Dynamical Locality and Theory Comparison

Dynamical locality asks whether an observable belongs to a region for an operational reason: does it remain unchanged under every admissible change of the metric made causally outside that region? The answer can agree with the usual embedding-based local net, but agreement is a substantive property of a theory, not another axiom written in different notation. It separates genuinely local degrees of freedom from global zero modes and topological charges that a kinematic net may inadvertently carry into every laboratory.

Required background. Flat-Spacetime Quantum Field Theory provides the algebraic meaning of fields, commutators, and local observables used below.

Helpful background. Partial Differential Equations and Green Functions supplies the causal propagation picture behind relative Cauchy evolution. Causality and Global Structure develops causal complements and globally hyperbolic subregions. Quantum Fields in Curved Spacetime provides the volume-level map linking local covariance, admissible states, and renormalized observables.

Two candidates for the local algebra

Let $\mathcal A:\mathsf{Loc}\to\mathsf{Alg}$ be a locally covariant theory and let $\iota_O:M|_O\to M$ embed a causally convex globally hyperbolic region. The kinematic algebra is simply the image

$$\mathcal A^{\mathrm{kin}}(M;O) =\mathcal A(\iota_O)\bigl(\mathcal A(M|_O)\bigr).$$

This definition is categorical: an observable is local because it came from the theory assigned to the smaller spacetime. It is the net used in local Einstein causality and behaves correctly for familiar free fields. Yet it does not by itself test whether the purported observable can detect geometric information outside $O$.

The dynamical construction makes that test. Choose a compact set $K\subset O$ and write $K^\perp=M\setminus J_M(K)$ for its causal complement. With the convention for relative Cauchy evolution, define

$$\mathcal A^\bullet(M;K) =\left\{A\in\mathcal A(M): \operatorname{rce}_M[h](A)=A \text{ for every admissible }h \text{ with }\operatorname{supp}h\subset K^\perp\right\}.$$

The dynamical algebra $\mathcal A^{\mathrm{dyn}}(M;O)$ is generated by these invariant subalgebras as $K$ ranges over suitable compact subsets of $O$. Thus an element is dynamically localized in $O$ when perturbing the geometry outside the causal influence of a compact core in $O$ cannot change it. Precise definitions use compact sets with multi-diamond neighborhoods so that the construction respects the causal and smooth structure; Fewster and Verch 2012, §§5–6 give the full formulation.

Equality as a theory-level criterion

A theory is dynamically local if

$$\mathcal A^{\mathrm{kin}}(M;O) =\mathcal A^{\mathrm{dyn}}(M;O)$$

for every admissible spacetime $M$ and every suitable region $O$. The two inclusions express different claims. Kinematic-to-dynamical inclusion says that observables constructed inside $O$ are insensitive to metric changes causally outside it. The reverse inclusion says that this response criterion does not hide additional, globally supplied observables inside $O$.

Einstein causality and the time-slice axiom do not imply this equality. A concrete counterexample is the massless minimally coupled scalar on a spacetime with a compact Cauchy surface. A nonzero constant solution has zero stress tensor and is unchanged by every compact metric perturbation. It therefore belongs to every response-defined subspace. Yet it cannot come from a sufficiently small diamond: a solution generated there has Cauchy data vanishing outside that diamond’s causal hull, whereas the constant solution does not. This produces an actual mismatch between the two localization rules. Conversely, equality does not say that every observable is a polynomial in a point field, nor does it choose a state or representation.

The comparison supports a rigorous version of the “same physics in all spacetimes” principle. If both theories are dynamically local, a natural transformation embedding one into the other that is an isomorphism on one spacetime is an isomorphism on every spacetime. The component maps must be theory morphisms in the stated category, not unrelated algebra isomorphisms. This is the SPASs result of Fewster and Verch 2012, Theorem 6.10, p. 42, PDF. Dropping dynamical locality permits the paper’s diagonal theories, whose field content changes with a global property of $M$.

Evidence through 2026-08-19

The evidence cutoff for this assessment is 2026-08-19. Results depend on the field content, the algebra chosen to represent it, and whether zero modes or topological sectors are retained. The matrix therefore compares formulations rather than attaching a single verdict to a classical equation.

