I realize now that paper is pulling together a set of building blocks the author and others have laid. Some links below.
http://arxiv.org/abs/1208.0354
Laplacians on discrete and quantum geometries
Gianluca Calcagni,
Daniele Oriti,
Johannes Thürigen
(Submitted on 1 Aug 2012 (
v1), last revised 17 May 2013 (this version, v2))
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.
http://arxiv.org/abs/1311.3340
Spectral dimension of quantum geometries
Gianluca Calcagni,
Daniele Oriti,
Johannes Thürigen
(Submitted on 13 Nov 2013 (
v1), last revised 18 Jun 2014 (this version, v2))
The spectral dimension is an indicator of geometry and topology of spacetime and a tool to compare the description of quantum geometry in various approaches to quantum gravity. This is possible because it can be defined not only on smooth geometries but also on discrete (e.g., simplicial) ones. In this paper, we consider the spectral dimension of quantum states of spatial geometry defined on combinatorial complexes endowed with additional algebraic data: the kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the effects of topology and discreteness of classical discrete geometries are studied in a systematic manner. We look for states reproducing the spectral dimension of a classical space in the appropriate regime. We also test the hypothesis that in LQG, as in other approaches, there is a scale dependence of the spectral dimension, which runs from the topological dimension at large scales to a smaller one at short distances. While our results do not give any strong support to this hypothesis, we can however pinpoint when the topological dimension is reproduced by LQG quantum states. Overall, by exploring the interplay of combinatorial, topological and geometrical effects, and by considering various kinds of quantum states such as coherent states and their superpositions, we find that the spectral dimension of discrete quantum geometries is more sensitive to the underlying combinatorial structures than to the details of the additional data associated with them.
http://arxiv.org/abs/1412.8390
Dimensional flow in discrete quantum geometries
Gianluca Calcagni,
Daniele Oriti,
Johannes Thürigen
(Submitted on 29 Dec 2014 (
v1), last revised 21 Apr 2015 (this version, v2))
In various theories of quantum gravity, one observes a change in the spectral dimension from the topological spatial dimension d at large length scales to some smaller value at small, Planckian scales. While the origin of such a flow is well understood in continuum approaches, in theories built on discrete structures a firm control of the underlying mechanism is still missing. We shed some light on the issue by presenting a particular class of quantum geometries with a flow in the spectral dimension, given by superpositions of states defined on regular complexes. For particular superposition coefficients parametrized by a real number 0<α<d, we find that the spatial spectral dimension reduces to ds≃α at small scales. The spatial Hausdorff dimension of such class of states varies between 1 and d, while the walk dimension takes the usual value dw=2. Therefore, these quantum geometries may be considered as fractal only when α=1, where the "magic number" dsspacetime≃2 for the spectral dimension of space\emph{time}, appearing so often in quantum gravity, is reproduced as well. These results apply, in particular, to special superpositions of spin-network states in loop quantum gravity, and they provide more solid indications of dimensional flow in this approach.
http://arxiv.org/abs/1409.3150
Group field theories for all loop quantum gravity
Daniele Oriti,
James P. Ryan,
Johannes Thürigen
(Submitted on 10 Sep 2014 (
v1), last revised 14 Feb 2015 (this version, v2))
Group field theories represent a 2nd quantized reformulation of the loop quantum gravity state space and a completion of the spin foam formalism. States of the canonical theory, in the traditional continuum setting, have support on graphs of arbitrary valence. On the other hand, group field theories have usually been defined in a simplicial context, thus dealing with a restricted set of graphs. In this paper, we generalize the combinatorics of group field theories to cover all the loop quantum gravity state space. As an explicit example, we describe the GFT formulation of the KKL spin foam model, as well as a particular modified version. We show that the use of tensor model tools allows for the most effective construction. In order to clarify the mathematical basis of our construction and of the formalisms with which we deal, we also give an exhaustive description of the combinatorial structures entering spin foam models and group field theories, both at the level of the boundary states and of the quantum amplitudes.
