Operational Locality: Couplings, Supports, and Protocols

A field measurement is operationally local only after the interaction, readout, and use of the record have all been placed in spacetime. Saying that a probe is “at $\mathbf x_A$” omits the very information that controls causality: its spatial extent, switching duration, tails, internal propagation, and any later classical communication. This page turns those data into a causal protocol and checks the resulting no-signaling claim.

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.

Chapter map. From a local coupling to a field instrument follows the complete measurement chain. The claim-validity table records which hypotheses license each conclusion, while three independent validity questions separates mathematical definition, causal implementation, and empirical interpretation.

Coupling regions, records, and two different graphs

Let a real system field $\Phi$ interact with two independent probe fields $\Psi_A$ and $\Psi_B$. A manifestly local model is

$$S_{\mathrm{int}} =-\sum_{i=A,B}\lambda_i\int d^4x\, \rho_i(x)\Phi(x)\Psi_i(x),$$

where each real $\rho_i\in C_0^\infty(\mathbb R^{1,3})$. Its coupling region is

$$K_i=\operatorname{supp}\rho_i,$$

not the center of the profile. Before the couplings, specify a system state $\omega$ and independent probe states $\sigma_A,\sigma_B$. Afterwards, measure probe observables $X_i=\Psi_i(h_i)$ with $h_i$ supported in an accessible readout region to the future of $K_i$. The compact-coupling scattering construction and its induced system observables are given in Fewster and Verch 2020, §§ 3.1–3.2, especially Theorem 3.3, PDF.

Two graphs are needed.

• The geometric graph contains an arrow $K_i\to K_j$ when $K_j$ can meet the causal future of $K_i$. It is fixed by the supports and the spacetime metric.
• The control graph contains an arrow $R_i\to K_j$ only when the choice made at $K_j$ depends on the earlier accessible record $R_i$. That arrow must be carried by an ordinary causal signal.

Correlated records do not by themselves create an arrow in either graph. A common field state can be a common cause of $R_A$ and $R_B$ even when the two coupling regions are spacelike.

Exact benchmark: two compact scalar probes

Work in Minkowski space with $c=\hbar=m_A=1$, so times and distances below are measured in units of the inverse $A$-probe mass. Define the standard bump

$$b(u)= \begin{cases} \exp[-1/(1-u^2)],& |u|<1,\\ 0,& |u|\ge 1. \end{cases}$$

Choose $\rho_i(t,\mathbf x)=N\,b(2t)b(|\mathbf x-\mathbf x_i|/R)$, where $N$ fixes $\int d^4x\,\rho_i(x)=1$, $R=1/4$, $\mathbf x_A=(-2,0,0)$, and $\mathbf x_B=(2,0,0)$. Thus

$$K_i\subset \left\{|t|\le\frac12, \ |\mathbf x-\mathbf x_i|\le\frac14\right\}.$$

For any $x\in K_A$ and $y\in K_B$,

$$|t_x-t_y|\le1, \qquad |\mathbf x-\mathbf y|\ge4-2R=\frac72,$$

so the largest possible squared interval is

$$(x-y)^2 \le 1^2-\left(\frac72\right)^2 =-\frac{45}{4}<0.$$

The regions are therefore strictly spacelike. Take $m_\Phi=0$, $m_B=6/5$, $\lambda_A=\lambda_B=0.02$ in these units, vacuum probe states, an arbitrary Hadamard system state, and fixed compact readout smearings $h_i$ in $J^+(K_i)$. Because $\int d^4x\,\rho_i=1$, each $\lambda_i$ has inverse-mass-squared dimension before units are fixed. These values define a reproducible first benchmark; the causal conclusion is independent of the masses and coupling strengths.

Local commutativity gives

$$[\rho_A(x)\Phi(x)\Psi_A(x), \rho_B(y)\Phi(y)\Psi_B(y)]=0$$

throughout $K_A\times K_B$. Consequently the two scattering maps commute. In the general measurement framework this is the symmetric causal-factorization identity

$$\Theta_{AB}=\Theta_A\Theta_B=\Theta_B\Theta_A,$$

and the two induced instruments may be composed in either order; see Fewster and Verch 2020, § 3.3, Eqs. (3.28)–(3.30), PDF.

The resulting dependency graph has the initial field state as the common parent and contains no arrow between the two couplings:

$$\begin{aligned} &\omega\longrightarrow K_A, \qquad \omega\longrightarrow K_B &&\text{(common field preparation)},\\ &\sigma_A\longrightarrow K_A\longrightarrow R_A\longrightarrow C,\\ &\sigma_B\longrightarrow K_B\longrightarrow R_B\longrightarrow C. \end{aligned}$$

Here $C\in J^+(R_A)\cap J^+(R_B)$ is a later comparison event. The graph has no cross-arrow $K_A\to K_B$ or $K_B\to K_A$: $R_A\to C\leftarrow R_B$ represents ordinary record transport, not retroactive signaling between the couplings.

Operational no signaling is a counterfactual statement. If the choice of the $A$ coupling is changed or the $A$ probe is omitted, while the preparation and the $B$ protocol are fixed, the unconditional distribution of $R_B$ is unchanged. It does not say that $R_A$ and $R_B$ are statistically independent. The stronger multi-observer statement and its precise geometric hypotheses are proved in Bostelmann, Fewster, and Ruep 2021, § IV, Theorem 2 and Eqs. (11)–(12), PDF.

