Ivan G. Avramidi (Department of Mathematics, New Mexico Tech)

Proceedings of the Conference: Heat kernel in quantum field theory, Nuclear Physics Proc. Suppl. 104 (2002) 3-32; 

Preprint: math-ph/0107018 67 pp.

ABSTRACT:

We give a short overview of the effective action approach in quantum field theory and quantum gravity and describe various methods for calculation of the asymptotic expansion of the heat kernel for second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold. We consider both Laplace type operators and non-Laplace type operators on manifolds without boundary as well as Laplace type operators on manifolds with boundary with oblique and non-smooth boundary conditions.
 

Lecture 1. "Effective Action Approach in Gauge Field Theories and Quantum Gravity"
Abstract:
An overview of the effective action approach in quantum field theory and quantum gravity.

Lecture 2. "Calculation of the Heat Kernel Asymptotic Expansion"
Abstract:
Review of basic techniques for calculation of the heat kernel asymptotic expansion for Laplace type partial diffferential operators acting o smooth sections of a vector bundle over a compact manifold without boundary.

Lecture 3. "Covariant Approximation Schemes for Calculation of the Heat Kernel"
Abstract:
Development of various approximation schemes for the heat kernel based on the behavior of the "background fields". In the "high-energy" limit, which is determined by rapidly varying background fields, the highest derivative terms are computed and a partial summation is carried out leading to closed formulas for the effective action. In the "low-energy" limit, which is determined by slowly varying background fields, an algebraic approach is developed that enables one to obtain closed non-perturbative formulas for the heat kernel that generate the whole asymptotic expansion.

Lecture 4. "Heat-kernel Asymptotics for Non-Laplace Type Operators"
Abstract:
The most general class of second-order operators (non-Laplace type operators), acting on sections of a vector bundle, with positive definite leading symbol, is studied and the calculation of the heat-kernel asymptotics is presented. The heat kernel and the resolvent are constructed explicitly in the leading order.

Lecture 5. "Heat-Kernel Asymptotics for the Grubb - Gilkey - Smith Boundary-Value Problem"
Abstract:
Laplace type partial differential operator acting on sections of a vector bundle over a compact Riemannian manifold with smooth boundary with the Grubb - Gilkey - Smith boundary conditions, which involve both normal and tangential derivatives on the boundary. The criterion of strong ellipticity is discussed and the first non-trivial coefficient of the asymptotic expansion of the trace of the heat kernel is computed.

Lecture 6. "Heat-Kernel Asymptotics for Non-Smooth Boundary Conditions"
Abstract:
The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with nonstandard singular boundary conditions, which include Dirichlet conditions on one part of the boundary and Neumann ones on another part of the boundary, is studied. The resolvent kernel and the heat kernel in the leading approximation are explicitly constructed and the leading heat-kernel asymptotics are computed.