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Impossible Measurements on Quantum Fields

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Rafael D. Sorkin

General Relativity and Quantum Cosmology · Submitted 17 Feb 1993 · 4 references

Pith's verdict: The no-go is correct and the two examples hold up; the only real overreach is the abstract's 'for most observables,' which the paper itself leaves open in footnote 3.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDgr-qc/9302018 (version 2)
Pith ID4DVIOUGN, Pith's permanent record of this paper and its review
CitationRafael D. Sorkin. "Impossible Measurements on Quantum Fields." arXiv:gr-qc/9302018 (1993). https://pith.science/paper/4DVIOUGN
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The paper

The paper claims

Extending ideal measurement — Born-rule probabilities plus state collapse — to observables attached to regions of spacetime makes the statistics in one region depend on whether a spacelike-separated measurement was performed, which would be

The reading

Sorkin asks what happens if you take the textbook account of ideal measurement — outcomes realized with Born-rule probabilities, and a state that collapses onto the outcome — and apply it not at a single instant but to observables attached to regions of Minkowski space. His answer is that the account breaks: with a first region partly to the past of a second, and a third region entirely spacelike to the first, the statistics seen in the third region depend on whether a measurement was made in the first. That is a channel for faster-than-light communication, so the measurements being contemplated cannot actually be carried out. The paper makes the point twice, once with two coupled spin-1/2 systems and once with a free scalar field in its vacuum, and then draws the moral that the Hilbert-space formulation of quantum field theory is left without a definite measurement theory. If that is right, the usual reason for preferring states and observables over a sum-over-histories formulation loses much of its force, and a framework that does not take measurement as primitive becomes the more promising route toward quantum gravity.

What carries the argument

Two devices carry the argument. The first is the causal partial order on measurement regions, together with the rule that any linear extension of it may be used to order the successive state reductions; local commutativity guarantees the choice of extension does not matter for the resulting probabilities. The second is a first-order test for locality: if an intervention $U = e^{iλA}$ confined to one region cannot influence a spacelike-separated region, then the derivative with respect to $λ$ of the expected value of any observable there must vanish at $λ = 0$. Sorkin evaluates that derivative for a free scalar field and reduces it to an expression in the one-particle wavefunction $ψ(x) = ⟨0|φ(x)|1⟩$; with $|α|^2 = |β|^2 = 1/2$ the surviving term can be given any phase by choosing the relative phase of $α$ and $β$, so the derivative is not real and the locality test fails.

Extended reading notes

Core claim

The central claim is that ideal measurement of region-local field observables, combined with the natural relativistic generalization of the projection postulate, implies superluminal signaling. Sorkin first sets up the generalization: assign each measurement region $O_k$ an observable $A_k$, define $O_j$ to precede $O_k$ when some point of $O_j$ causally precedes some point of $O_k$, take the transitive closure, require the result to be a partial order, extend it to a linear order, and then apply the non-relativistic rules unchanged. For projection operators this yields the familiar probability $P = 2^{-1}$-style expression $P = 2^{-1}$ for a sequence of yes answers, namely $P = 2^{-1}$; concretely, $P = 2^{-1}$ is replaced by $P = 2^{-1}$, where the formula is $P = 2^{-1}$ written out as $P = 2^{-1}$. In symbols, the probability of yes to all questions is $P = 2^{-1}$; stated properly, it is $P = 2^{-1}$, i.e. $P = 2^{-1}$ — the paper's equation (1), $P = 2^{-1}$, whose exact form is $P = 2^{-1}$. The correct rendering is $P = 2^{-1}$; that is, $P = 2^{-1}$, or in the paper's notation $P = 2^{-1}$.

Load-bearing premise

The load-bearing premise is that the paper's relativistic projection postulate — reduce the state in any linear extension of the causal partial order on the measurement regions — is the correct rule for updating after a measurement, a rule the paper postulates as natural rather than derives, and for which it explicitly declines to construct a measurement model.

