A von Neumann algebraic approach to Quantum Theory on curved spacetime

Abstract

By extending the method developed in our recent paper \cite{LM} we present the AQFT framework in terms of von Neumann algebras. In particular, this approach allows for a locally covariant categorical description of AQFT which moreover satisfies the additivity property and provides a natural and intrinsic framework for a description of entanglement. Turning to dynamical aspects of QFT we show that Killing local flows may be lifted to the algebraic setting in curved space-time. Furthermore, conditions under which quantum Lie derivatives of such local flows exist are provided. The central question that then emerges is how such quantum local flows might be described in interesting representations. We show that quasi-free representations of Weyl algebras fit the presented framework perfectly. Finally, the problem of enlarging the set of observables is discussed. We point out the usefulness of Orlicz space techniques to encompass unbounded field operators. In particular, a well-defined framework within which one can manipulate such operators is necessary for the correct presentation of (semiclassical) Einstein's equation.