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

##### Fulltext (public)

0906.0005

(Preprint), 2MB

PRD84_104039.pdf

(Any fulltext), 3MB

##### Supplementary Material (public)

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##### Citation

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.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-013B-6

##### 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.