A Modular Operator Approach to Entanglement of Causally Closed Regions

TL;DR

This paper explores quantum entanglement in causally separated regions using conformal quantum mechanics, von Neumann algebras, and modular operators, revealing new links between thermal time flow and spacetime emergence.

Contribution

It introduces a novel approach connecting entanglement in causal diamonds with conformal quantum mechanics and modular theory, deriving a holographic entropy formula.

Findings

Entanglement between causally separated regions demonstrated via modular operators.

A holographic entropy formula for conformal quantum mechanics causal diamonds derived.

Thermal time flow linked to physical time flow, supporting spacetime emergence hypothesis.

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.