The generally covariant locality principle – a new paradigm for local quantum field theory.

*(English)*Zbl 1047.81052In the paper under review a new general approach to quantum field theory based on covariance functors is introduced. More specifically, a locally covariant quantum filed theory is defined as a covariant functor between the category of globally hyperbolic, oriented and time oriented, four-dimensional manifolds (spacetimes) with morphisms given by isometric embeddings respecting the orientations and the category of unital \(C^\ast\)-algebras with morphisms given by faithful unital \(\ast\)-homomorphisms. It is shown that the free Klein-Gordon field theory as well as the Haag-Kastler algebraic quantum field theory are special cases of this approach.

As a next step, a state space of the quantum field theory is defined by a “dual” contravariant functor between the category of spacetimes and the category of convex, transformation invariant subsets of state spaces of the corresponding \(C^\ast\)-algebras. State spaces which have the property that their “local folia” are invariant under the factorial actions of isometric embeddings of spacetime manifolds are characterized. State spaces of this kind fulfill the condition of local definiteness. It is shown that the Hadamard condition induces such a state space.

It is further proved that the quantum field theories with time-slice axiom enjoy special dynamics in the form of automorphic actions which describe the reaction of the quantum field theory on local perturbations of the spacetime metric. It is shown that the functional derivative of such evolution is divergence-free.

Finally, it is shown that the construction of locally covariant Wick-polynomials by Hollands and Wald provides a solution to a cohomological problem.

As a next step, a state space of the quantum field theory is defined by a “dual” contravariant functor between the category of spacetimes and the category of convex, transformation invariant subsets of state spaces of the corresponding \(C^\ast\)-algebras. State spaces which have the property that their “local folia” are invariant under the factorial actions of isometric embeddings of spacetime manifolds are characterized. State spaces of this kind fulfill the condition of local definiteness. It is shown that the Hadamard condition induces such a state space.

It is further proved that the quantum field theories with time-slice axiom enjoy special dynamics in the form of automorphic actions which describe the reaction of the quantum field theory on local perturbations of the spacetime metric. It is shown that the functional derivative of such evolution is divergence-free.

Finally, it is shown that the construction of locally covariant Wick-polynomials by Hollands and Wald provides a solution to a cohomological problem.

Reviewer: Jan Hamhalter (Praha)

##### MSC:

81T05 | Axiomatic quantum field theory; operator algebras |

81T20 | Quantum field theory on curved space or space-time backgrounds |