Adversarial controls: tails and overlap

First replace $b(2t)$ by the Gaussian $e^{-t^2/(2T^2)}$. Its support is all of time, so the two complete coupling supports are not causally disjoint even when their nominal centers are spacelike. Truncating at $|t|\le4T$ discards the fraction

$$\frac{\int_{|t|>4T}dt\,e^{-t^2/(2T^2)}} {\int_{-\infty}^{\infty}dt\,e^{-t^2/(2T^2)}} =\operatorname{erfc}(2\sqrt2) \simeq6.33425\times10^{-5}.$$

That number is a profile-tail fraction, not yet a bound on a detector probability: quantum fields are unbounded operators. The exact claim must be downgraded to an approximate, state- and observable-specific statement after a signaling estimator or another controlled seminorm is evaluated. Exponential effective-light-cone bounds for noncompact detector profiles are developed in de Ramón, Papageorgiou, and Martín-Martínez 2023, § III, Eq. (43), and § VI with Appendix B, Eqs. (B8)–(B9), PDF.

Second keep compact profiles but move $\mathbf x_B$ onto $\mathbf x_A=(-2,0,0)$ while retaining the same time interval. Then the supports overlap, so the support-separation certificate above disappears: some pairs of points are timelike or coincident, and causal factorization no longer guarantees that the interaction densities or induced instruments commute. Special interactions may still commute accidentally, but that must be proved from their dynamics rather than inferred from geometry. The surviving universal claim is only that the combined interaction is confined to $K_A\cup K_B$. If instead $K_B$ is translated wholly into $J^+(K_A)$, the correct graph has $K_A\to K_B$ and causal signaling is allowed.

These controls expose two different failures. Gaussian tails weaken an exact support statement even when their numerical effect is tiny. Overlap changes the causal graph itself.

Common pitfalls

Using correlation as evidence of signaling. Vacuum and thermal states correlate spacelike observables. Signaling is tested by varying a controlled intervention at $A$ and comparing the unconditional $B$ marginal with everything else fixed.

Localizing only the interaction center. A center and a width do not define a compact support. Exact causal factorization needs the full spacetime coupling region; a noncompact profile needs a quantitative approximate-locality analysis.

Forgetting the record worldline. A selective subensemble is available at $B$ only after the outcome label reaches the data-analysis event. Before that message arrives, the operational state is the nonselective average.

Exercises

1. Certify spacelike separation

Repeat the benchmark with centers separated by a distance $D$, temporal half-width $T/2$, and spatial radius $R$. Find a simple sufficient condition for the two supports to be spacelike.

Solution

The largest temporal difference is $T$ and the smallest spatial distance is $D-2R$. Every pair is spacelike if

$$T^2-(D-2R)^2<0,$$

or $D>2R+T$. Equality permits null-related boundary points and is not strict spacelike separation. For the benchmark, $D=4$, $R=1/4$, and $T=1$, so $4>3/2$.

2. Prove independence of the remote marginal

Let $U_A$ and $U_B$ commute, let $E_B$ be a $B$-probe effect, and assume $[U_A,E_B]=0$. Show that inserting or omitting $U_A$ cannot change the unconditional probability of $E_B$.

Solution

With both couplings, the Heisenberg effect is

$$U_A^\dagger U_B^\dagger E_B U_B U_A =U_B^\dagger U_A^\dagger E_B U_A U_B =U_B^\dagger E_B U_B.$$

Its expectation in any initial state is therefore the same as if $U_A$ were omitted. The argument concerns the nonselective $B$ probability; conditioning on a communicated $A$ outcome can change a later joint statistic.

3. Diagnose a shared controller

A clock in $J^-(K_A)\cap J^-(K_B)$ chooses both coupling strengths, producing correlated settings. Does that create a signaling arrow $A\to B$?

Solution

No. The clock is a shared parent of both setting nodes. A signaling test varies the local $A$ intervention while holding the shared preparation and the $B$ intervention fixed, then compares $B$‘s unconditional statistics. Correlated settings may violate an independence assumption in a Bell experiment, but they do not create an $A\to B$ causal path.

4. Quantify but do not overinterpret a Gaussian tail

For a normalized Gaussian time profile, compute the fraction outside $nT$ and explain why it is not by itself a probability-error bound.

Solution

The fraction is

$$\operatorname{erfc}\!\left(\frac{n}{\sqrt2}\right).$$

At $n=4$ it is about $6.33\times10^{-5}$. A response is a quadratic pairing with a Wightman distribution, not an integral of the profile alone. Its error also depends on the field state, detector gap, spatial smearing, and measured effect, so a response-specific estimate is still required.

References

• Bostelmann, H., Fewster, C. J., and Ruep, M. H. (2021). “Impossible Measurements Require Impossible Apparatus.” Physical Review D 103, 025017. DOI. Open PDF.
• de Ramón, J., Papageorgiou, M., and Martín-Martínez, E. (2023). “Causality and Signalling in Non-Compact Detector–Field Interactions.” Physical Review D 108, 045015. 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.