DeWitt-Schwinger Renormalization and Vacuum Polarization in d Dimensions

Abstract

Calculation of the vacuum polarization, $<ϕ^2(x)>$, and expectation value of the stress tensor, $ $, has seen a recent resurgence, notably for black hole spacetimes. To date, most calculations of this type have been done only in four dimensions. Extending these calculations to $d$ dimensions includes $d$-dimensional renormalization. Typically, the renormalizing terms are found from Christensen's covariant point splitting method for the DeWitt-Schwinger expansion. However, some manipulation is required to put the correct terms into a form that is compatible with problems of the vacuum polarization type. Here, after a review of the current state of affairs for $<ϕ^2(x)>$ and $ $ calculations and a thorough introduction to the method of calculating $<ϕ^2(x)>$, a compact expression for the DeWitt-Schwinger renormalization terms suitable for use in even-dimensional spacetimes is derived. This formula should be useful for calculations of $<ϕ^2(x)>$ and $ $ in even dimensions, and the renormalization terms are shown explicitly for four and six dimensions. Furthermore, use of the finite terms of the DeWitt-Schwinger expansion as an approximation to $<ϕ^2(x)>$ for certain spacetimes is discussed, with application to four and five dimensions.