DeWitt-Schwinger Renormalization and Vacuum Polarization in d Dimensions

Calculation of the vacuum polarization, $<\phi^2(x)>$, and expectation value of the stress tensor, $<T_{\mu\nu}(x)>$, 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 $<\phi^2(x)>$ and $<T_{\mu\nu}(x)>$ calculations and a thorough introduction to the method of calculating $<\phi^2(x)>$, 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 $<\phi^2(x)>$ and $<T_{\mu\nu}(x)>$ 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 $<\phi^2(x)>$ for certain spacetimes is discussed, with application to four and five dimensions.