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Quantum Field Theory on Curved Noncommutative Spacetimes

We summarize our recently proposed approach to quantum field theory on noncommutative curved spacetimes. We make use of the Drinfel'd twist deformed differential geometry of Julius Wess and his group in order to define an action functional for a real scalar field on a twist-deformed time-oriented, connected and globally hyperbolic Lorentzian manifold. The corresponding deformed wave operator admits unique deformed retarded and advanced Green's operators, provided we pose a support condition on the deformation. The solution space of the deformed wave equation is constructed explicitly and can be canonically equipped with a (weak) symplectic structure. The quantization of the solution space of the deformed wave equation is performed using *-algebras over the ring C[[λ]]. As a new result we add a proof that there exist symplectic isomorphisms between the deformed and the undeformed symplectic R[[λ]]-modules. This immediately leads to *-algebra isomorphisms between the deformed and the formal power series extension of the undeformed quantum field theory. The consequences of these isomorphisms are discussed.