A note on Haag duality

Haag duality is a remarkable property in QFT stating that the commutant of algebra observables localized some region spacetime exactly associated to causally disconnected region. It strong condition on local structure and has direct consequences entanglement measures. was first shown hold for free scalar field causal diamonds by Araki 1964 later many authors different ways. In particular, Eckmann Osterwalder (EO) used Tomita-Takesaki modular theory give proof. However, it not straightforward relate this proof works Araki, since they rely two forms canonical commutation relations (CCR), called Segal Weyl formulations, while EO work as starting point assumes holds so-called “first quantization” formulation. our purpose introduce more easy-to-read but still rigorous self-contained fashion, show how stated formulations both second quantizations (and their immediate combination). This permits understand setting duality. There nothing essentially new manuscript, with exception what we consider simplification uses adjoint S⁎ Tomita operator S instead introducing several auxiliary operators. We hope note will be useful those seeking where comes from QFT.