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Abstract The fermion propagator is derived in detail from the model of fermion coupled to loop quantum gravity (LQG). As an ingredient of the propagator, the vacuum state is defined as the ground state of some effective fermion Hamiltonian under the background geometry given by a coherent state resembling the classical Minkowski spacetime. Moreover, as a critical feature of LQG, the superposition over graphs is employed to define the vacuum state. It turns out that the graph superposition leads to the propagator being the average of the propagators of the lattice field theory over various graphs so that all fermion doubler modes are suppressed in the propagator. This resolves the doubling problem in LQG. Our result suggests that the superposition nature of quantum geometry should, on the one hand, resolve the tension between fermion and the fundamental discreteness and, on the other hand, relate to the continuum limit of quantum gravity. 

Abstract BackgroundRecent developments to segment and characterize the regions of interest (ROI) within medical images have led to promising shape analysis studies. However, the procedures to analyze the ROI are arbitrary and vary by study. A tool to translate the ROI to analyzable shape representations and features is greatly needed. ResultsWe developed SAFARI (shape analysis for AI-segmented images), an open-source package with a user-friendly online tool kit for ROI labelling and shape feature extraction of segmented maps, provided by AI-algorithms or manual segmentation. We demonstrated that half of the shape features extracted by SAFARI were significantly associated with survival outcomes in a case study on 143 consecutive patients with stage I–IV lung cancer and another case study on 61 glioblastoma patients. ConclusionsSAFARI is an efficient and easy-to-use toolkit for segmenting and analyzing ROI in medical images. It can be downloaded from the comprehensive R archive network (CRAN) and accessed athttps://lce.biohpc.swmed.edu/safari/.