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.