The generally covariant locality principle -- A new paradigm for local quantum physics
Romeo Brunetti, Klaus Fredenhagen, Rainer Verch
TL;DR
This work reframes quantum field theory as a locally covariant theory, formulating QFT as a covariant functor from the category of globally hyperbolic spacetimes to a category of algebras, thereby encoding locality and general covariance at a structural level. It provides a concrete Klein-Gordon field example, showing how Weyl algebras arise from a spacetime-dependent symplectic space and how causality and the time-slice axiom are preserved in this functorial setting. The paper further develops states via microlocal (Hadamard) conditions, introduces relative Cauchy-evolution to describe metric-induced dynamics, and outlines a covariant, cohomological construction of Wick polynomials, all within a framework that naturally recovers traditional algebraic QFT on a fixed spacetime. The formalism yields a robust notion of local definiteness through invariant state foliations and offers a principled path to covariant perturbative and nonperturbative constructions in curved spacetimes, with the Klein-Gordon field as a guiding exemplar.
Abstract
A new approach to the model-independent description of quantum field theories will be introduced in the present work. The main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity, thus giving rise to the concept of a "locally covariant quantum field theory". Such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories, on one side, of globally hyperbolic spacetimes with isometric embeddings as morphisms and, on the other side, of *-algebras with unital injective *-endomorphisms as morphisms. Moreover, locally covariant quantum fields can be described in this framework as natural transformations between certain functors. The usual Haag-Kastler framework of nets of operator-algebras over a fixed spacetime background-manifold, together with covariant automorphic actions of the isometry-group of the background spacetime, can be re-gained from this new approach as a special case. Examples of this new approach are also outlined. In case that a locally covariant quantum field theory obeys the time-slice axiom, one can naturally associate to it certain automorphic actions, called ``relative Cauchy-evolutions'', which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics. The functional derivative of a relative Cauchy-evolution with respect to the spacetime metric is found to be a divergence-free quantity which has, as will be demonstrated in an example, the significance of an energy-momentum tensor for the locally covariant quantum field theory. Furthermore, we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness.