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The metric field of general relativity is almost fully determined by its causal structure. Yet, in spin foam models of quantum gravity, the role played by the causal structure is still largely unexplored. The goal of this paper is to clarify how causality is encoded in such models. The quest unveils the physical meaning of the orientation of the two-complex and its role as a dynamical variable. We propose a causal version of the EPRL spin foam model and discuss the role of the causal structure in the reconstruction of a semiclassical space–time geometry. 

We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-Abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-Abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-Abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the Abelian case, new phenomena arise from the interplay of locality and non-Abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2) symmetry. 

We show that bosonic and fermionic Gaussian states (also known as``squeezed coherent states’’) can be uniquely characterized by theirlinear complex structure J J which is a linear map on the classical phase space. This extendsconventional Gaussian methods based on covariance matrices and providesa unified framework to treat bosons and fermions simultaneously. PureGaussian states can be identified with the triple (G,\Omega,J) ( G , Ω , J ) of compatible Kähler structures, consisting of a positive definitemetric G G ,a symplectic form \Omega Ω and a linear complex structure J J with J^2=-\mathbb{1} J 2 = − 1 .Mixed Gaussian states can also be identified with such a triple, butwith J^2\neq -\mathbb{1} J 2 ≠ − 1 .We apply these methods to show how computations involving Gaussianstates can be reduced to algebraic operations of these objects, leadingto many known and some unknown identities. We apply these methods to thestudy of (A) entanglement and complexity, (B) dynamics of stablesystems, (C) dynamics of driven systems. From this, we compile acomprehensive list of mathematical structures and formulas to comparebosonic and fermionic Gaussian states side-by-side.