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The exploration of the universe has recently entered a new era thanks to the multi-messenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give us information about their sources in the universe and the properties of the intergalactic medium. Moreover, multi-messenger astronomy opens up the possibility to search for phenomenological signatures of quantum gravity. On the one hand, the most energetic events allow us to test our physical theories at energy regimes which are not directly accessible in accelerators; on the other hand, tiny effects in the propagation of very high energy particles could be amplified by cosmological distances. After decades of merely theoretical investigations, the possibility of obtaining phenomenological indications of Planck-scale effects is a revolutionary step in the quest for a quantum theory of gravity, but it requires cooperation between different communities of physicists (both theoretical and experimental). This review, prepared within the COST Action CA18108 “Quantum gravity phenomenology in the multi-messenger approach”, is aimed at promoting this cooperation by giving a state-of-the art account of the interdisciplinary expertise that is needed in the effective search of quantum gravity footprints in the production, propagation and detection of cosmic messengers. © 2022 The Authors

In this note we carry out the counting of states for a black hole in loop quantum gravity, however assuming an equidistant area spectrum. We find that this toy-model is exactly solvable, and we show that its behavior is very similar to that of the correct model. Thus this toy-model can be used as a nice and simplifying `laboratory' for questions about the full theory.

We give a definition for the Wigner function for quantum mechanics on the Bohr compactification of the real line and prove a number of simple consequences of this definition. We then discuss how this formalism can be applied to loop quantum cosmology. As an example, we use the Wigner function to give a new quantization of an important building block of the Hamiltonian constraint.

Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms. While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.

The present paper is the companion of [1] in which we proposed a scheme that tries to derive the Quantum Field Theory (QFT) on Curved Spacetimes (CST) limit from background independent Quantum General Relativity (QGR). The constructions of [1] make heavy use of the notion of semiclassical states for QGR. In the present paper, we employ the complexifier coherent states for QGR recently proposed by Thiemann and Winkler as semiclassical states, and thus fill the general formulas obtained in [1] with life. We demonstrate how one can, under some simplifying assumptions, explicitely compute expectation values of the operators relevant for the gravity-matter Hamiltonians of [1] in the complexifier coherent states. These expectation values give rise to effective matter Hamiltonians on the background on which the gravitational coherent state is peaked and thus induce approximate notions of n-particle states and matter propagation on fluctuating spacetimes. We display the details for the scalar and the electromagnetic field. The effective theories exhibit two types of corrections as compared to the the ordinary QFT on CST. The first is due to the quantum fluctuations of the gravitational field, the second arises from the fact that background independence forces both geometry and matter to propagate on a spacetime that is the product of the real line and a (random) graph. Finally we obtain explicit numerical predictions for non-standard dispersion relations for the scalar and the electromagnetic field. They should, however, not be taken too seriously, due to the many ambiguities in our scheme, the analysis of the physical significance of which has only begun. We show however, that one can classify these ambiguities at least in broad terms

In this article and a companion paper we address the question of how one might obtain the semiclassical limit of ordinary matter quantum fields (QFT) propagating on curved spacetimes (CST) from full fledged Quantum General Relativity (QGR), starting from first principles. We stress that we do not claim to have a satisfactory answer to this question, rather our intention is to ignite a discussion by displaying the problems that have to be solved when carrying out such a program. In the present paper we propose a scheme that one might follow in order to arrive at such a limit. We discuss the technical and conceptual problems that arise in doing so and how they can be solved in principle. As to be expected, completely new issues arise due to the fact that QGR is a background independent theory. For instance, fundamentally the notion of a photon involves not only the Maxwell quantum field but also the metric operator - in a sense, there is no photon vacuum state but a "photon vacuum operator"! While in this first paper we focus on conceptual and abstract aspects, for instance the definition of (fundamental) n-particle states (e.g. photons), in the second paper we perform detailed calculations including, among other things, coherent state expectation values and propagation on random lattices. These calculations serve as an illustration of how far one can get with present mathematical techniques. Although they result in detailed predictions for the size of first quantum corrections such as the gamma-ray burst effect, these predictions should not be taken too seriously because a) the calculations are carried out at the kinematical level only and b) while we can classify the amount of freedom in our constructions, the analysis of the physical significance of possible choices has just begun

