Gianluca Calcagni, Daniele Oriti, Johannes Thürigen
We extend discrete calculus to a bra-ket formalism for arbitrary (p-form) fields on discrete geometries, based on cellular complexes. We then provide a general definition of discrete Laplacian using both the primal cellular complex and its topological 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.
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http://arxiv.org/abs/1208.0354
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