The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory

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.

Quantum Fields in Curved Spacetime

“…The Hadamard condition requires A k → 1, B k → 0 at large k. If we assume that A k ≈ 1, B k ≈ 0 for all "short wavelength modes" (i.e., k/a(t 1 ) > H 0 ) at some time t 1 < t 0 during the inflationary era, then we have A k ≈ 1, B k ≈ 0 for all modes 19 whose physical wavelength is smaller than H −1 0 exp[(t 0 − t 1 )H 0 ] at the end of inflation. For (t 0 − t 1 )H 0 60, this encompasses all wavelengths relevant for cosmology.…”

“…However, modes whose wavelength was larger than the Hubble radius at the end of inflation but is smaller than the Hubble radius in the present universe are highly relevant to cosmology. 19 For m 2 > 0, the analogous modes are obtained by setting the index of the Hankel function to…”

“…To define the local and covariant requirement on Q, we first note that our construction of W (M , g) is local and covariant in the following sense [19]: Let (M , g) and (M , g ) be globally hyperbolic spacetimes and suppose that there is a causality preserving isometric embedding ψ : M → M . Then there exists a corresponding canonical homomorphism α(ψ) :…”

Quantum Field Theory: Where We Are

“…It is realized as a covariant functor which associates to every globally hyperbolic spacetime its algebra of observables and which maps isometric embeddings of spacetimes to homomorphic embeddings of algebras. The interpretation of the theory is done in terms of covariant fields, which are mathematically defined as natural transformations from a geometrically defined functor which associates to every spacetime its test function space to the functor describing the quantum field theory [14].…”

Action Ward Identity and the Stückelberg-Petermann Renormalization Group

“…On the other side, the path integral approach is intrinsically nonlocal; therefore difficulties arise for the treatment of infrared problems and of finite temperatures, and theories on curved back grounds or with external fields cannot properly be formulated. In algebraic quantum field theory, concepts for dealing with these problems have been developed [13,2,4,14,5,17]. We therefore started a program for a perturbative construction of the net of local algebras of observables [4,7].…”