Quantum Space-Time: theory and phenomenology Scuola di dottorato Vito Volterra Dottorato di Ricerca in Fisica – XXXI Ciclo Candidate Michele Ronco ID number 1417121 Thesis Advisors Prof. Giovanni Amelino-Camelia Prof. Valeria Ferrari A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics October 2018 Thesis not yet defended Quantum Space-Time: theory and phenomenology Ph.D. thesis. Sapienza – University of Rome © 2013 Michele Ronco. All rights reserved This thesis has been typeset by LATEX and the Sapthesis class. Version: January 28, 2019 Author’s email: [email protected] Dedicated to Adriano v Abstract Modern physics is based on two fundamental pillars: quantum mechanics and Einstein’s general relativity. Even if, when taken separately, they can claim success in describing satisfactorily a plenty of physical phenomena, so far any attempt to make them compatible with each other failed. It is a central goal of theoretical physics to find a common approach to coherently merge quantum theory and general relativity. Such an effort is not only motivated by the conceptual necessity of completeness that imposes us to look for a unified theoretical framework that gives a consistent picture at all scales, but also by the existence of physical regimes we can not fully describe without a quantum theory of gravity, e.g. the first instants of early universe cosmology. This problem has remained open for more than eighty years now and keeps challenging physicists that, in the struggle to find a solution, have proposed a myriad of models, none of whom can claim full success. In fact, mainly due to the lack of experimental hints, the landscape of quantum gravity currently looks like a variegated compound of approaches that start from different conceptual premises and use different mathematical formalisms. In the majority of cases, it is not clear whether different models reach compatible predictions or even if they produce observable outcomes at all. Given the impossibility to achieve a unique acknowledged theory, it is of pivotal importance to seek insightful connections between different approaches. Such a strategy may help identifying few promising hot spots that may catalyze forthcoming efforts in the quantum gravity research community. In particular, more synergy between top-down and bottom-up models is certainly needed. Besides shedding light on some formal aspects of the models and eventually giving further support to specific ideas, reducing the gap between full-fledged quantum gravity proposals and simpler models that try to capture at least some expected features might produce tangible progress in the field of quantum gravity phenomenology. Indeed, it is now well-established that some effects introduced genuinely at the Planck scale by heuristic models can be efficiently investigated in ongoing and forthcoming experiments. Moreover, the exciting era of multi-messenger and multi-wavelength has started where several satellites, telescopes, and new generation detectors are furnishing us with an incomparable amount of data to probe the structure of gravity on cosmological scales and in new regimes which had remained inaccessible. Finding ways to rigorously derive Planckian testable effects from quantum gravity theories is then needed to enter another phase of maturity of quantum gravity phenomenology, i.e. the passage from the search for Planck-scale signals to the falsification of actual theories. This thesis represents a small step in this direction. To put this plan into action, we start recognizing that, despite the aforementioned heterogeneity, there is the common expectation that near the Planck scale our description of the spacetime as a smooth continuum, a picture shared by both general relativity and quantum mechanics, should break down and be replaced by some “fuzzy" structure we generically refer to as quantum spacetime. Again there are different ways to implement such an idea in different models, however we feel that the most relevant feature that characterizes spacetime quantization from a physical point of view is the associated departure from classical spacetime symmetries that most vi significantly encode spacetime’s properties. In this regard, in the literature of the last three decades there has been much interest in the development of deformations of the Poincaré symmetries of special relativity, which most notably took the form of quantum groups or Hopf algebras, with the aim of modeling Planck scale physics. However, almost the totality of these studies is confined to the limit where gravitational effects are negligible, i.e. a sort of “quantum Minkowski regime". With the objective to bridge quantum gravity and, in general, beyond general relativity theories with quantum or non-standard Minkowski spacetime models we here devote our attention to the symmetry content of general relativity, synthesized in the hypersurface deformation algebra, and explore possible deformations caused by non- classical spacetime effects. Candidate modifications of the algebra of diffeomorphisms have been already obtained in some recent analyses, others will be derived in this thesis for the first time. We then translate modifications of the hypersurface deformation algebra into corresponding deviations from special relativistic symmetry with the main objective of looking for phenomenological opportunities. In particular, studying the Minkowski limit of deformed diffeomorphism algebras, we shall infer two much studied Planckian phenomena, namely modified dispersion relation and the running of spacetime dimensions with the probed scale. In this thesis we focus in particular on four different paths toward the character- ization of non-classical (to be meant in a general sense as non-standard) spacetime properties: noncommutative geometry, loop quantum gravity, multifractional geome- try, and non-Riemannian geometry; only the second being widely recognized as a candidate full-fledged quantum gravity theory. We first motivate why these two phenomenological Planck scale effects, i.e. di- mensional reduction and modifications of particles’ dispersion in vacuum, can be ascribed to spacetime fuzziness or quantization intended as an intrinsic obstruction to the measurability of spacetime distances below the Planck scale, an effect which can be deduced from the heuristic combination of general relativistc and quantum mechanical principles. Modified dispersion relation is derived rigorously in the framework of noncommutative geometries and we discuss two different noncommu- tative models which are of interest for this thesis: θ-Minkowski and κ-Minkowski. The phenomenon of dimensional flow is instead presented from the perspective of multifractional geometry. Within this framework we show how dimensional flow and spacetime fuzziness are deeply connected. We illustrate how the assumption of an anomalous scaling of the spacetime dimension in the ultraviolet and a slow change of the dimension in the infrared is enough to produce a scale-dependent deformation of the integration measure with also a fuzzy spacetime structure. We also compare the multifractional correction to lengths with the types of Planckian uncertainty for distance and time measurements. This may offer an explanation why dimensional flow is encountered in almost the totality of quantum gravity models. We then introduce the (classical) hypersurface deformation algebra and construc- tively present two different ways of deriving it which we designate as representations of the algebra: the gravitational constraint representation, where the brackets are reproduced by the time and spatial diffeomorphism generators, and the Gaussian vector field representation, in which the algebra can be read off from the Lie bracket involving the components of a certain class of vector fields. Using this second realization, we study possible Drinfeld twists of space-time diffeomorphisms with vii Hopf-algebra techniques. We consider both deformed and twisted diffeomorphisms and compute the associated hypersurface deformation algebra. We then turn our attention to recent loop-quantum-gravity-inspired studies that have motivated a restricted class of modifications of the algebra of gravitational constraints. We discuss these new results in the light of the possibility to identify an effective quantum-spacetime picture of loop quantum gravity, applicable in the Minkowski regime, where the large-scale (coarse-grained) spacetime metric is flat. We show that these symmetry-algebra results are consistent with a description of spacetime given in terms of the κ-Minkowski noncommutative spacetime, whose relevance for the study of the quantum-gravity problem has already been proposed for independent reasons. We exploit this unexpected link to extract viable testable pre- dictions out of loop quantum gravity models. These loop-quantum-gravity-inspired corrections to spacetime symmetries are used to analyze both the consequences on particle propagation and on dimensional running. Adopting a different strategy, we also construct a set of three operators suitable for identifying coordinate-like quanti- ties on a spin-network configuration on the kinematical Hilbert space. Computing their action on coherent coarse-grained states, we are able to study some relevant properties such us the spectra, which are discrete. After that we scrutinize the symmetry structure of multifractional theories with either weighted or q- derivatives. These theories have the property that the spacetime dimensions are anomalous