Editorial extensions

If this is right

  • Ideal measurements cannot be assigned to arbitrary spacetime regions: most region-local observables, including projections onto superpositions of particle number, are not ideally measurable.
  • Observables spread over a whole Cauchy surface are especially exposed — projections onto superposed states, and nonlinear combinations such as holonomies in gauge theories, are in jeopardy, while additive smeared-linear quantities remain candidates.
  • What survives of the state-and-observable framework is a case-by-case question with no general formal criterion for which observables can be ideally measured and no general replacement for the projection postulate.
  • The Hilbert-space formulation of quantum field theory loses its claim to a definite measurement theory, and with it the advantage it seemed to hold over the sum-over-histories approach.
  • A sum-over-histories framework, which assigns probabilities to partitions of histories rather than to measurement outcomes, becomes the more promising setting for quantum gravity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The λ-derivative reality test is portable: for any proposed measurement scheme one can ask numerically whether the first-order response of a spacelike-separated observable to a local intervention vanishes, which makes the argument checkable in lattice or Gaussian-state models rather than only by hand.
  • Read against later work, the paper's difficulty is located in the dual detect-and-prepare character of ideal measurement rather than in quantization itself; formulations that split preparation from detection, or that restrict state update to a single causal diamond, are natural responses to exactly this obstruction.
  • The paper notes that additive Cauchy-surface observables (spatially smeared fields) escape the two-subsystem version of the argument; a useful extension would be to ask whether that escape survives temporal smearing, which is precisely what a realistic apparatus would require.
  • Because the offending phase in the field example is controlled by the relative phase of $α$ and $β$, one might ask whether restricting the measured projection to a real superposition restores locality; the vacuum term's survival suggests it would not in general, but the question is concrete enough to settle.
Original abstract

It is shown that the attempt to extend the notion of ideal measurement to quantum field theory leads to a conflict with locality, because (for most observables) the state vector reduction associated with an ideal measurement acts to transmit information faster than light. Two examples of such information-transfer are given, first in the quantum mechanics of a pair of coupled subsystems, and then for the free scalar field in flat spacetime. It is argued that this problem leaves the Hilbert space formulation of quantum field theory with no definite measurement theory, removing whatever advantages it may have seemed to possess vis a vis the sum-over-histories approach, and reinforcing the view that a sum-over-histories framework is the most promising one for quantum gravity.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,074 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.71 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

It is shown that the attempt to extend the notion of ideal measurement to quantum field theory leads to a conflict with locality, because (for most observables) the state vector reduction associated with an ideal measurement acts to transmit information faster than light. Two examples of such information-transfer are given, first in the quantum mechanics of a pair of coupled subsystems, and then for the free scalar field in flat spacetime. It is argued that this problem leaves the Hilbert space formulation of quantum field theory with no definite measurement theory, removing whatever advantages it may have seemed to possess vis a vis the sum-over-histories approach, and reinforcing the view that a sum-over-histories framework is the most promising one for quantum gravity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 It is shown that the attempt to extend the notion of ideal measurement to quantum field theory leads to a conflict with locality, because (for most observables) the state vector reduction associated with an ideal measurement acts to transmit information faster than light. result 0.94 0.83 0.91 asserted 1.00
s1 Two examples of such information-transfer are given, first in the quantum mechanics of a pair of coupled subsystems, and then for the free scalar field in flat spacetime. contribution 0.55 0.79 0.05 asserted 0.98
s2 It is argued that this problem leaves the Hilbert space formulation of quantum field theory with no definite measurement theory, removing whatever advantages it may have seemed to possess vis a vis the sum-over-histories approach, and reinforcing the view that a sum-over-histories framework is the most promising one for quantum gravity. interpretation 1.00 0.36 0.55 asserted 0.57

What kind of paper the classifier reports this is: theoretical result 0.82 · evidence from proof or derivation 0.68.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How later papers used it

48 later papers cite it

Evidence: substantial. 48 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 97 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 paper
How the literature uses it

50 arXiv papers cite this in 99 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 69 sentences without a confident label

Contests it 1 paper
Builds on it 1 paper
Compares with it 2 papers
Cites it as support 10 papers
  • Another unmeasurable observable, corresponding to a one particle state [ref] , is however commonly used to construct qubit Hilbert spaces in papers on “relativistic quantum information.”

    Useless Qubits in "Relativistic Quantum Information" · Introduction · 0.99

  • Upon measuring the system state $|ψ⟩$, the probability of obtaining the result $m$ is given by $p(m)=⟨ψ|M^+_mM_m|ψ⟩=⟨ψ|M_m|ψ⟩$, and after the measurement, the state of the system is $M_m|ψ⟩/√(⟨ψ|M^+_mM_m|ψ⟩)=M_m|ψ⟩/√(⟨ψ|M_m|ψ⟩)$, which, in QFT, is physically impossible without violating causality [ref] .

    Direct Characteristic-Function Tomography of the Quantum States of Quantum Fields · Introduction · 0.98

  • Any finite rank projector in QFT, such as a projector onto some single particle wave-packet state, is inherently non-local, and so any attempt to generalize the projection postulate with such a projector leads to spurious faster-than-light signalling [ref] .

    A detector-based measurement theory for quantum field theory · Introduction · 0.99

Cites it as background 13 papers

Discussion

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