Einstein causality of the Tomonaga-Schwinger slice-based measurement update rule beyond the instantaneous, irreversible, interaction-picture idealisations

The Tomonaga-Schwinger interaction picture assigns quantum states to Cauchy hypersurfaces of a globally hyperbolic spacetime, and a recent analysis (Fedida, Ann. Phys. 485, 170314 (2026)) showed that an Aharonov-Albert-style generalisation of the Lüders rule to such slices obeys a state-independent anyonic commutation relation for selective measurements over spacelike-separated regions, implies quantum no-signalling for non-selective measurements in an instantaneous-measurement idealisation, and averts the immediate contradictions of Sorkin's impossible measurements as long as the collapse-inducing measurement operators are irreversible (non-invertible). The author explicitly leaves open whether these Einstein-causality conclusions survive outside the idealised regime. This record states that open problem: (i) whether quantum no-signalling for non-selective measurements persists for realistic finite-duration measurements, beyond the weak-* instantaneous limit in which it is proven; (ii) whether the problematic case of invertible collapse operators in Sorkin-type configurations can be cured, or instead forces the conclusion that collapse-inducing measurements must be fundamentally irreversible or that the slice-based state update rule itself fails; and (iii) how far the commutation and no-signalling statements extend to theories and spacetimes where the interaction picture does not exist.

Background

In non-relativistic quantum mechanics, the Lüders rule prescribes that a selective measurement with Kraus operator $M_j$ (the measurement elements of a positive-operator-valued measure, POVM, i.e. a resolution of the identity into positive operators that generate the outcome probabilities) maps a pure state $|\psi\rangle$ to $M_j|\psi\rangle/\|M_j|\psi\rangle\|$, while a non-selective measurement (outcome ignored) maps a density operator $\rho$ to $\sum_j M_j \rho M_j^\dagger$. Relativity makes this prescription subtle: there is no preferred simultaneity hyperplane, and a rule that updates the state 'everywhere at once' appears to be in tension with Einstein causality, the requirement that operations in a spacetime region cannot influence outcome statistics in spacelike-separated regions (quantum no-signalling). Hellwig and Kraus proposed updating states along light cones; Aharonov and Albert introduced instead an 'instantaneous' collapse localised at a spacetime point in relativistic quantum mechanics and found deep difficulties with multiparticle states. Sorkin later exhibited his 'impossible measurements': in a configuration where regions U and V are spacelike separated while a third region W lies in the causal future of U and the causal past of V, a joint measurement in W followed by the Lüders update makes the outcome statistics of measurements in V depend on whether an observable in U was measured, apparently enabling superluminal signalling even though all local operators commute at spacelike separation (microcausality).

The Tomonaga-Schwinger picture offers a covariant arena for such questions: a globally hyperbolic spacetime (one possessing Cauchy hypersurfaces, i.e. surfaces that every inextendible causal curve intersects exactly once, so that data on them determine the entire evolution) is foliated into such hypersurfaces, the quantum state is assigned hypersurface by hypersurface, and evolution is generated by functional derivatives with respect to deformations of the surface. Consistency requires the dynamics to be integrable (independent of the deformation path) and the state on a given hypersurface to be independent of the foliation used to reach it. This interaction picture is known to exist only in restricted settings: as Torre and Varadarajan showed for free fields, and as is generic for interacting theories and many spacetime topologies, the Hilbert spaces of different Cauchy hypersurfaces need not be unitarily equivalent, so a single Hilbert-space interaction picture generally does not exist.

