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High Energy Physics - Theory, hep-th,General Relativity and Quantum Cosmology, gr-qc,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
Abstract:
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.
