A Modular Operator Approach to Entanglement of Causally Closed Regions

DOI10.1007/S10773-022-05211-WarXiv2201.01898WikidataQ114226192 ScholiaQ114226192MaRDI QIDQ6387605

Publication date: 5 January 2022

Abstract: Quantum entanglement is shown for causally separated regions along the radial direction by using a conformal quantum mechanical correspondence with conformal radial Killing fields of causal diamonds in Minkowski space. In particular, the theory of local von Neumann algebras and Tomita Takesaki modular operators is applied in the entanglement structure of causal diamonds in conformal quantum mechanics. The entanglement of local states in their respective causal regions is shown through the measures of concurrence and entanglement entropy using the Tomita Takesaki modular conjugation operator. A holographic entropy formula is derived for the conformal quantum mechanics causal diamond correspondence. A new connection is made between the thermal time flow defined by the modular group of automorphisms to the physical time flow in a causal diamond via the aforementioned correspondence. The thermal interpretation of these results via two-point thermal Green's functions and modular group flow supports the idea of a possible emergent theory of spacetime.

Mathematics Subject Classification ID

Finite automorphism groups of algebraic, geometric, or combinatorial structures (20B25) Relativistic cosmology (83F05) General theory of von Neumann algebras (46L10) Conformal metrics (hyperbolic, Poincaré, distance functions) (30F45) Modular correspondences, etc. (11F32) Minkowski geometries in nonlinear incidence geometry (51B20) Quantum coherence, entanglement, quantum correlations (81P40) Causal inference from observational studies (62D20) Entanglement measures, concurrencies, separability criteria (81P42)

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