Operational Locality: Couplings, Supports, and Protocols

A field-information protocol is operationally local only after every intervention has a spacetime support, every record has an accessible worldline, and the dependence of later operations on earlier records is declared. A pointlike detector label is insufficient: switching tails, spatial smearing, shared controls, and classical communication all contribute to the physical support.

Required background. Local regions and algebras supplies isotony and causal complements. Operator-valued distributions explains why fields must be smeared before they enter an interaction.

Helpful background. Local preparation and operational independence distinguishes algebraic independence from a realizable preparation protocol.

A protocol is a supported causal circuit

Consider a real scalar field and two probes $A$ and $B$. In an interaction picture, take

$$H_I(t)=\lambda_A\chi_A(t)\,\mu_A(t)\Phi(F_A,t) +\lambda_B\chi_B(t)\,\mu_B(t)\Phi(F_B,t),$$

with $\Phi(F_i,t)=\int d^d\mathbf x\,F_i(\mathbf x)\phi(t,\mathbf x)$. The spacetime test function $f_i(t,\mathbf x)=\chi_i(t)F_i(\mathbf x)$, not the nominal detector position, determines the coupling region $K_i=\operatorname{supp}f_i$. A complete protocol specifies the probe states, the coupling constants, the functions $f_i$, the probe observables read out, the readout regions, and any conditional control.

There are two distinct graphs. The geometric graph has an arrow $K_i\to K_j$ when $K_j$ meets the causal future of $K_i$. The control graph has an arrow when the choice at $j$ uses a record produced at $i$. A valid implementation requires each control arrow to be carried by an ordinary causal signal. Correlation between records creates neither kind of arrow.

Operational locality is checked stage by stage: support belongs to the interaction, the instrument belongs to the coupled probe and its readout, and the final claim belongs to accessible records. The diagram is schematic.

Two compact probes

Suppose $K_A$ and $K_B$ are compact and spacelike separated. Microcausality gives

$$[\Phi(f_A),\Phi(f_B)]=0,$$

and, for the corresponding localized scattering maps, causal factorization gives $\Theta_A\Theta_B=\Theta_B\Theta_A$. The two probe records may nevertheless be correlated because the initial field state need not factorize across the regions. Operational no signaling asks a counterfactual question: after averaging over $A$‘s uncommunicated outcome, can a change of $A$‘s instrument alter $B$‘s marginal distribution? For properly localized spacelike instruments, it cannot.

If $K_B$ lies to the future of $K_A$, the factorization becomes ordered rather than commutative. Then the induced channel from $A$‘s preparation choice to $B$‘s readout can be nontrivial. This is causal signaling, not a failure of relativistic locality.

A reproducible specification should record at least

$$(K_i,f_i,\lambda_i,H_{P_i},\sigma_i,E_i),$$

where $H_{P_i}$ is the free probe Hamiltonian, $\sigma_i$ its initial state, and $E_i$ the readout POVM. It should also state the perturbative order and a bound on numerical quadrature or truncation error.

Tails change the claim

A Gaussian $f_i$ has no compact support. It may be an excellent numerical approximation, but exact spacelike separation is then false. Choose compact core regions $K_i^{(R)}$ and decompose $f_i=f_i^{(R)}+\delta f_i^{(R)}$. A quantitative approximate-locality statement needs a norm or observable-specific bound on the effect of $\delta f_i^{(R)}$; merely drawing separated ellipses is not enough.

The same issue appears with conditional operations. If a rare outcome at $A$ is used to sort $B$‘s data only after a classical message arrives, the conditional correlation is operationally available in the joint future. Without that message, $B$ observes the unconditional marginal. Treating the inaccessible sorted ensemble as $B$‘s local state creates an apparent superluminal effect.

Support tails and postselection enter different branches of the analysis: tails weaken exact localization, whereas unavailable outcome sorting invalidates an operational signaling claim. The map is schematic.

The compact-coupling formulation and its causal factorization are developed in Fewster and Verch 2020, §§ 3–5. It also explains why induced observables can be localized in the causal hull of the interaction without assuming a tensor factor for the local field algebra. The apparatus-level obstruction to an idealized measurement with an impossible localization region is analyzed by Bostelmann, Fewster, and Ruep 2021, §§ II–IV.

Check your understanding

Let $K_A$ and $K_B$ be spacelike, but let a common clock controller in $J^-(K_A)\cap J^-(K_B)$ choose both switching functions. Does correlation between the settings demonstrate signaling from $A$ to $B$?

Solution

No. The common controller is a shared cause. To test signaling, vary $A$‘s local intervention while holding the preparation and $B$‘s intervention fixed, then compare $B$‘s unconditional statistics. Correlated settings may invalidate a Bell-style setting-independence assumption, but they do not create a causal arrow from $A$ to $B$.

References

• Bostelmann, H., Fewster, C. J., and Ruep, M. H. (2021). “Impossible Measurements Require Impossible Apparatus.” Physical Review D 103, 025017. DOI. Open PDF.
• Fewster, C. J., and Verch, R. (2020). “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378, 851–889. DOI. Open PDF.