The exploration of the universe has recently entered a new era thanks to the multimessenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give us information about their sources in the universe and the properties of the intergalactic medium. Moreover, multi-messenger astronomy opens up the possibility to search for phenomenological signatures of quantum gravity. On the one hand, the most energetic events allow us to test our physical theories at energy regimes which are not directly accessible in accelerators; on the other hand, tiny effects in the propagation of very high energy particles could be amplified by cosmological distances. After decades of merely theoretical investigations, the possibility of obtaining phenomenological indications of Planck-scale effects is a revolutionary step in the quest for a quantum theory of gravity, but it requires cooperation between different communities of physicists (both theoretical and experimental). This review, prepared within the COST Action CA18108 ‘‘Quantum gravity phenomenology in the multi-messenger approach", is aimed at promoting this cooperation by giving a state-of-the art account of the interdisciplinary expertise that is needed in the effective search of quantum gravity footprints in the production, propagation and detection of cosmic messengers., Talent Scientific Research Program of College of Physics, Sichuan University 1082204112427, Fostering Program in Disciplines Possessing Novel Features for Natural Science of Sichuan University 2020SCUNL209, 1000 Talent program of Sichuan province 2021, Xunta de Galicia, European Commission European Union ERDF, "Maria de Maeztu'' Units of Excellence program MDM-2016-0692, Red Tematica Nacional de Astroparticulas RED2018-102661-T, La Caixa Foundation 100010434, European Commission 847648 LCF/BQ/PI21/11830030 754510, Ministry of Education, Science & Technological Development, Serbia 451-03-9/2021-14/200124, FSR Incoming Postdoctoral Fellowship Ministry of Education, Science and Technological Development, Serbia 451-03-9/2021-14/200124, University of Rijeka grant uniri-prirod-18-48, Croatian Science Foundation (HRZZ) IP-2016-06-9782, Villum Fonden 29405 DGA-FSE 2020-E2117R, European Regional Development Fund through the Center of Excellence (TK133) "The Dark Side of the Universe'' European Regional Development Fund (ESIF/ERDF), Ministry of Education, Youth & Sports - Czech Republic CoGraDS-CZ.02.1.01/0.0/0.0/15 003/0000437, Blavatnik grant, Basque Government IT-97916 Basque Foundation for Science (IKERBASQUE), European Space Agency C4000120711 4000132310, FNRS (Belgian Fund for Research), Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnologica (PAPIIT), Universidad Nacional Autonoma de Mexico TA100122, National University of La Plata X909 DICYT 042131GR, National Research, Development & Innovation Office (NRDIO) - Hungary 123996, FQXi, Swiss National Science Foundation (SNSF), European Commission 181461 199307, Netherlands Organization for Scientific Research (NWO) 680-91-119 15MV71, Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT) Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research (KAKENHI) 20H01899 20H05853 JP21F21789, Estonian Research Council PRG356, Julian Schwinger Foundation, Generalitat Valenciana Excellence PROMETEO-II/2017/033 PROMETEO/2018/165, Istituto Nazionale di Fisica Nucleare (INFN), European ITN project HIDDeN H2020-MSCA-ITN-2019//860881-HIDDeN, Swedish Research Council, European Commission 2016-05996 