Abstract
This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using stacks of categories, this proposal is realized concretely for the simplest case of 1-dimensional spacetimes, leading to a stack of smooth 1-dimensional AQFTs. Concrete examples of smooth AQFTs, of smooth families of smooth AQFTs and of equivariant smooth AQFTs are constructed. The main open problems that arise in upgrading this approach to higher dimensions and gauge theories are identified and discussed.
Highlights
Introduction and SummaryAn m-dimensional algebraic quantum field theory (AQFT) is a functor A : Locm → ∗AlgC from a suitable category of m-dimensional Lorentzian spacetimes to the category of associative and unital ∗-algebras over C
The algebra A(M ) that is assigned by this functor to a spacetime M is interpreted as the algebra of quantum observables of the theory A that can be measured in M. Such functors are required to satisfy a list of physically motivated axioms, see, e.g., [10,15], which includes most notably the Einstein causality axiom. Even though this axiomatic definition of AQFTs is widely used in the relevant research community and has led to interesting model-independent results, we would like to point out the following issue that is usually not discussed: Suppose that we consider a family of spacetimes {Ms ∈ Locm}s∈R that depends smoothly on a parameter s ∈ R
Evaluating an AQFT A : Locm → ∗AlgC on this smooth family results in a family of algebras {A(Ms) ∈ ∗AlgC}s∈R which, will in general not be smooth in any appropriate sense because smoothness is not covered by the usual AQFT axioms
Summary
An m-dimensional algebraic quantum field theory (AQFT) is a functor A : Locm → ∗AlgC from a suitable category of m-dimensional Lorentzian spacetimes to the category of associative and unital ∗-algebras over C. Such functors are required to satisfy a list of physically motivated axioms, see, e.g., [10,15], which includes most notably the Einstein causality axiom Even though this axiomatic definition of AQFTs is widely used in the relevant research community and has led to interesting model-independent results, we would like to point out the following issue that is usually not discussed: Suppose that we consider a family of spacetimes {Ms ∈ Locm}s∈R that depends smoothly (in some appropriate sense as explained in this paper) on a parameter s ∈ R. These are later used for constructing explicit examples, which illustrate our proposed approach to smooth AQFT. Open Problem 6.1 poses the question of existence of smoothly parameterized retarded/advanced Green operators for vertical normally hyperbolic operators on smooth families of Lorentzian spacetimes, which goes beyond the standard results developed, e.g., in [1] and might be of interest to researchers in hyperbolic PDE theory
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