Laplacians on discrete and quantum geometries
Aug, 2012
43 pages
Published in:
- Class.Quant.Grav. 30 (2013) 125006
e-Print:
- 1208.0354 [hep-th]
Report number:
- AEI-2012-078
View in:
Citations per year
Abstract: (arXiv)
We extend discrete calculus for arbitrary (-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.Note:
- 43 pages, 2 multiple figures. v2: discussion improved, references added, minor typos corrected
- Models of Quantum Gravity
- Lattice Models of Gravity
- geometry: discrete
- dimension: spectral
- field theory: scalar
- quantum geometry
- duality
- quantum gravity: loop space
- topological
- heat kernel
References(102)
Figures(6)
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