The analytic approach to recursion relations

Fulling, S. A.

doi:10.1016/s0747-7171(08)80008-9

Cited by 9 publications

(4 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Christensen ( 1998 ) {MathTensor} reported work to support DeWitt’s approach to quantum field theory in curved spacetime, which required calculation to high orders of the behaviour of the geodesic interval as . This was also addressed by Rodionov and Taranov ( 1987 , 1988a ) {Reduce} and by Fulling ( 1991 ) who used bespoke C programs.…”

Section: Applicationsmentioning

confidence: 94%

“…Done incautiously, one would end up evaluating twice and so on (cf. Fulling 1991 ). So in a “lazy evaluation” strategy, where a value is computed only when it is required, and otherwise only the way to compute it is stored (this is used in calculating geodesics in GRWorkbench and in the power series packages of Reduce and Sheep), storing those values that have been computed may also improve efficiency.…”

Section: The Nature Of Computer Algebra Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Computer algebra in gravity research

MacCallum

2018

Living Rev Relativ

View full text Add to dashboard Cite

The complicated nature of calculations in general relativity was one of the driving forces in the early development of computer algebra (CA). CA has become widely used in gravity research (GR) and its use can be expected to grow further. Here the general nature of computer algebra is discussed, along with some aspects of CA system design; features particular to GR’s requirements are considered; information on packages for CA in GR is provided, both for those packages currently available and for their predecessors; and applications of CA in GR are outlined.

show abstract

Section: Applicationsmentioning

confidence: 94%

Section: The Nature Of Computer Algebra Systemsmentioning

confidence: 99%

Computer algebra in gravity research

MacCallum

2018

Living Rev Relativ

View full text Add to dashboard Cite

show abstract

“…The goal is to find covariant series expressions for the Hadamard and DeWitt coefficients. Several methods have been previously applied for doing such calculations, both by hand and using computer algebra [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. However, this effort has been focused primarily on the calculation of the diagonal coefficients.…”

Section: Semi-recursive Approach To Covariant Expansionsmentioning

confidence: 99%

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

Ottewill

Wardell

2011

Phys. Rev. D

View full text Add to dashboard Cite

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, σ (x, x ), the square root of the Van Vleck determinant, ∆ 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 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 [a7(x, x)] and V (x, x ) to order (σ 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. * adrian.ottewill@ucd.ie † barry.wardell@aei.mpg.de 1 The Hadamard and DeWitt coefficients also appear in the literature under several other guises. They may be called DeWitt, Gilkey, Minakshisundaram, Schwinger or Seeley coefficients, or any combination thereof (yielding acronyms such as DWSC, DWSG and HDMS).In the coincidence limit, it has been proposed that they be called Hadamard-Minakshisundaram-DeWitt (HaMiDeW) [8] coefficients. For the remainder of this paper, we will refer to them as either DeWitt (for the coefficients a k A B ) or Hadamard (for the coefficients Vr A B ) coefficients.arXiv:0906.0005v3 [gr-qc]

show abstract

“…Although this method has a more restricted domain of applicability than the first, it is also more efficient. It has recently been extended by Schimming [53] and Rodionov and Taranov [52]; see also Fulling [51]. Yet another method has been given by Drager [16].…”

Section: Linearizationmentioning

confidence: 99%