Transport Equation Approach to Calculations of Hadamard Green functions and non-coincident DeWitt coefficients

TL;DR

This paper develops transport equations for key bi-tensors in curved spacetime, enabling high-order covariant expansions and efficient numerical calculations relevant to quantum field theory and gravitational physics.

Contribution

It introduces a transport equation framework for bi-tensors like the world-function and Van Vleck determinant, with implementations for high-order expansions and numerical integration.

Findings

High-order covariant series expansions computed efficiently

Numerical integration scheme for transport equations demonstrated in specific spacetimes

Mathematica implementation achieves rapid calculations of complex bi-tensor limits

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.