The theoretical challenge of making Quantum Field Theory and General Relativity work together takes the name of Quantum Gravity (QG). The search for QG is far form conclusion but it has led to several impressive proposals. However, in the absence of experimental hints, researches started wondering how QG can make progress. We believe that this can be possible if we: construct bridges between different approaches and quest for model-independent results. Reaching a synergy between top-down and bottom-up approaches would allow to obtain both guidance on the most significant formal structures and develop insight on how to handle those structures, hopefully also in terms of experimental tests. A contact point may reside in the modifications of the hypersurface deformation algebra (HDA), recently derived in loop quantum gravity (LQG). In fact, taking the Minkowski limit, those corrections give rise to a Deformed Special Relativity (DSR) scenario. We have found that this LQG-inspired DSR model describes the symmetries of a non-commutative spacetime. First, we aim to extend the domain of validity of these results. Then, encouraged by them, we wish to pursue an almost complementary path: deriving the HDA for non-commutative manifold. This represents a long-standing challenge, whose solution would impact on many fields of research such as string theory. Another strategy is to look for features that are shared by different QG programs. Among them, pride of place is held by the reduction of dimensions near the QG scale, as realised e.g. in multifractal and miltifractional models. Our objective is to investigate the relation of dimensional reduction with spacetime fuzziness, a feature that is directly inherited by the coexistence of quantum mechanical and general relativistic principles. Besides shedding light on the recurrence of dimensional flow in QG studies, establishing such a relation would open up the possibility to test QG running dimensions in uncertainty measurements.
From a broad perspective the main novelty of our project is our intent of searching for bridges between different QG approaches. This point of view has been often hoped in the literature and also recently advocated by Smolin. However, so far there has been no specific research program in this direction. In the past, theoretical physics often took advantage from contaminations. We believe that this would be the case also for QG if we established connections between top-down and bottom-up approaches. Indeed, the former are able to solve technical problems (e.g. Renormalizability), while the latter try to figure out a way to make contact with present and forthcoming observations (see astrophysical tests of modified dispersion relations).
Our main idea is that of using the HDA as a bridge between top-down models, introducing quantum deformations of GR symmetries (i.e. of the HDA), and bottom-up models, which are often confined to the flat limit of QG. Specifically, on the basis of some encouraging results we already obtained, we wish to prove that space-time non-commutativity with DSR symmetries provides the Minkowski regime of QG approaches that modify the HDA, such as LQG. This has been often conjectured since the very beginning of the DSR program by Amelino-Camelia, but never concretely realised. On the one hand, this would finally allow us to have some physical predictions for LQG. On the other, we would have a consistent non-zero-curvature regime for DSR (and non-commutative) models (a remote possibility for some, see e.g. early critics by Gibbons-Gielen).
Correspondingly, we wish to see how GR symmetries should be modified in non-commutative manifolds. Wess and his colleagues succeeded in formulating the so-called "twisted diffeomorphisms", but there are reasons to believe they are spurious symmetries. Then, this still remains an open challenge. Currently, there are basically two proposals: a reduction of the group of diffeomoprhisms (leading to unimodular gravity), or Konstevich covariant star product. None of them has been fully successful. As already said, a consistent formulation of Non-commutative Gravity would be of great importance also for string theory, where a noncommutative structure of space-time arises in the quantization of the open string in the presence of a non-vanishing B-field. Technically, in this regard, the novelty we propose consists in deriving the HDA for (locally) non-commutative spaces by generalising properly the Gaussian-vector-field method introduced by Weinstein and already partially generalised by Bojowald. The transposition of this procedure (that extracts the Poission brackets between gravitational constraints from those of a couple of Gaussian vector fields) to non-commutativity will be done in collaboration with Bojowald.
The last part has as a main focus the look for model-independent features in QG approaches. Indeed, waiting for experimental inputs, these rare results that are shared by different formalisms might provide a unique opportunity to guide the next steps to be taken in this field. In fact, finding the same prediction regardless of the specific technique adopted might signal the presence of a deep and novel feature, as it already happened other times in the history of physics. Particularly, the recurring feature we want to focus on is dimensional reduction. This phenomenon is the central element of multifractal and multifractional geometries models, whose key assumption is an anomalous scaling of the spacetime dimension in the ultraviolet and a slow change of the dimension in the infrared. They have been recently proposed as an approach to the QG problem by Calcagni. As newly born models, they have already received some obstinate attention due to their potential in giving a physical meaning to several concepts scattered in quantum gravity. In particular, not only they allow one to control the change of spacetime dimensionality typical of all quantum gravities analytically, but they also recognize this feature as a treasure trove for phenomenology, since it leaves an imprint in observations at virtually all scales. At the same time, with respect to other mainstream theories, they still represent an almost unexplored territory. Within this new framework we see the potentiality to connect dimensional flow and spacetime fuzziness, two much-studied aspects of quantum gravity. Connecting these two aspects might have important consequences both within multifractional models, where it may fix some free parameters, and for the broader QG community, since it might set the stage for a role for dimensional flow in QG phenomenology. In fact, while space-time uncertainties in distance measurements can be tested thanks to gravitational-wave interferometers (see e.g. works by Amelino-Camelia), dimensional reduction is still evading any experimental signature (see a recent review by Carlip for critical discussions).