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absence of transverse algebraic structure ambient apex area density asymptotic automorphism group BMS group Bondi-Metzner-Sachs boundary algebras bulk algebra cast a causal causal disjoint causal horizon causal shadow cast causal shadow property change of spacetime characteristic data classical setting D-supported test-function spaces diffeomorphisms double cone algebra double cone holography e-dependent extended chiral theory finite formula forward lightcone free scalar field fuzzy modular groups act geometrically holographic encoding holographic image holographic lightcone holographic projection horizon algebra horizon dD hypersurface identical lightfront restricted lightlike lightray direction limit localization entropy lower horizon mantle Metzner-Sachs modular group acts modular Hamiltonian modular inclusion modular localization Moebius non-geometric null null-surface Poincare group pointlike field pseudo-differential Publicações Rua Dr quantization quantum mechanical quantum physics region result Rio de Janeiro spacelike spacetime encoding spacetime indexing subalgebras subregions substructure of dD subsymmetries symmetry group tensor-factorization theorem Unruh vacuum fluctuations vacuum polarization wedge algebra wedge situation wedge-localized Xavier Sigaud