Laplacians on discrete and quantum geometries

Abstract

We extend discrete calculus for arbitrary ($p$-form) fields on embedded lattices to abstract discrete geometries based on combinatorial complexes. We then provide a general definition of discrete Laplacian using both the primal cellular complex and its combinatorial dual. The precise implementation of geometric volume factors is not unique and, comparing the definition with a circumcentric and a barycentric dual, we argue that the latter is, in general, more appropriate because it induces a Laplacian with more desirable properties. We give the expression of the discrete Laplacian in several different sets of geometric variables, suitable for computations in different quantum gravity formalisms. Furthermore, we investigate the possibility of transforming from position to momentum space for scalar fields, thus setting the stage for the calculation of heat kernel and spectral dimension in discrete quantum geometries.

43 pages, 2 multiple figures. v2: discussion improved, references added, minor typos corrected

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• Multiscale spacetimes from first principles
• Spectral dimension of quantum geometries
• Do We Really Understand the Cosmos?
• ABC of multi-fractal spacetimes and fractional sea turtles
• Emergence of Spacetime in a restricted Spin-foam model
• The mass of simple and higher-order networks
• Dirac gauge theory for topological spinors in 3+1 dimensional networks
• Generalised spectral dimensions in non-perturbative quantum gravity
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• Curvature effects in the spectral dimension of spin foams
• Toward matter dynamics in spin foam quantum gravity
• Quantum walk on a spin network
• Multisymplectic effective General Boundary Field Theory
• Random Lattice Gauge Theories and Differential Forms
• Free Energy on a Cycle Graph and Trigonometric Deformation of Heat Kernel Traces on Odd Spheres
• Semiclassical states in quantum gravity: Curvature associated to a Voronoi graph
• Natural discrete differential calculus in physics