Heat kernel expansion for higher order minimal and nonminimal operators

Abstract

We build a systematic calculational method for the covariant expansion of the two-point heat kernel $\hat{K}(\tau |x,x^{\prime })$ for generic minimal and non-minimal differential operators of any order. This is the expansion in powers of dimensional background field objects -- the coefficients of the operator and the corresponding spacetime and vector bundle curvatures, suitable in renormalization and effective field theory applications. For minimal operators whose principal symbol is given by an arbitrary power of the covariant Laplacian $(-\square {)}^M$, $M>1$, this result generalizes the well-known Schwinger--DeWitt (or Seeley--Gilkey) expansion to the infinite series of positive and negative fractional powers of the proper time ${\tau }^{1/M}$, weighted by the generalized exponential functions of the dimensionless argument $-\sigma (x,x^{\prime })/2{\tau }^{1/M}$ depending on the Synge world function $\sigma (x,x^{\prime })$. The coefficients of this series are determined by the chain of auxiliary differential operators acting on the two-point parallel transport tensor, which in their turn follow from the solution of special recursive equations. The derivation of these operators and their recursive equations are based on the covariant Fourier transform in curved spacetime. The series of negative fractional powers of $\tau$ vanishes in the coincidence limit $x^{\prime }=x$, which makes the proposed method consistent with the heat kernel theory of Seeley--Gilkey and generalizes it beyond the heat kernel diagonal in the form of the asymptotic expansion in the domain $\sigma (x,x^{\prime })\ll {\tau }^{1/M}$, $\tau \to 0$.${\nabla }^a\sigma (x,x^{\prime })\ll {\tau }^{1/2M}$, $\tau \to 0$.

arXiv comment24 pages, 2 figures

References in corpus (10)

• Heat kernel expansion: user's manual
• Conformally Covariant Differential Operators: Properties and Applications
• Hořava gravity is asymptotically free (in 2+1 dimensions)
• Gravitational effective action and entanglement entropy in UV modified theories with and without Lorentz symmetry
• Beta functions of (3+1)-dimensional projectable Horava gravity
• Diagram technique for the heat kernel of the covariant Laplace operator
• Heat kernel: proper time method, Fock-Schwinger gauge, path integral representation, and Wilson line
• Heat kernel for higher-order differential operators and generalized exponential functions
• Heat Kernel Expansion for Operators of the Type of the Square Root of the Laplace Operator
• New Heat Kernel Method in Lifshitz Theories

Cited by in corpus (11)

• Asymptotic freedom and safety in quantum gravity
• Special Functions for Heat Kernel Expansion
• Notes on peculiarities of Schwinger--DeWitt technique: one-loop double poles, total-derivative terms and determinant anomalies
• Schwinger--DeWitt expansion for the heat kernel of nonminimal operators in causal theories
• Radiative corrections to the $R$ and $R^2$ invariants from torsion fluctuations on maximally symmetric spaces
• Commutator technique for the heat kernel of minimal higher derivative operators
• Effective actions in supersymmetric gauge theories: heat kernels for non-minimal operators
• Functorial properties of Schwinger-DeWitt expansion and Mellin-Barnes representation
• On heat coefficients, multiplicative anomaly and 4D Casimir energy for GJMS operators
• Heat kernel approach to the one-loop effective action for nonlinear electrodynamics
• Pseudodifferential calculus in Schwinger--DeWitt formalism: UV and IR parts