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absence of transverse algebra acts ambient area density area law behavior Bekenstein black hole causal horizon chiral algebra chiral theory commutation relation conceptual conformal invariance conformal theories covariance derivation double cone field theoretic finite number FÍSICA formula free field geometric global algebra global heat bath Hamiltonian heat bath thermal Hilbert space holographic projection interpretation KMS state associated lightfront lightfront algebra lightfront projection lightlike lightray localization entropy localization-entropy Longo Math modular group modular Hamiltonian modular inclusions modular localization modular theory Moebius monade Neumann operator algebra parametrization partial charge particle partition function Phys quantum mechanical quantum physics relativistic box Schroer short distance spacetime split property standard heat bath subalgebra symmetry temperature duality relation tensor tensor-factorizes theorem thermal aspects thermal manifestations thermodynamic limit thermodynamic limit sequence transverse direction transverse vacuum transversely extended chiral unitary vacuum fluctuations vacuum polarization vacuum representation vector volume factor wedge algebra wedge region wedge-localization Wiesbrock