We propose physically motivated spacetime uncertainty relations (STUR) for flat Friedmann-Lemaître cosmologies. We show that the physical features of these STUR crucially depend on whether a particle horizon is present or not. In particular, when this is the case we deduce the existence of a maximal value for the Hubble rate (or equivalently for the matter density), thus providing an indication that quantum effects may rule out a pointlike big bang singularity. Finally, we costruct a concrete realisation of the corresponding quantum Friedmann spacetime in terms of operators on a Hilbert space.

We study the metric aspect of the Moyal plane from Connes' noncommutative geometry point of view. First, we compute Connes' spectral distance associated with the natural isometric action of R 2 on the algebra of the Moyal plane A. We show that the distance between any state of A and any of its translated is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes' spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasize on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) A by C 2 . We show that on the set of states obtained by translation of an arbitrary state of A, this distance is given by Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital & non-degenerate spectral triples. Applied to the Doplicher-Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes' spectral distance and the DFR quantum length coincide on the set of states of optimal localization.

We derive new spacetime uncertainty relations (STUR) at the fundamental Planck length L P from quantum mechanics and general relativity (GR), both in flat and curved backgrounds. Contrary to claims present in the literature, our approach suggests that no minimal uncertainty appears for lengths, but instead for minimal space and fourvolumes. Moreover, we derive a maximal absolute value for the energy density. Finally, some considerations on possible commutators among quantum operators implying our STUR are done.

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although on the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular on the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d L , which coincides exactly with the spectral distance d D on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator -together with their translations -d L and d D coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d L and d D as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.

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Background: Long-acting injectable (LAI) aripiprazole was found to be efficacious in schizophrenia. In common clinical practice, the use of LAIs is often restricted to chronic patients with frequent relapses and poor adherence. Recently, some investigators advanced the idea of early LAI use also in young people with schizophrenia at their first psychotic episode (FEP). Objective: Our study aimed to assess the effect of LAI aripiprazole once monthly (AOM) in the treatment of FEP in patients aged 18-26 years. Methods: We included 50 patients with DSM-5 schizophrenia as assessed with SCID, and used the Clinical Global Impressions Scale-Severity of Illness (CGI-S) and the Positive and Negative Syndrome Scale (PANSS) to assess symptom severity and the World Health Organization Quality of Life (WHOQOL), the Short Form Health Survey (SF-36) and the Personal and Social Performance Scale (PSP) to assess quality of life (QoL) and global health perception at baseline and 3, 6, 9, and 12 months after the first AOM injection. Results: AOM was associated with a progressive improvement, compared to baseline, of both positive (p < 0.001) and negative (p < 0.001) symptoms and in general psychopathology (p < 0.001) and decrease in global severity (p < 0.001). We also observed progressive improvement in QoL and social and personal functioning. Treatment adherence was 78% at study endpoint. Our results support that AOM may improve psychotic symptoms, QoL and social functioning in young FEP patients. Further studies should compare AOM to its oral formulation in the treatment of young patients with schizophrenia at the outset of their illness.

Within the framework of algebraic quantum field theory, we propose a new method of constructing local generators of (global) gauge symmetries in field theoretic models, starting from the existence of unitary operators implementing locally the flip automorphism on the doubled theory. We show, in the simple example of the internal symmetries of a multiplet of free scalar fields, that through the pointlike limit of such local generators the conserved Wightman currents associated with the symmetries are recovered.

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