What are the limitations of ideal measurements in relativistic QFT?

Statement

In quantum mechanics, ideal (projective) measurements are idealized operations that instantaneously collapse the wavefunction onto an eigenstate of the measured observable. In relativistic QFT, such instantaneous operations are problematic: they can signal superluminally, violate causality, or be inconsistent with the haag-kastler-axioms. Which observables in QFT admit ideal measurements, which admit approximate measurements, and which are fundamentally impossible to measure? What are the implications for the Bell-CHSH program, where the “Alice” and “Bob” operations are modeled as measurements of weyl-operators?

Why It Matters

The Bell-CHSH inequality tests in QFT are typically formulated as thought experiments: Alice and Bob each measure a dichotomic observable localized in their respective spacetime regions, and the correlations of their outcomes are compared with the classical bound of 2. But what does it mean to “measure” a Weyl operator $W(f)=\exp (i\varphi (f))$ in relativistic QFT? The field operator $\varphi (f)$ is an unbounded operator, and $W(f)$ is bounded and unitary, but performing a projective measurement of $W(f)$ in real time requires a physical interaction between the field and an apparatus that:

1. Does not disturb the field outside the localization region of $f$.
2. Does not create superluminal signals.
3. Produces a definite, reproducible outcome.

The Sorkin-Bostelmann-Fewster arguments show that certain idealized field measurements are impossible — they would violate causality. The Unruh-DeWitt detector model provides an operational, causal alternative to ideal measurement. Understanding which measurements are possible shapes the experimental interpretation of Bell-CHSH violations in QFT: if Alice’s measurement is not operationally realizable, the violation is a formal fact about correlators, not a testable prediction.

This question has implications for:

• The physical interpretation of all Bell-inequality violations computed in the program.
• The design of realistic detector models (e.g., unruh-dewitt-detectors) that approximate ideal measurements.
• The relationship between entanglement measures (which may require ideal measurements) and physical observables.

What We Know

The Sorkin no-go theorem (1993):

• Sorkin showed that ideal projective measurements of local observables in QFT can lead to superluminal signaling: if Alice performs an ideal measurement of a local observable $O_A$ in region ${\mathcal{R}}_A$, then a measurement by Bob in a spacelike separated region ${\mathcal{R}}_B$ can detect Alice’s intervention, violating causality.
• The mechanism is the collapse postulate: after Alice’s measurement, the state changes globally (not just in ${\mathcal{R}}_A$), and this global change can be detected by Bob.

Fewster-Verch measurement framework:

• Fewster and Verch have developed an algebraic framework for “selective” and “non-selective” measurements in AQFT that is causally consistent. In this framework, measurements are modeled as local unitary evolutions (not projections), and the “outcome” is obtained by coupling the field to a probe system.
• Not all observables admit such causal measurements: the class of “measurable” observables (in the Fewster-Verch sense) is strictly smaller than the full local algebra. In particular, observables that are intrinsically “non-selective” (e.g., the vacuum projector) are not directly measurable.

Unruh-DeWitt detector as operational measurement:

• The unruh-dewitt-detectors model couples a two-level system (the detector) to the field via a local interaction $H_{\text{int}}(t)=\lambda \chi (t)\mu (\tau )\otimes \varphi (x(\tau ))$. After the interaction, measuring the detector’s two-level observable gives indirect information about the field. This is causally consistent.
• Bell-CHSH experiments using Unruh-DeWitt detectors have been studied in the program (JHEP 2024, Universe 2024), and the resulting Bell violations are related to but different from the direct Weyl operator correlators.

What is not known:

• A complete classification of which observables in local QFT (specifically type-iii-von-neumann-algebras) are “measurable” in the Fewster-Verch sense.
• Whether the specific Weyl operators $W(f)$ used in the Bell-CHSH program admit operational measurements — or whether they are formal objects whose correlators can only be inferred indirectly.
• How the gap between ideal and operational measurements affects the quantitative Bell violation: does the Unruh-DeWitt model always give a smaller violation than the ideal Weyl operator computation?

Possible Approaches

1. Classification of measurable operators in type III algebras: Using the Fewster-Verch framework, determine which elements of the local von Neumann algebra $\mathcal{A}(\mathcal{O})$ admit causally consistent measurements. In particular, check whether weyl-operators $W(f)$ with $\text{supp}(f)\subset \mathcal{O}$ are measurable. This is a mathematical physics question with a potentially sharp answer.

2. Comparison of Weyl and Unruh-DeWitt Bell violations: In the cases where both computations are available (scalar field, Rindler geometry), compare the Bell-CHSH violation from the ideal Weyl computation with the Unruh-DeWitt detector computation. Determine the ratio as a function of the coupling strength $\lambda$, the detector switching function $\chi (t)$, and the field mass $m$. This comparison quantifies the “measurement gap.”

3. Regularization and finite-time measurements: Replace the instantaneous projection with a finite-duration von Neumann measurement (Gaussian in time). Compute the Bell violation as a function of measurement duration $T$ and show that the ideal result is recovered as $T\to 0^{+}$, with corrections of order $T/\tau$ where $\tau$ is the natural timescale of the field (inverse mass). This provides a smooth interpolation between ideal and physical measurements.

4. Bostelmann-Fewster “impossible measurements” catalogue: Systematically apply the Bostelmann-Fewster analysis to the specific observables used in the Bell-CHSH program. Identify which are possible and which are not, and for the impossible ones, find the closest operationally achievable observable and estimate the Bell violation for that observable.

5. Connection with the split property: The split-property (which holds for free massive fields) ensures that for any two spacelike separated bounded regions, the joint algebra factors as a product. This is related to the ability to perform independent measurements in the two regions. For massless fields or theories without a mass gap, the split property may fail, complicating the measurement picture.

Related Questions

• higher-spin-bell-inequalities — Ideal measurement limitations are more severe for higher-spin fields, which have more constrained localization properties
• embezzlement-cost-relative-entropy — Operational limitations on measurements bound what information can be extracted from embezzlement protocols
• entanglement-as-confinement-probe — If entanglement measures require impossible measurements, the phase-transition probe program needs operational reformulation
• bell-inequalities-with-gribov-horizon — Measurement limitations in gauge theories are compounded by the Gribov problem