Spectral analysis of a massless charged scalar field with cutoffs

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.

The first order expansion of a ground state energy of the $ϕ^4$ model with cutoffs

“…For the existence of the ground states of singular perturbation models, refer to e.g. [22,23]. In addition, the asymptotic completeness of the one-dimensional φ 4 model with a spatial cutoff was shown in [8].…”

Time-ordered exponential on the complex plane and Gell-Mann—Low formula as a mathematical theorem

“…From the late 1990s to the 2000s, several important methods to prove the existence of ground states were developed in the study of a quantum system consisting of quantum particles and a Bose field (for example, see [3,4,10,11,15]). These methods have been improved by many authors to be also applicable to systems of interacting quantum fields [1,5,6,7,16,17,18,13]. Once field operators and the ground state are given, we can define the n-point correlation function Ω, T φ (1) (x 1 ) · · · φ (n) (x n ) Ω non-perturbatively.…”