Final Report Summary - QGNC (Non-commutative geometry and quantum gravity) Work on this project produced several scientific results, mainly relevant for the study of non-commutative geometry and algebra quantum field theory. Concerning non-commutative geometry the equality, first noticed by Rieffel, between Connes distance (in the commutative case) and the Wasserstein distance in the theory of optimal transport was extended to the locally compact case. And a proposal was made for some novel directions to follow in order to develop a theory of optimal transport in non-commutative geometry. Connes' distance was also applied in deformation quantisation. Recall that the Moyal algebra is the non-commutative deformation - via a star product - of the algebra of Schwartz function on R2n. The project also produced a computation of the distance between a certain class of states of the Moyal algebra (corresponding to the eigenstates of the quantum harmonic oscillator). And general results were obtained on the Moyal plane: the distance between any state of the Moyal algebra and any of its translation is precisely the amplitude of the translation. The project also resulted in a proposal of general framework for comparing Connes distance in the Moyal plane to the quantum length that has been defined in various model of quantum spacetime (like the Doplicher-Fredenhagen-Robert (DFR) model) as the spectrum of a suitable length operator. Applied to the eigenstates of the quantum harmonic oscillator, it was found that the length has the spectrum of an operator and Connes distance correspond to two different ways of integrating the same non-commutative line element on a non-commutative space. Work done during the project also showed that the product of the Moyal space with the spectral triple of C2 - restricted to coherent states - is orthogonal in the sense of Pythagoras theorem. Concerning modular flow in algebraic quantum field theory the project applied the thermal time hypothesis of Connes-Rovelli to double-cone regions in a bi-dimensional conformal field theory with boundary. It was found that the modular flow associated to Longo's ad hoc state was purely geometrical (as for a qft in Minkowski spacetime), while the modular flow associated to the vacuum of the two-dimensional (2D) boundary conformal field theory (CFT) combines the geometrical action with a term that mixes the components of the field on the edge of the double cone. This result is particularly interesting for it gives an explicit illustration of Connes theorem, according to which the modular flow defined by two distinct states are unitarily equivalent. Here, it was found that the action of Connes cocycle was purely non-geometrical. And part of the project was also devoted to exploring the implication of these results the nature of time.
