Shape from sound: toward new tools for quantum gravity

TL;DR

This paper explores using spectral geometry to connect differential geometry and functional analysis, enabling shape determination from vibrational spectra, with potential applications in quantum gravity.

Contribution

It introduces a regularized, step-by-step approach to spectral geometry, demonstrating the feasibility of reconstructing 2D shapes from spectra.

Findings

Successfully reconstructs 2D shapes from vibrational spectra

Regularization makes spectral geometry more manageable

Provides a new tool for quantum gravity research

Abstract

To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at least in the case of the euclidean signature, is the discipline of spectral geometry. It aims at describing curved manifolds in terms of the spectra of their canonical differential operators. As an immediate benefit, this would offer a clean gauge-independent identification of the metric's degrees of freedom in terms of invariants that should be ready to quantize. However, spectral geometry is itself hard and has been plagued by ambiguities. Here, we regularize and break up spectral geometry into small finite-dimensional and therefore manageable steps. We constructively demonstrate that this strategy works at least in two dimensions. We can now calculate the shapes of 2-dimensional objects from their vibrational spectra.