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

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Cited by in corpus (15)

• Hadamard renormalized scalar field theory on anti-de Sitter space-time
• A Mode-Sum Prescription for Vacuum Polarization in Odd Dimensions
• A Mode-Sum Prescription for Vacuum Polarization in Even Dimensions
• Self-energy of a scalar charge near higher-dimensional black holes
• Stress-energy tensor of the quantized massive fields in Schwarzschild-Tangherlini spacetimes. The back reaction
• Twist decomposition of proton structure from BFKL and BK amplitudes
• Vacuum polarization on the brane
• Vacuum polarization of the quantized massive scalar field in Reissner-Nordström spacetime
• Semiclassical lukewarm black holes
• Stress-energy tensor of quantized massive fields in static wormhole spacetimes
• Inside the Schwarzschild-Tangherlini black holes
• Vacuum polarization for varying quantum scalar field parameters in Schwarzschild-anti-de Sitter spacetime
• Black Hole Quantum Vacuum Polarization in Higher Dimensions
• Quantum vacuum polarization around a Reissner-Nordström black hole in five dimensions
• Classical and Quantum Thermodynamic Systems in Curved Spacetime