DeWitt-Schwinger Renormalization and Vacuum Polarization in d Dimensions

Abstract

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

21 pages, 2 tables, 3 figures. References added, rewritten to clarify some points, corrections performed, our claim in the first version that there is an error in Anderson's calculations is incorrect

References in corpus (10)

• Stress Tensor from the Trace Anomaly in Reissner-Nordstrom Spacetimes
• Enhanced Black Hole Horizon Fluctuations
• Irreducible forms for the metric variations of the action terms of sixth-order gravity and approximated stress-energy tensor
• Vacuum polarization for lukewarm black holes
• Vacuum polarization in asymptotically anti-de Sitter black hole geometries
• Scalar Casimir Effect on a D-dimensional Einstein Static Universe
• Some general properties of the renormalized stress-energy tensor for static quantum states on (n+1)-dimensional spherically symmetric black holes
• Vacuum polarization of massive spinor fields in static black-string backgrounds
• Enhanced Geometry Fluctuations in Minkowski and Black Hole Spacetimes
• Quantum-corrected ultraextremal horizons and validity of WKB in massless limit