Renormalization for a Scalar Field in an External Scalar Potential

Abstract

The Pauli--Villars regularization procedure confirms and sharpens the conclusions reached previously by covariant point splitting. The divergences in the stress tensor of a quantized scalar field interacting with a static scalar potential are isolated into a three-parameter local, covariant functional of the background potential. These divergences can be naturally absorbed into coupling constants of the potential, regarded as a dynamical object in its own right; here this is demonstrated in detail for two different models of the field-potential coupling. here is a residual dependence on the logarithm of the potential, reminiscent of the renormalization group in fully interacting quantum field theories; these terms are finite but numerically dependent on an arbitrary mass or length parameter, which is purely a matter of convention. This work is one step in a program to elucidate boundary divergences by replacing a sharp boundary by a steeply rising smooth potential.

arXiv comment19 pages; invited issue of Symmetry on Casimir Interactions

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• Regularization versus renormalization: Why are Casimir energy differences so often finite?
• Vacuum energy density and pressure inside a soft wall
• Casimir energy due to inhomogeneous thin plates
• Scarf for Lifshitz
• Casimir-Polder forces in inhomogeneous backgrounds