The strong-coupling expansion and the ultra-local approximation in field theory

Abstract

We first discuss the strong-coupling expansion in ( λ ϕ 4 ) d theory and quantum electrodynamics in a d-dimensional Euclidean space. In a formal representation for the Schwinger functional, we treat the Gaussian part of the action as a perturbation with respect to the remaining terms. In this way, we develop a perturbative expansion around the ultra-local model, where fields defined at different points of Euclidean space are decoupled. We examine the singularities of the strong-coupling perturbative expansion, analysing the analytic structure of the zero-dimensional generating functions in the coupling constant complex planes. We also discuss the ultra-local generating functional in a non-polynomial model in field theory, defined by the following interaction Lagrangian density: L II ( g 1 , g 2 ; ϕ ) = g 1 ( cosh ( g 2 ϕ ( x ) ) - 1 ) . Finally, we use the strong-coupling perturbative expansion to compute the renormalized vacuum energy of the strongly coupled ( λ ϕ 4 ) d theory, assuming that the scalar field is defined in a region bounded by two parallel hyperplanes, where we are imposing Dirichlet–Dirichlet boundary conditions.