Dynamical Locality and Kinematic–Dynamic Nets

Dynamical locality asks whether two ways of assigning observables to a region agree. The kinematic algebra is obtained by applying the theory to that subspacetime. The dynamical algebra consists of observables unchanged by every metric perturbation in the region’s causal complement. Equality is a substantive theorem: it fails in theories with invisible zero modes or topological observables.

Required background. Isotony, Additivity, Duality, and Primitive Causality supplies regional subalgebras. Locally Covariant QFT as a Functor supplies kinematic inclusions. Time-Slice Axiom and Relative Cauchy Evolution supplies background perturbations.

Helpful background. Topology, Zero Modes, and Global Sectors prepares the obstructions.

Kinematic and dynamical subobjects

Let $O$ be a nonempty causally convex globally hyperbolic open subset of $M$, with inclusion $\iota_O:M|_O\to M$. The kinematic subobject is

$$\mathcal A^{\mathrm{kin}}(M;O) =\operatorname{im}\mathcal A(\iota_O).$$

To define the dynamical alternative, fix a compact set $K\subset M$. Let $\mathscr H(M;K^\perp)$ be the admissible smooth metric perturbations supported in the causal complement $K^\perp=M\setminus J_M(K)$. In a target category admitting the required equalizers, set

$$\mathcal A^\bullet(M;K) =\bigcap_{h\in\mathscr H(M;K^\perp)} \operatorname{Eq}\!\left(\operatorname{rce}_M[h],\mathrm{id}\right).$$

Thus an element belongs to $\mathcal A^\bullet(M;K)$ exactly when all exterior metric perturbations fix it. The dynamical subobject $\mathcal A^{\mathrm{dyn}}(M;O)$ is generated by the $\mathcal A^\bullet(M;K)$ for suitable compact $K\subset O$ possessing neighborhoods with controlled Cauchy development. The categorical definition uses unions of subobjects rather than an informal set union Fewster and Verch 2012, § 5, pp. 29–35.

A theory is dynamically local when it obeys time-slice and the canonical inclusions of $\mathcal A^{\mathrm{kin}}(M;O)$ and $\mathcal A^{\mathrm{dyn}}(M;O)$ into $\mathcal A(M)$ are isomorphic as subobjects for every object $M$ and every allowed nonempty $O$ Fewster and Verch 2012, Definition 6.2, p. 35. The statement is equality inside the theory, not merely equality of dimensions or abstract isomorphism types.

Massive scalar theorem and zero-mode failure

For the real Klein–Gordon field with $m>0$, exterior-rce invariance forces the associated classical solution to be localizable in the causal hull of $K$. Energy estimates and the stress–energy response identify the dynamically invariant subspace with the kinematic one. The classical result is Fewster and Verch 2012, Theorem 3.4, p. 12; their quantization arguments yield dynamical locality for the corresponding infinitesimal Weyl and Weyl theories under the stated weak nondegeneracy and categorical preservation hypotheses Fewster and Verch 2012, Theorem 5.3 and Theorem 6.2, pp. 23–26.

The minimally coupled massless scalar fails. Locally constant solutions are unchanged by every compact metric perturbation, so they enter dynamical subobjects even when they cannot be generated kinematically in a small region Fewster and Verch 2012, Theorem 3.6, pp. 12–13. Passing to the massless current—quotienting the constant shift gauge freedom—restores dynamical locality on connected spacetimes in dimensions $n\geq2$ and on the multi-component category for $n\geq3$, but not in the exceptional two-dimensional multi-component case Fewster and Verch 2012, Theorem 4.3, p. 17.

This gives a precise comparison with Massless Zero Modes and the Invariant-State Obstruction: the locally constant mode is not a harmless bookkeeping term. It is invariant under exterior dynamics and defeats kinematic–dynamic equality unless the theory’s gauge quotient removes it.

Topological obstruction and independent checks

The same adversarial pattern occurs for the universal Maxwell theory. Cohomological flux observables can be fixed by all exterior metric perturbations yet encode global topology. Assigning such an observable to every small region makes the dynamical net too large. For the universal theory, injectivity and dynamical locality fail; a reduced theory removes the radical/topological observables and becomes dynamically local Fewster and Lang 2016, Theorem 5.6, p. 16, and Theorem 6.5 and Corollary 6.6, pp. 20–21. The cost is real: the reduced theory no longer retains those topological charges.

Two checks are independent of the proof. If $O_1\subset O_2$, both nets must be isotonic. If a Cauchy morphism transports $O$ and all allowed compact sets, both nets must transform coherently. A proposed equality that is true only after choosing coordinates, or that retains a global observable in every arbitrarily small region, fails these tests.

The compact-set definition is essential. Requiring invariance only under perturbations outside the open set $O$ can miss the way causal propagation thickens supports, while taking all closed subsets without regular neighborhoods can destroy the union construction. The admissible compact family used by Fewster and Verch supplies multi-diamonds whose causal hulls remain inside $O$. Additivity over that family then compares the two subobjects at the correct localization scale.

Nor is dynamical locality the same as Einstein causality. Causality compares algebras of two disjoint regions; dynamical locality compares two constructions of the algebra of one region. A theory may satisfy causal commutation and still fail dynamically because a zero mode is assigned everywhere.

Exercise

Why does an element fixed by every $\operatorname{rce}_M[h]$ for perturbations outside all compact $K\subset O$ not automatically belong to the kinematic algebra of $O$?

Solution

Exterior invariance is only the definition of dynamical localization. A global zero mode or topological charge can be insensitive to every local metric perturbation without being generated from fields supported in $O$. Equality with the kinematic algebra is therefore the content of dynamical locality, and the massless scalar and universal Maxwell theory furnish counterexamples when the relevant modes are retained.

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. “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.