Towards Spectral Geometric Methods for Euclidean Quantum Gravity

TL;DR

This paper explores how spectral geometry can be used to understand shape changes in Riemannian manifolds, proposing a generalized linearized approach and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a generalized linearized spectral geometry method allowing unequal shape and spectral degrees of freedom, enabling better shape reconstruction from spectral data.

Findings

Small shape changes can be reconstructed from spectral changes with enough eigenvalues.

Isospectral non-isometric shapes are rare, suggesting spectral data often uniquely encodes shape.

The approach is numerically validated on planar domains, showing practical applicability.

Abstract

The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis respectively. Of particular interest in this regard is the field of spectral geometry, which studies to which extent the shape of a Riemannian manifold is describable in terms of the spectra of differential operators defined on the manifold. Spectral geometry is hard because it is highly nonlinear, but linearized spectral geometry, i.e., the task to determine small shape changes from small spectral changes, is much more tractable, and may be iterated to approximate the full problem. Here, we generalize this approach, allowing, in particular, non-equal finite numbers of shape and spectral degrees of freedom. This allows us to study how well the shape degrees of freedom are encoded in the eigenvalues. We apply this strategy numerically to a class of planar domains and find that the reconstruction of small shape changes from small spectral changes is possible if enough eigenvalues are used. While isospectral non-isometric shapes are known to exist, we find evidence that generically shaped isospectral non-isometric shapes, if existing, are exceedingly rare.