title: Shape from sound: toward new tools for quantum gravity.
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 two-dimensional objects from their vibrational spectra.
authors.affiliation: Department of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA.
authors.forename: David
authors.initials: D
authors.lastname: Aasen
authors.affiliation: Department of Physics, Princeton University, Princeton, New Jersey 08544, USA.
authors.forename: Tejal
authors.initials: T
authors.lastname: Bhamre
authors.affiliation: Departments of Applied Mathematics and Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 and Centre for Quantum Computing Technology, Department of Physics, University of Queensland, St. Lucia, Queensland 4072, Australia.
authors.forename: Achim
authors.initials: A
authors.lastname: Kempf