Description
Spectral geometry offers a powerful means to describe the geometric information of manifolds using the spectra of operators defined on them. While much work has been done on the spectral geometry of Riemannian manifolds, comparatively little work has been done on Lorentzian manifolds. Causal sets, being a discrete realization of Lorentzian manifolds, provide an ideal setting for such a study. I will review spectral geometry in causal set theory and discuss what geometric information can be obtained from the spectra of various operators on causal sets. I will show results indicating that one can indeed “hear” some physical aspects of a causal set.
From a physical point of view, these studies are motivated by their application in quantum gravity. If one were to uniquely distinguish causal sets through a set of spectra, one could sum over these spectra in a path sum. Spectra are relabeling invariant and lower dimensional compared to the adjacency matrices (such as the link and causal matrices) that are typically used to describe a causal set, making them beneficial to work with both in principle and in practice. Eigenvalues and eigenvectors of certain operators have also been central to defining quantum fields on causal sets. I will highlight some of these applications and comment on future directions of this area of research.
