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

Adrian C. Ottewill
Barry Wardell
Journal / Anthology

Physical Review D
Year: 2012
Volume: 84
Issue: 10

Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bitensors, including derivatives of the world function, x; x0, the square root of the Van Vleck determinant, 1=2x; x0, and the tail term, Vx; x0, appearing in the Hadamard form of the Green function. These bitensors 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 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 a7x; x and Vx; x0 to order a20 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 Vx; x0 in Nariai and Schwarzschild spacetimes.