Theory or position	Evidence and result	Scope, assumptions, or adverse case
Massive minimally coupled Klein–Gordon field	The kinematic and dynamical nets agree, giving a model of dynamical locality and a clean response interpretation.	The cited theorem fixes $m\gt0$ and $\xi=0$; it is not a claim for every curvature coupling or matrix-valued potential.
Massless minimally coupled scalar	The naive field algebra fails the criterion on spacetimes with compact Cauchy surfaces because locally constant solutions behave as global zero modes.	The obstruction depends on algebra and geometry; passing to the massless current or quotienting the shift symmetry changes the theory being compared.
Universal Maxwell theory	Cohomological electric and magnetic fluxes produce a nontrivial radical and spoil dynamical locality on general spacetime categories.	The adverse case is topological rather than a failure of Maxwell hyperbolicity; restricting topology would conceal rather than resolve it.
Reduced Maxwell formulation	Quotienting the topological radical gives dynamically local classical and quantum theories in the stated four-dimensional globally hyperbolic category.	The price is deliberate exclusion of global flux observables, so equivalence with the universal theory does not extend to every observable sector.

The scalar results are established in Fewster and Verch 2012, free scalar field, §§3–6. The Maxwell comparison and its cohomological obstruction are developed in Fewster and Lang 2016, §§3–6, especially Theorem 6.5 and Corollary 6.6. Universal Maxwell theory has an additional problem on the full category: embeddings can erase topological classes, so the induced maps need not be injective. The reduced theory repairs this as well as dynamical locality. Retaining or quotienting global fluxes changes the observable content; the two formulations must not be called equivalent on all sectors.

What the criterion does not establish

Dynamical locality does not prove that a theory exists nonperturbatively, that its preferred state is known, or that its local von Neumann algebras have a particular type. It is formulated at the level of a locally covariant algebraic theory and its response to compactly supported metric perturbations. Boundary conditions, nonglobally hyperbolic geometry, gauge edge modes, and extended operators require additional data before the response test is even well defined.

Nor does the criterion yield an unconditional ban on natural states. Assume dynamical locality, extended locality, and a natural state whose GNS representation is faithful and Reeh–Schlieder on a spacetime with noncompact Cauchy surfaces. The theorem then forces the theory to be trivial; the conclusion and hypotheses appear in Fewster and Verch 2012, Definition 6.12 and Theorem 6.13, pp. 43–44, PDF. Symmetric individual backgrounds may still admit ground, KMS, or invariant states; the theorem concerns a single compatible assignment across the entire category.

An unresolved question is how broadly the response-defined idea should be extended when the observable algebra includes generalized symmetries, gauge boundaries, defects, or nonperturbative interacting sectors. Evidence does not establish one universal algebra class for those cases. The regime axes—topology, boundary condition, mass, gauge reduction, and observable class—must remain visible in any claim of locality.

What could change these conclusions

The conclusions could change if a new theorem proves dynamical locality, or a no-go result proves its failure, for an interacting or gauge model with explicitly stated algebra class and spacetime category. A new counterexample involving boundaries, edge modes, generalized symmetries, or a different state-independent response construction would also change the map. Independent calculations that show a purported topological obstruction disappears without quotienting physical charges would change the interpretation, as would a proof that reduced and universal formulations have equivalent observable content in a category where they are presently distinct.

Exercise

Suppose $A\in\mathcal A(M)$ is fixed by $\operatorname{rce}_M[h]$ for every admissible perturbation supported in $K^\perp$, where $K\subset O$. What localization statement follows directly, and what additional fact is required before one may call $A$ kinematically local in $O$?

Solution

The invariance condition places $A$ in $\mathcal A^\bullet(M;K)$ and therefore in the algebra generated by all such compact cores inside $O$, namely $\mathcal A^{\mathrm{dyn}}(M;O)$. It does not by itself show that $A$ lies in the image of $\mathcal A(M|_O)$ under the inclusion morphism. That second conclusion requires the dynamical-locality equality for this theory, spacetime, and region. The distinction is exactly why response invariance is a test rather than a renamed kinematic definition.

References

• Fewster, Christopher J., and Benjamin Lang. “Dynamical Locality of the Free Maxwell Field.” Annales Henri Poincaré 17 (2016): 401–436. DOI. Open PDF.
• Fewster, Christopher J., and Rainer Verch. “Algebraic Quantum Field Theory in Curved Spacetimes.” In Advances in Algebraic Quantum Field Theory, 125–189. Springer, 2015. 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.
• Fewster, Christopher J., and Rainer Verch. “Dynamical Locality of the Free Scalar Field.” Annales Henri Poincaré 13 (2012): 1675–1709. DOI. Open PDF.