Diagrams for heat kernel expansions

Abstract

A diagramatic heat kernel expansion technique is presented. The method is especially well suited to the small derivative expansion of the heat kernel, but it can also be used to reproduce the results obtained by the approach known as covariant perturbation theory. The new technique gives an expansion for the heat kernel at coincident points. It can also be used to obtain the derivative of the heat kernel, and this is useful for evaluating the expectation values of the stress-energy tensor.Pacs numbers: 03.70.+k, 98.80.Cq

The method of covariant symbols in curved space-time

“…A first application would be to compute the heat kernel, not using the standard Seeley-DeWitt expansion, which orders operator by their dimension, but the covariant derivative expansion. Explicit calculations along this line exist only for the minimal case (i.e., Klein-Gordon theories with a trivial gauge sector) [27,53]. The non-minimal, but flat space-time, calculation of [19] can be extended to the curved case and results for traced and untraced coefficients will be presented elsewhere.…”

Quantum corrections to the kinetic term in the Randall–Sundrum model

“…As in reference [14], we arrange the time ordered exponential in such a way that we can use Wick's theorem. Defining 15) with Z(t) = (4πt) −2 n e −ω 2 n t , gives…”

“…The calculation adapts a diagramatic method for obtaining derivative expansions of a heat kernel [14,15], which is closely related to the 'covariant perturbation theory' first devised by Vilkovisky [16,17,18,19]. In this case the fine tunings are protected at one loop order.…”

Derivative expansion of the heat kernel in curved space

“…Nevertheless the series expansion in powers of the proper time is available with computable coefficients, the so-called HadamardMinakshisundaram-DeWitt-Seeley (HMDS) or heat kernel coefficients [4,20]. These coefficients have been computed with different techniques in several setups and to dimension ten [21,22,23,24,25,26,27]. For reviews on spectral geometry and field theory on curved space see [28,29,30,31,32,33,34,35,36].…”