Fedida (Ann. Phys. 485, 170314 (2026)) analysed a slice-based generalisation of the Lüders rule (sometimes called a side-cone update, because the updated state is assigned to hypersurfaces lying on the boundary of the causal cone of the measured region) in this Tomonaga-Schwinger setting: upon a selective measurement with outcome $j$ in a precompact region $U$ (precompact meaning spatially bounded), the state on any acausal Cauchy hypersurface (acausal meaning that no two of its points are causally connected) containing the future boundary $B^+(U)$ of $U$ is obtained by applying the Kraus operator $M_j(U)$ (with the dynamics transported between slices). Under the assumptions of unitary integrable Tomonaga-Schwinger dynamics, foliation-independence of the rays on acausal hypersurfaces, and unitary equivalence of the Hilbert spaces of all such hypersurfaces, the paper proves: (1) a state-independent anyonic commutation relation for selective measurements, $M_{1,m}(U)M_{2,n}(V) = e^{i\varphi_{mn}(U,V)} M_{2,n}(V)M_{1,m}(U)$ for spacelike-separated precompact regions ('anyonic' meaning commutation up to a phase, $[A,B]_\varphi := AB - e^{i\varphi}BA$), which propagates to ordinary (bosonic) commutation of the POVM effects; (2) quantum no-signalling for non-selective measurements, obtained in an instantaneous-measurement idealisation through a weak-* limit in which the measurement region degenerates onto a single Cauchy hypersurface; (3) in Sorkin configurations, immediate contradictions are averted precisely because collapse-inducing measurements are assumed irreversible, i.e. their Kraus operators are non-invertible; and (4) a slice-indexed generalisation (Hilbert spaces attached to individual hypersurfaces with unitary isomorphisms between them) that survives beyond a global interaction picture, at the cost of dynamics-dependent commutation phases.

The adjacent literature on Sorkin's problem establishes that microcausality alone does not prevent signalling by postulated updates: Borsten, Jubb and Kells gave a necessary and sufficient condition for an ideal measurement to be non-signalling and showed that bipartite product observables can signal; Bostelmann, Fewster and Ruep showed in a probe-coupling framework that state updates are causal when the system-probe coupling is local, so that 'impossible measurements require impossible apparatus'; Jubb characterised exactly which state updates (quantum operations, including one-parameter, hence finite-duration, Kraus families) are causal in real scalar QFT, finding only smeared fields and the identity admissible; detector-based measurement theories and the positive formalism (Oeckl's covariant, boundary-based reformulation of quantum theory, in which measurement processes are represented by local, compositional objects) reconstruct causality from local interactions. All of these frameworks analyse measurements implemented by physical local interactions; none analyses the postulated instantaneous slice collapse of the Tomonaga-Schwinger rule, which is precisely the object whose consistency is at issue.

Problem Statement

Determine whether the slice-based (side-cone, Aharonov-Albert-style) state update rule in the Tomonaga-Schwinger picture remains consistent with Einstein causality outside the idealised regime in which its consistency has been proven, where 'consistent with Einstein causality' means both spacelike no-signalling (outcome statistics in a spacelike-separated region are insensitive to non-selective measurements elsewhere, for all states) and absence of the Sorkin-type contradiction (statistics in a region V not depending on whether a measurement was performed in a spacelike-separated region U, in the three-region Sorkin configuration with an intermediary region W in the future of U and past of V). Concretely: (i) Finite-duration measurements: generalise the non-selective state update beyond the weak-* instantaneous limit in which it is currently defined (measurements occupying precompact regions with nonempty temporal extent, with the collapse distributed over the Cauchy hypersurfaces crossing the region), and either prove that quantum no-signalling of the same operational form still holds, or exhibit an explicit counterexample in which finite-duration measurements enable spacelike signalling. (ii) Invertible collapse operators: decide whether the invertible-Kraus case of the Sorkin configuration can be cured within the Tomonaga-Schwinger picture - for example by a principled restriction of the admissible measurement class, a modification of the slice-based update rule that does not reintroduce signalling or foliation dependence, or an additional physical constraint - or prove a no-go showing that invertibility necessarily breaks Einstein causality, which would elevate the paper's irreversibility assumption to a physical necessity or reveal a fundamental limitation of the slice-based update rule. (iii) Beyond the interaction picture: extend the commutation and no-signalling statements to theories and spacetimes where the Hilbert spaces of different Cauchy hypersurfaces are not unitarily equivalent (so the Tomonaga-Schwinger interaction picture does not exist), for instance within algebraic quantum field theory or relativistic quantum instrument theory, and clarify whether the operational Einstein-causality statements survive in those frameworks in a form that recovers the instantaneous-limit results. The three strands are the objective, and each is independently resolvable and carries its own completion test: strand (i) is resolved by a proof that the instantaneous-limit no-signalling statement also holds for non-selective finite-duration measurements, or by an explicit counterexample in which finite-duration measurements enable spacelike signalling; strand (ii) is resolved by a principled cure of the invertible-Kraus case that does not reintroduce signalling or foliation dependence, or by a no-go proof that invertibility necessarily breaks Einstein causality; strand (iii) is resolved, within one named framework (algebraic quantum field theory or relativistic quantum instrument theory), by determining whether the commutation and no-signalling statements are recovered in a form that recovers the instantaneous-limit results. Strands (i) and (ii) are the sharp author-declared questions and are mandatory; strand (iii) is the programmatic extension set out by the source paper and is required in the bounded form just stated.

