Problems of dynamics in generally covariant quantum field theory

Problems connected with the structural aspects of dynamics are addressed in the context of the algebraic approach to generally covariant quantum field theory. It is argued that the dynamical structure of observables in the generally covariant context becomes fundamentally state dependent. This makes it necessary to relate the entire dynamics to state-dependent automorphisms of the algebra of observables. The relevant states are highly correlated on large scales, so that we may not have exact accuracy for the identification of their observables in terms of a (quasi) local net of algebras. This feature is controlled by a scale fluctuation of the total observables around a point which is used to obtain a description of a one-parameter group of state-dependent automorphisms in terms of the modular group. In general, it is not clear whether the action of the latter group has a dynamical interpretation. We comment on a duality principle which could provide a straightforward means to obtain an ''asymptotic'' interpretation of the modular group on small scales.