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  Transport equation approach to calculations of Hadamard Green functions and non-coincident DeWitt coefficients

Ottewill, A. C., & Wardell, B. (2011). Transport equation approach to calculations of Hadamard Green functions and non-coincident DeWitt coefficients. Physical Review D, 84(10): 104039. doi:10.1103/PhysRevD.84.104039.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-000F-013B-6 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-000F-013C-4
Genre: Journal Article
Other : A Transport Equation Approach to Calculations of Hadamard Green functions and non-coincident DeWitt coefficients (Preprint)

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Ottewill, Adrian C., Author
Wardell, Barry1, Author              
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1Astrophysical Relativity, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24013              

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Free keywords: General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
 Abstract: Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bi-tensors, including derivatives of the world-function, \sigma(x,x'), the square root of the Van Vleck determinant, \Delta^{1/2}(x,x'), and the tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These bi-tensors are central to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity. Their transport equations may be used either in a semi-recursive approach to determining their covariant Taylor series expansions, or as the basis of numerical calculations. To illustrate the power of the semi-recursive approach, we present an implementation in \textsl{Mathematica} which computes very high order covariant series expansions of these objects. Using this code, a moderate laptop can, for example, calculate the coincidence limit a_7(x,x) and V(x,x') to order (\sigma^a)^{20} in a matter of minutes. Results may be output in either a compact notation or in xTensor form. In a second application of the approach, we present a scheme for numerically integrating the transport equations as a system of coupled ordinary differential equations. As an example application of the scheme, we integrate along null geodesics to solve for V(x,x') in Nariai and Schwarzschild spacetimes.

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 Dates: 2009-05-292011-11-252011
 Publication Status: Published in print
 Pages: 32 pages, 5 figures. Final published version
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 Table of Contents: -
 Rev. Method: -
 Identifiers: arXiv: 0906.0005
DOI: 10.1103/PhysRevD.84.104039
URI: http://arxiv.org/abs/0906.0005
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Title: Physical Review D
  Other : Phys. Rev. D.
Source Genre: Journal
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Publ. Info: Lancaster, Pa. : Published for the American Physical Society by the American Institute of Physics
Pages: - Volume / Issue: 84 (10) Sequence Number: 104039 Start / End Page: - Identifier: ISSN: 0556-2821
CoNE: /journals/resource/111088197762258