DeWitt-Schwinger Renormalization and Vacuum Polarization in d Dimensions

Details DeWitt-Schwinger Renormalization and Vacuum Polarization in d Dimensions DeWitt-Schwinger Renormalization and Vacuum Polarization in d Dimensions

2008-11-24

arxiv.org/abs/0811.3962v2

Phys.Rev.D80:064017,2009

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

21 pages, 2 tables, 3 figures. References added, rewritten to clarify some points, corrections performed, our claim in the fir

Scientific paper

10.1103/PhysRevD.80.064017

Calculation of the vacuum polarization, $<\phi^2(x)>$, and expectation value of the stress tensor, $$, has seen a recent resurgence, notably for black hole spacetimes. To date, most calculations of this type have been done only in four dimensions. Extending these calculations to $d$ dimensions includes $d$-dimensional renormalization. Typically, the renormalizing terms are found from Christensen's covariant point splitting method for the DeWitt-Schwinger expansion. However, some manipulation is required to put the correct terms into a form that is compatible with problems of the vacuum polarization type. Here, after a review of the current state of affairs for $<\phi^2(x)>$ and $$ 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 $$ 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.

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