Locally Covariant QFT and Dynamical Locality

Locally covariant QFT describes one theory across a category of spacetimes. Its central question is whether transporting observables, changing the background, and identifying local physics remain mutually consistent.

Enter this chapter

Start with the spacetime category and functor. Introduce relative Cauchy evolution only after understanding why time-slice maps are invertible. Helpful background. Local Covariance, Isometries, and Boundaries previews covariance across backgrounds. Relative Cauchy Evolution and Background Response supplies the physical response problem. Local Field Algebras, Causality, and the Time-Slice Property supplies the fixed-spacetime algebraic setting.

Topics

1. Globally Hyperbolic Spacetimes and the Loc Categories — Defines admissible spacetime objects and causally compatible embeddings.
2. Locally Covariant QFT as a Functor — Makes the algebra assignment respect identities and composition.
3. Natural Transformations, Fields, and Subtheory Embeddings — Expresses fields and comparisons of theories as natural transformations.
4. Time-Slice Axiom and Relative Cauchy Evolution — Constructs the response automorphism using Cauchy isomorphisms.
5. Stress–Energy Response and Background Variation — Relates differentiable background response to stress–energy generators.
6. Dynamical Locality and Kinematic–Dynamic Nets — Compares geometric localization with insensitivity to exterior perturbations.
7. The Same-Physics-in-All-Spacetimes Principle — States when agreement propagates coherently across backgrounds.
8. No-Natural-State Results and Covariant State Spaces — Separates a covariant family of admissible states from a preferred state.
9. Local Covariance with Boundaries and Background Structures — Records which extra structures objects and embeddings must preserve.
10. LCQFT, Nets, Fields, and Factorization: Typed Comparisons — Identifies directional comparisons and the hypotheses needed for converses.

How the ideas fit together

Functoriality transports observables; Einstein causality constrains spacelike images; time-slice controls dynamics. These independent requirements underlie relative Cauchy evolution Brunetti, Fredenhagen, and Verch 2003, §§ 2.2, 2.5, and 4 (PDF). Dynamical locality adds equality between localization by embeddings and localization by response to exterior changes. Its same-physics and natural-state results require their own hypotheses Fewster and Verch 2012, §§ 5–6 (PDF). Agreement of algebras on Minkowski space alone does not provide a natural equivalence.

Hypotheses and licensed conclusions

Domains, hypotheses, directional conclusions, and failure tests in locally covariant QFT

Object and domain	Required hypotheses	Licensed conclusion	Excluded converse or upgrade	Adversarial check
Embedding between oriented, time-oriented globally hyperbolic spacetimes	Same dimension; isometric; orientation- and time-orientation-preserving; causally convex open image	A Loc morphism that preserves causal propagation and composes with other admissible embeddings	An arbitrary isometric open embedding is not admissible, and a Loc morphism need not be Cauchy	Let a target causal curve leave and re-enter the image and test Green-operator intertwining
Functor $\mathcal A:\mathsf{Loc}\to\mathsf{Phys}$	Identity and composition; injective $*$-maps; Einstein causality and time-slice imposed separately	Coherent observable transport, spacelike commutation, and isomorphisms for Cauchy morphisms	Functoriality alone supplies neither causality nor time-slice, and no preferred state follows	Use a non-causally-convex embedding and compare the transported Klein–Gordon commutator
Field or theory transformation	A component on every object and a commuting naturality square for every Loc morphism	A locally covariant field, subtheory embedding, or natural equivalence with a fixed direction	Objectwise formulas or one component isomorphism do not form a natural transformation or equivalence	Choose curvature-renormalization constants independently on embedded spacetimes
Compact metric perturbation and its represented derivative	Both metrics globally hyperbolic; four Cauchy maps; time-slice; for the derivative, a Hadamard representation, common domain, and differentiability	A relative-Cauchy automorphism; in the free scalar theorem, weak generation by the smeared renormalized stress tensor	Finite rce does not imply a norm derivative or an abstract inner derivation	Pass through a nonglobally hyperbolic metric, or add a nonconserved curvature term and test a compact diffeomorphism
Kinematic and dynamical regional subobjects	Time-slice, admissible exterior metric perturbations, and target equalizers/unions; model-specific support theorem	Equality for the massive scalar and stated reduced gauge theories; regional localization is recovered dynamically	Exterior invariance alone does not prove kinematic localization	Retain a locally constant massless mode or a universal Maxwell flux observable
Natural transformation between dynamically local theories	Dynamical locality for both theories and one component isomorphism; categorical subobject hypotheses	The transformation is a natural isomorphism on all spacetimes	One-spacetime agreement between unrestricted LCQFTs does not imply same physics everywhere	Construct a diagonal theory whose species number changes with the spacetime class
Natural state or boundary-enriched theory	For the no-go theorem: dynamical locality, faithful Reeh–Schlieder GNS on a noncompact-Cauchy spacetime, and extended locality; for boundaries: fixed well-posed boundary data preserved by arrows	The natural state forces trivial rce and, with the full hypotheses, a trivial theory; a fixed-boundary propagator may define a restricted functor	No-natural-state does not mean no Hadamard states, and a fixed Robin construction is not generic boundary covariance	Confuse a covariant state space with one natural section, or change the Robin parameter while keeping the same propagator map

Structured table data (JSON).

Review the chapter

Does a nonempty Hadamard state space on every spacetime determine a preferred natural state?

Solution

No. Nonemptiness supplies choices separately. A natural state must also be compatible with every admissible embedding. The latter requirement is much stronger and can conflict with nontrivial dynamics under the no-natural-state theorem’s additional assumptions.

Where to go next

Continue to thermal AQFT for state selection relative to a chosen dynamics, or microlocal QFT for Hadamard regularity.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Physics.” Communications in Mathematical Physics 237 (2003): 31–68. DOI; Open PDF.
• Fewster, Christopher J., and Rainer Verch. “Dynamical Locality and Covariance: What Makes a Physical Theory the Same in All Spacetimes?” Annales Henri Poincaré 13 (2012): 1613–1674. DOI; Open PDF.