Commutator technique for the heat kernel of minimal higher derivative operators

TL;DR

This paper introduces a new commutator-based technique for deriving the asymptotic heat kernel expansion of minimal higher derivative operators, simplifying calculations in gauge theories and quantum gravity.

Contribution

It develops a novel commutator algebra method converting Fourier integral approaches into a more straightforward, symbolic algorithm for heat kernel expansion of higher derivative operators.

Findings

Provides a systematic three-step algorithm for heat kernel expansion

Expresses coefficients in terms of second order operator coefficients

Facilitates computer implementation for symbolic calculations

Abstract

We suggest a new technique of the asymptotic heat kernel expansion for minimal higher derivative operators of a generic $2M$-th order, $F(\nabla)=(-\Box)^M+\cdots$, in the background field formalism of gauge theories and quantum gravity. This technique represents the conversion of the recently suggested Fourier integral method of generalized exponential functions [Phys. Rev. D105, 065013 (2022), arXiv:2112.03062] into the commutator algebra of special differential operators, which allows one to express expansion coefficients for $F(\nabla)$ in terms of the Schwinger-DeWitt coefficients of a minimal second order operator $H(\nabla)$. This procedure is based on special functorial properties of the formalism including the Mellin-Barnes representation of the complex operator power $H^M(\nabla)$ and naturally leads to the origin of generalized exponential functions without directly appealing to the Fourier integral method. The algorithm is essentially more straightforward than the Fourier method and consists of three steps ready for computer codification by symbolic manipulation programs. They begin with the decomposition of the operator into a power of some minimal second order operator $H(\nabla)$ and its lower derivative "perturbation part" $W(\nabla)$, $F(\nabla)=H^M(\nabla)+W(\nabla)$, followed by considering their multiple nested commutators. The second step is the construction of special local differential operators -- the perturbation theory in powers of the lower derivative part $W(\nabla)$. The final step is the so-called procedure of their {\em syngification} consisting in a special modification of the covariant derivative monomials in these operators by Synge world function $\sigma(x,x')$ with their subsequent action on the HaMiDeW coefficients of $H(\nabla)$.