fixed total charge spaces. Moreover, under certain conditions, the Hamiltonian is bounded below, self-adjoint and has a ground ground state for an arbitrarily coupling constant. A relation between the total charge of the ground state and a number operator bound is also revealed.