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A Quantum Texture of Spacetime and Fields Sergio Doplicher Abstract The motivations which impose a quantum picture of Spacetime at the Planck scale are discussed; they lead to a simplest model of Quantum Spacetime, which is introduced; we describe the Quantum Geometry resulting from it. We then outline features, promises and problems of Quantum Field Theory on Quantum Spacetime, and some of its possible applications to Cosmology. References [1] Sergio Doplicher, Klaus Fredenhagen, John E. Roberts: “The quantum struc- ture of spacetime at the Planck scale and quantum fields”, Commun. Math. Phys. 172 (1995), 187-220; [2] Luca Tomassini, Stefano Viaggiu: “Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime”, Class. Quantum Grav. 28 (2011), 075001; [3] Luca Tomassini, Stefano Viaggiu: “Building non commutative spacetimes at the Planck length for Friedmann flat cosmologies”, Class. Quantum Grav. 31 (2014), 185001; [4] Sergio Doplicher, Gerardo Morsella, Nicola Pinamonti: “On Quantum Space- time and the horizon problem”, J. Geom. Phys. 74 (2013), 196-210; [5] Dorothea Bahns, Sergio Doplicher, Klaus Fredenhagen, Gherardo Piacitelli: “Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators”, Commun. Math. Phys. 308 (2011), 567-589; [6] Sergio Doplicher: “The Principle of Locality. Effectiveness, fate and chal- lenges”, J. Math. Phys. 51 (2010) 015218; 1 [7] Gherardo Piacitelli: “Twisted Covariance as a Non Invariant Restriction of the Fully Covariant DFR Model”, Commun. Math. Phys. 295 (2010) 701-729; [8] Gherardo Piacitelli: “Non Local Theories: New Rules for Old Diagrams”, J. High Energy Phys. 8 (2004) 31-43; [9] Dorothea Bahns: “Ultraviolet Finiteness of the averaged Hamiltonian on the noncommutative Minkowski space”, arXiv:hep-th/0405224. [10] D. Bahns , S. Doplicher, K. Fredenhagen, G. Piacitelli: “Ultraviolet Finite Quantum Field Theory on Quantum Spacetime”, Commun. Math. Phys. 237 (2003) 221-241; [11] Sergio Doplicher: “Spacetime and Fields, a Quantum Texture”, Proceedings of the 37th Karpacz Winter School of Theoretical Physics, 2001, 204-213; arXiv:hep-th/0105251; [12] D. Bahns , S. Doplicher, G.Morsella, G. Piacitelli: “Quantum Spacetime and Algebraic Quantum Field Theory”, arXiv:1501.03298.

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