References

1. Samuel Fedida, Einstein causality of quantum measurements in the Tomonaga-Schwinger picture, Annals of Physics 485, 170314 (2026). DOI: 10.1016/j.aop.2025.170314, https://doi.org/10.1016/j.aop.2025.170314 (preprint arXiv:2506.14693, https://arxiv.org/abs/2506.14693)
2. Rafael D. Sorkin, Impossible measurements on quantum fields, in B. L. Hu and T. A. Jacobson (eds.), Directions in General Relativity, Vol. 2 (Cambridge University Press, 1993), pp. 293-305. arXiv:gr-qc/9302018, https://arxiv.org/abs/gr-qc/9302018
3. Yakir Aharonov and David Z. Albert, Can we make sense out of the measurement process in relativistic quantum mechanics?, Physical Review D 24, 359 (1981). DOI: 10.1103/PhysRevD.24.359, https://doi.org/10.1103/PhysRevD.24.359
4. K.-E. Hellwig and K. Kraus, Formal description of measurements in local quantum field theory, Physical Review D 1, 566 (1970). DOI: 10.1103/PhysRevD.1.566, https://doi.org/10.1103/PhysRevD.1.566
5. Charles G. Torre and Madhavan Varadarajan, Functional evolution of free quantum fields, Classical and Quantum Gravity 16, 2651 (1999). DOI: 10.1088/0264-9381/16/8/306, https://doi.org/10.1088/0264-9381/16/8/306
6. Matthias Lienert and Roderich Tumulka, Born's rule for arbitrary Cauchy surfaces, Letters in Mathematical Physics 110, 753 (2020). DOI: 10.1007/s11005-019-01239-y, https://doi.org/10.1007/s11005-019-01239-y
7. Leron Borsten, Ian Jubb, and Graham Kells, Impossible measurements revisited, Physical Review D 104, 025012 (2021). DOI: 10.1103/PhysRevD.104.025012, https://doi.org/10.1103/PhysRevD.104.025012 (preprint arXiv:1912.06141, https://arxiv.org/abs/1912.06141)
8. Henning Bostelmann, Christopher J. Fewster, and Maximilian H. Ruep, Impossible measurements require impossible apparatus, Physical Review D 103, 025017 (2021). DOI: 10.1103/PhysRevD.103.025017, https://doi.org/10.1103/PhysRevD.103.025017 (preprint arXiv:2003.04660, https://arxiv.org/abs/2003.04660)
9. Ian Jubb, Causal state updates in real scalar quantum field theory, Physical Review D 105, 025003 (2022). DOI: 10.1103/PhysRevD.105.025003, https://doi.org/10.1103/PhysRevD.105.025003 (preprint arXiv:2106.09027, https://arxiv.org/abs/2106.09027)
10. Emma Albertini and Ian Jubb, Are ideal measurements of real scalar fields causal?, arXiv:2306.12980 (2023). https://arxiv.org/abs/2306.12980
11. Maria Papageorgiou and Doreen Fraser, Eliminating the 'Impossible': Recent progress on local measurement theory for quantum field theory, Foundations of Physics 54, 26 (2024). DOI: 10.1007/s10701-024-00756-8, https://doi.org/10.1007/s10701-024-00756-8
12. Christopher J. Fewster and Rainer Verch, Measurement in quantum field theory, arXiv:2304.13356 (2023). https://arxiv.org/abs/2304.13356
