Spectral Geometry and Quantum Gravity

TL;DR

This paper investigates the mathematical structure of boundary contributions in quantum gravity, focusing on universal functions derived from spectral geometry and their role in semiclassical approximations.

Contribution

It introduces new universal functions related to boundary invariants involving tangential derivatives, computed for the first time, advancing understanding of quantum gravity boundary conditions.

Findings

Derived boundary invariants involving tangential derivatives.

Computed universal functions independent of conformal rescaling.

Enhanced understanding of one-loop semiclassical approximations.

Abstract

Recent progress in quantum field theory and quantum gravity relies on mixed boundary conditions involving both normal and tangential derivatives of the quantized field. In particular, the occurrence of tangential derivatives in the boundary operator makes it possible to build a large number of new local invariants. The integration of linear combinations of such invariants of the orthogonal group yields the boundary contribution to the asymptotic expansion of the integrated heat-kernel. This can be used, in turn, to study the one-loop semiclassical approximation. The coefficients of linear combination are now being computed for the first time. They are universal functions, in that are functions of position on the boundary not affected by conformal rescalings of the background metric, invariant in form and independent of the dimension of the background Riemannian manifold. In Euclidean quantum gravity, the problem arises of studying infinitely many universal functions.