Uniqueness, axiom equivalences, and extension to realistic quantum field theories for the curved Born rule on arbitrary Cauchy surfaces

Lienert and Tumulka (Lett. Math. Phys. 110, 753-804 (2020)) proved the curved Born rule - the statement that detectors placed along an arbitrary Cauchy surface Sigma in Minkowski space-time register the particle configuration with distribution density $|\psi_\Sigma|^2$ - starting only from the usual Born and collapse rules on the horizontal hyperplanes of a fixed Lorentz frame, plus interaction locality and propagation locality of a postulated hypersurface evolution. The paper's own outlook, restated as open in the 2022 follow-up by Lill and Tumulka, leaves a precise open core: (1) whether a hypersurface evolution satisfying the two locality properties is uniquely determined, up to unitary equivalence, by its restriction to spacelike hyperplanes (or, given Poincare covariance, to horizontal hyperplanes), and whether such an evolution extending a given Hamiltonian evolution between horizontal surfaces exists; (2) two axiom-equivalence conjectures - that factorization of the projection-valued measure is equivalent to a direct-integral decomposition over single-point Hilbert spaces, and that interaction locality plus no-creation-from-vacuum implies propagation locality - together with the conjecture that excluding double detections is technically unnecessary in the original horizontal detection scheme; and (3) the extension of the derivation beyond the ideal-detector class to quantum field theories with negative-energy sectors and ultraviolet problems, and the question of how Malament-type no-go theorems constrain ideal detectors when the energy is bounded below.

Background

In non-relativistic quantum mechanics, Born's rule asserts that measuring the positions of all particles at time $t$ yields a configuration distributed with density $|\Psi_t|^2$ , and the collapse rule prescribes how the wave function changes upon detection. In special relativity, 'at time $t$ ' means 'on a horizontal hyperplane in a chosen Lorentz frame', and it is natural to ask whether the same probability law holds on any Cauchy surface - a hypersurface intersected by every inextendible causal curve exactly once - so that detector probabilities do not depend on how Minkowski space-time is foliated. The statement that detectors placed along a Cauchy surface $\Sigma$ register the particle configuration with distribution density $|\Psi_\Sigma|^2$ (relative to the natural volume measure on the configuration space $\Gamma(\Sigma)$ of finite point sets in $\Sigma$ ) is called the curved Born rule.

Lienert and Tumulka proved the curved Born rule as a theorem for any quantum theory equipped with a hypersurface evolution: for every Cauchy surface $\Sigma$ there is a Hilbert space $\mathscr{H}_\Sigma$ of wave functions $\Psi_\Sigma$ , a projection-valued measure (PVM, that is, a countably additive assignment of orthogonal projections to measurable subsets) $P_\Sigma$ on $\Gamma(\Sigma)$ , and unitary evolution operators $U^{\Sigma'}_\Sigma$ between surfaces. The derivation assumes only the Born and collapse rules on the horizontal hyperplanes of a fixed frame, together with two locality properties of the evolution: interaction locality (IL) - if two surfaces intersect in a region $A$ , the evolution between them acts as the identity on the sector of particles located in $A$ , expressing absence of faster-than-light interaction; and propagation locality (PL) - a wave function concentrated in a region $A$ of $\Sigma$ evolves to one concentrated in the region reachable from $A$ at most at light speed, so wave functions do not spread superluminally. The proof approximates $\Sigma$ by small horizontal pieces and models detection as alternating unitary evolution and projections; double detections (one world-line crossing several detector pieces) are excluded by tracing out detected particles, a technical device the authors conjecture to be unnecessary. Examples of such evolutions come from multi-time wave functions - solutions of one Schrödinger equation per particle with one time coordinate per space-time point - and from the Tomonaga-Schwinger formulation of quantum field theory, in which the state vector is transported between arbitrary spacelike hypersurfaces rather than propagated in a fixed time coordinate.

