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. There 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.
Highlights
We have recently found the same thing for an external scalar potential [7]
The first main thrust of the present paper is to develop the method for a theory with an external scalar potential, noting its essential consistency with the point splitting conducted in [7]
In the previous paper [7], divergences were regularized by point splitting, and a “renormalized”
Summary
Quantum field theories are notoriously afflicted by infinities, known as divergences or as cutoff dependences. The present paper belongs to a different tradition, that where the quantized field satisfies a linear equation of motion against a nontrivial background, which is treated classically. This “external condition” may be a reflecting boundary (as in the Casimir effect), a space-time geometry (as in cosmology), a strong applied electromagnetic field The identification and removal of divergences usually takes place in two steps, regularization and renormalization The first of these terms refers to modification of the theory, or of formal expressions arising within the theory, so that they are finite, but somewhat arbitrary and not physically correct.
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