Bell Nonlocality with Quantum Fields

Bell nonlocality in QFT is the failure of a local-hidden-variable model for correlations between bounded observables in spacelike regions. It is compatible with no signaling. A field Bell test must specify local setting choices, bounded readouts, spacelike supports, a trial definition, setting independence, detector efficiency, and postselection rules; entanglement or a nonzero vacuum correlator alone does not imply a Bell violation.

Required background. Field communication supplies the localized apparatus channel. Spacelike joint measurements supplies order-independent local instruments.

Helpful background. Entanglement witnesses and tomography limits distinguishes a witness from a complete state claim.

CHSH with local field observables

Choose Hermitian contractions $A_0,A_1\in\mathcal A(O_A)$ and $B_0,B_1\in\mathcal A(O_B)$ with spectra in $[-1,1]$, where $O_A$ and $O_B$ are spacelike. The CHSH operator is

$$\mathcal B=A_0(B_0+B_1)+A_1(B_0-B_1).$$

Every local-hidden-variable model obeys

$$|\langle\mathcal B\rangle|\le2,$$

while quantum theory allows up to $2\sqrt2$. Spacelike commutation $[A_i,B_j]=0$ ensures measurement compatibility across the regions; it does not force the state to factorize.

This four-setting inequality and its operational assumptions originate in Clauser, Horne, Shimony, and Holt 1969, Eqs. (1)–(4), pp. 880–882.

Bell testing is a correlation task on spacelike branches, not a communication task. Its local setting operations must nevertheless satisfy the same support and record-access requirements. The diagram is schematic.

Bounded smeared observables

The smeared field $\Phi(f)$ is unbounded, so use bounded functions or Weyl operators. For example, a dichotomic observable can be constructed through functional calculus as $A_0=\operatorname{sgn}\Phi(f_0)$ with $f_0$ supported in $O_A$, provided domains and zero-eigenspace conventions are controlled, or from Hermitian combinations of Weyl operators. Choose the four smearings and evaluate their correlators in the declared field state.

Summers and Werner proved that maximal Bell violation is generic for suitable spacelike local algebras in QFT; see Summers and Werner 1987, Theorems 3.1 and 4.1, pp. 252–257. This is a structural existence result. It does not say that an arbitrary finite-energy detector realizes the required observables with high efficiency or finite statistics.

Loophole-aware protocol

For each trial, choose settings independently in the two regions, implement supported instruments, time-tag every eligible event according to a rule fixed in advance, and compare records only in the common future. Estimate $S_{\mathrm{CHSH}}$ with a confidence method valid for the trial dependence and stopping rule.

Two adversarial changes are decisive. First, condition on coincident detections chosen after viewing the outcomes; this can open a detection or coincidence loophole. Second, allow the switching profiles to overlap causally; then a setting at one wing may influence the other. A reported violation after either change no longer establishes the original spacelike Bell claim without additional analysis.

No signaling is tested separately by verifying that each wing’s marginal is independent of the remote setting. A CHSH violation with stable marginals is the expected quantum pattern, not a contradiction.

Postselection and support overlap can turn a formal CHSH value into an invalid Bell inference. The appropriate causal check is remote-setting independence of unconditional marginals. The map is schematic.

Evidence boundary

As assessed through 2026-08-10, algebraic QFT establishes strong existence theorems for Bell-violating local observables, and explicit field models provide calculable examples. Those results are not interchangeable with a loophole-free experimental Bell test using localized field observables. State which level supports each claim.

References

• Clauser, J. F., Horne, M. A., Shimony, A., and Holt, R. A. (1969). “Proposed Experiment to Test Local Hidden-Variable Theories.” Physical Review Letters 23, 880–884. DOI.
• Summers, S. J., and Werner, R. (1987). “Maximal Violation of Bell’s Inequalities Is Generic in Quantum Field Theory.” Communications in Mathematical Physics 110, 247–259. DOI.