13. José Polo-Gómez, Luis J. Garay, and Eduardo Martín-Martínez, A detector-based measurement theory for quantum field theory, Physical Review D 105, 065003 (2022). DOI: 10.1103/PhysRevD.105.065003, https://doi.org/10.1103/PhysRevD.105.065003
14. José de Ramón, Maria Papageorgiou, and Eduardo Martín-Martínez, Relativistic causality in particle detector models: Faster-than-light signaling and impossible measurements, Physical Review D 103, 085002 (2021). DOI: 10.1103/PhysRevD.103.085002, https://doi.org/10.1103/PhysRevD.103.085002 (preprint arXiv:2102.03408, https://arxiv.org/abs/2102.03408)
15. Robin Simmons, Maria Papageorgiou, Marios Christodoulou, and Časlav Brukner, Factorisation conditions and causality for local measurements in QFT, arXiv:2511.21644 (2025). https://arxiv.org/abs/2511.21644
16. Robert Oeckl, Causal measurement in quantum field theory: spacetime, Physical Review D 113, 085002 (2026). DOI: 10.1103/5gl8-3j6p, https://doi.org/10.1103/5gl8-3j6p (preprint arXiv:2511.06566, https://arxiv.org/abs/2511.06566)
17. Robert Oeckl and Adamantia Zampeli, Towards local and compositional measurements in quantum field theory, arXiv:2505.10968 (2025). https://arxiv.org/abs/2505.10968
18. José Polo-Gómez, T. Rick Perche, and Eduardo Martín-Martínez, State updates and useful qubits in relativistic quantum information, arXiv:2506.18906 (2025). https://arxiv.org/abs/2506.18906
19. Jesse Huhtala and Iiro Vilja, Phenomenological constraints on 'impossible' measurements, arXiv:2507.06767 (2025). https://arxiv.org/abs/2507.06767
20. M. G. M. Moreno, Alejandro Fonseca, and Márcio M. Cunha, Finite-time collapse process and non-local correlations are incompatible with non-signaling theories, arXiv:1809.00383 (2018). https://arxiv.org/abs/1809.00383
21. Stephen D.H. Hsu, Relativistic covariance and nonlinear quantum mechanics: Tomonaga-Schwinger analysis, Physics Letters B 872, 140053 (2026). DOI: 10.1016/j.physletb.2025.140053, https://doi.org/10.1016/j.physletb.2025.140053 (preprint arXiv:2511.15935, https://arxiv.org/abs/2511.15935)
22. Lajos Diósi, Comment on 'Relativistic covariance and nonlinear quantum mechanics: Tomonaga-Schwinger analysis', arXiv:2602.06845 (2026). https://arxiv.org/abs/2602.06845
23. Vincenzo Fiorentino and Stefan Weigert, Beyond the projection postulate and back: Quantum theories with generalized state-update rules, Physical Review A 113, 012204 (2026). DOI: 10.1103/2zpm-jsh7, https://doi.org/10.1103/2zpm-jsh7 (preprint arXiv:2506.06207, https://arxiv.org/abs/2506.06207)
24. Kaixun Tu and Qing Wang, Causality and the interpretation of quantum mechanics, arXiv:2402.05450 (2024). https://arxiv.org/abs/2402.05450
25. Arash Zaghi, Born's rule from contextual relative-entropy minimization, Entropy 27, 898 (2025). DOI: 10.3390/e27090898, https://doi.org/10.3390/e27090898