The Role of Dilations in Diffeomorphism Covariant Algebraic Quantum Field Theory

The quantum analogue of general relativistic geometry should be implementable on smooth manifolds without an a priori metric structure, whence the kinematical covariance group acts by diffeomorphisms. Here I approach quantum gravity (QG) in the view of constructive, algebraic quantum field theory (QFT). Comparing QG with usual QFT, the algebraic approach elucidates the analogies and peculiarities. As usual, an isotonic net of ∗-algebras is taken to encode the quantum field operators. For QG, the kinematical covariance group acts via diffeomorphisms on the open sets of the manifold, and via algebraic isomorphisms on the algebras. The algebra of observables is only covariant under a (dynamical) subgroup of the general diffeomorphism group. After an algebraic implementation of the dynamical subgroup of dilations, small and large scale cutoffs may be introduced algebraically. So the usual a priori conflict of cutoffs with general covariance is avoided. Even more, these cutoffs provide a natural local cobordism for topological quantum field theory. A new commutant duality between the minimal and maximal algebra allows to extract the modular structure from the net of algebras. The outer modular isomorphisms are then again related to dilations, which (under certain conditions) may provide a notion of time. ∗Lecture given at MG8, Jerusalem 1997