European Research Council (ERC) European Commission 668679, Advanced ERC grant TReX, Ministry of Education, Universities and Research (MIUR) 2017X7X85K, Fonds de la Recherche Scientifique - FNRS 4.4501.18, Ministry of Research, Innovation and Digitization - Romania PN19-030102-INCDFM PN-III-P4ID-PCE-2020-2374, United States Department of Energy (DOE) DE-SC0020262, Ministry of Science, ICT & Future Planning, Republic of Korea 075-15-2020-778, German Academic Scholarship Foundation German Research Foundation (DFG) 408049454 420243324 425333893 445990517 Germany's Excellence Strategy (EXC 2121 "Quantum Universe'') 390833306 390837967 Federal Ministry of Education & Research (BMBF) 05 A20GU2 05 A20PX1, Centro de Excelencia "Severo Ochoa'' SEV-2016-0588, CERCA program of the Generalitat de Catalunya, Agencia de Gestio D'Ajuts Universitaris de Recerca Agaur (AGAUR) Generalitat de Catalunya 2017-SGR-1469 2017-SGR-929 ICCUB CEX2019-000918-M, National Science Centre, Poland 2019/33/B/ST2/00050 2017/27/B/ST2/01902, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPQ) 306414/2020-1, Dicyt-USACH 041931MF, National Science Fund of Bulgaria KP-06-N 38/11 RCN ROMFORSK 302640, Comunidad de Madrid 2018-T1/TIC-10431 2019-T1/TIC-13177 S2018/NMT-4291, UK Research & Innovation (UKRI), Science & Technology Facilities Council (STFC) ST/T000759/1 ST/P000258/1 ST/T000732/1 ST/V005596/1, Portuguese Foundation for Science and Technology UIDB/00618/2020 UIDB/00777/2020 UIDP/00777/2020 CERN/FIS-PAR/0004/2019 PTDC/FIS-PAR/29436/2017 PTDC/FISPAR/31938/2017 PTDC/FIS-OUT/29048/2017 SFRH/BD/137127/2018, Centre National de la Recherche Scientifique (CNRS), LabEx UnivEarthS ANR-10-LABX-0023 ANR18-IDEX-0001, Junta de Andalucia European Commission A-FQM-053-UGR18, Natural Sciences and Engineering Research Council of Canada (NSERC) RGPIN-2021-03644, National Science Centre Poland Sonata Bis 2019/33/B/ST2/00050 DEC-2017/26/E/ST2/00763, Natural Sciences and Engineering Research Council of Canada (NSERC) DGIID-DGA 2015-E24/2, Spanish Research State Agency and Ministerio de Ciencia e Innovacion MCIN/AEI PID2019-104114RB-C32 PID2019-105544GB-I00 PID2019-105614GB-C21 PID2019106515GB-I00 PID2019-106802GB-I00 PID2019-107394GB-I00 PID2019-107844GB-C21 PID2019-107847RB-C41 MCIN/AEI PGC2018-095328-B-I00 PGC2018-094856-B-I00 PGC2018-096663-B-C41 PGC2018-096663-B-C44 PGC2018-094626-BC21 PGC2018-101858-B-I00 FPA2017-84543-P FPA2016-76005-C2-1-P, Spanish 'Ministerio de Universidades' BG20/00228 Spanish Government PID2020-115845GBI00 Generalitat de Catalunya Comunidad de Madrid S2018/NMT-4291 Spanish Government PID2019-105544GB-I00, Perimeter Institute for Theoretical Physics, Government of Canada through the Department of Innovation, Science and Economic Development, Province of Ontario through the Ministry of Colleges and Universities, Centre National de la Recherche Scientifique (CNRS), Netherlands Organization for Scientific Research (NWO), Fundamental Questions Institute (FQXi), European Cooperation in Science and Technology (COST) CA18108, Research Council of University of Guilan, Iniziativa Specifica TEONGRAV Iniziativa Specifica QGSKY Iniziativa Specifica QUAGRAP Iniziativa Specifica GeoSymQFT, the Spanish Research State Agency and Ministerio de Ciencia e Innovacion MCIN/AEI PID2020-115845GBI00 PID2019-108485GB-I00 PID2020-113334GB-I00 PID2020-113701GB-I00 PID2020-113775GB-I00 PID2020-118159GB-C41 PID2020-118159GA-C42 PRE2019-089024, Rothchild grant UID/MAT/00212/2020 FPU18/04571