The framework deliberately restricts to a class of hypersurface evolutions in which ideal detectors and a factorization of the PVM exist, roughly the analogue of a tensor product $\mathscr{H}_A \otimes \mathscr{H}_{\Sigma\setminus A}$ over disjoint regions of a surface. Relevant quantum field theories lie outside this class - the realistic theories have energies bounded below, and interacting ones suffer ultraviolet divergences - and it remains an open problem how to prove the curved Born rule there. The framework also assumes the existence of ideal particle detectors; results of the kind proven by Malament and Hegerfeldt limit such detectors under a positive-energy condition (a relativistic quantum theory with energy bounded below cannot possess a sharply localized position observable under natural assumptions), so they do not apply to the source's models, which do not exclude negative energies (its free Dirac example makes no distinction between positive and negative energies), and in the exclusively positive-energy case they are expected to impose fundamental limitations on the accuracy of detectors. The source paper states several further unresolved items: whether the hypersurface evolution is uniquely determined by the horizontal evolution plus (IL) and (PL); whether factorization of the PVM is equivalent to a measurable field of single-point Hilbert spaces with a direct-integral decomposition of $\mathscr{H}_A$ over the configuration space of $A$ ; whether (IL) together with the no-creation-from-vacuum property (NCFV: the vacuum subspace evolves to the vacuum subspace) implies (PL); and whether an alternative piecewise-flat detection scheme approximating $\Sigma$ by tilted hyperplane pieces in local Lorentz frames yields the same curved Born rule without excluding double detections. The piecewise-flat detection scheme, the last of these items, was proven by Lill and Tumulka (2022, reference 3); the other three delimit the present problem. A SpringerBriefs review by Lienert, Petrat, and Tumulka summarizes the framework.

Problem Statement

Within the hypersurface-evolution framework of Lienert and Tumulka's 'Born's rule for arbitrary Cauchy surfaces', establish the uniqueness, the axiom economy, and the physical range of the curved Born rule. The common objective is to determine whether the foliation-independent measurement framework is internally fixed by its flat-surface content, is formulated with minimal assumptions, and applies to realistic quantum field theories; the three requested strands concern the same definitions and examples introduced in the source paper. Each of the three strands is an independent research objective, and resolving any one of them, by proof or by counterexample, constitutes a resolution of this task; the same holds for each sub-question or conjecture enumerated within a strand.

Uniqueness and existence of hypersurface evolution. Decide whether a hypersurface evolution satisfying interaction locality (IL) and propagation locality (PL) is uniquely determined, up to unitary equivalence, by its restriction to spacelike hyperplanes; and whether, given Poincaré covariance, it is uniquely determined by its restriction to the horizontal hyperplanes of a fixed Lorentz frame. In the source paper's formulation: is the hypersurface wave function $\psi_\Sigma$ already determined by the (unitary and collapse) evolution of $\psi_{\Sigma_t}$ on the horizontal surfaces together with (IL) and (PL)? Settle also the related existence question raised there: given a Hamiltonian defining the unitary evolution between horizontal surfaces, does a hypersurface evolution satisfying (IL) and (PL) that extends it exist? The known partial result - that two such evolutions agreeing on hyperplanes yield the same Born distribution on every Cauchy surface - does not settle uniqueness of the evolution itself.
Axiom equivalences. Decide the conjectures left open in the source paper: (a) the factorization axiom for the PVM (axiom (iii) there) is equivalent to the existence of a measurable field of single-point Hilbert spaces $\mathscr{H}_x$ with the direct-integral decomposition $\mathscr{H}_A=\int_{\Gamma(A)}^{\oplus}dq\,\bigotimes_{x\in q}\mathscr{H}_x$ ; (b) (IL) together with no-creation-from-vacuum (NCFV) implies (PL); and (c) in the original horizontal detection scheme, excluding double detections is technically unnecessary because double detections do not change the detection probabilities in the limit $\varepsilon\to 0$ .
Range of validity. Extend the derivation of the curved Born rule beyond the class of hypersurface evolutions for which ideal detectors and a PVM factorization exist - a general extension is required, and it must at least cover quantum field theories with negative-energy sectors (such as the electron-positron Dirac field) and theories with ultraviolet problems - and decide whether the Malament-type no-go theorems, which apply only when the energy is bounded below, obstruct the curved Born rule in the positive-energy regime or leave it intact.

References

Matthias Lienert and Roderich Tumulka, "Born's rule for arbitrary Cauchy surfaces", Letters in Mathematical Physics 110, 753-804 (2020). DOI: 10.1007/s11005-019-01239-y, https://doi.org/10.1007/s11005-019-01239-y (preprint arXiv:1706.07074, https://arxiv.org/abs/1706.07074)
Matthias Lienert, Sören Petrat, and Roderich Tumulka, "Born's Rule for Arbitrary Cauchy Surfaces", chapter 6, pages 61-69, in: Multi-time Wave Functions, SpringerBriefs in Physics, Springer (2020). DOI: 10.1007/978-3-030-60691-6_6, https://doi.org/10.1007/978-3-030-60691-6_6
Sascha Lill and Roderich Tumulka, "Another Proof of Born's Rule on Arbitrary Cauchy Surfaces", Annales Henri Poincare 23, 1489-1524 (2022). DOI: 10.1007/s00023-021-01130-4, https://doi.org/10.1007/s00023-021-01130-4 (preprint arXiv:2104.13861, https://arxiv.org/abs/2104.13861)
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Sascha Lill, Lukas Nickel, and Roderich Tumulka, "Consistency Proof for Multi-time Schrödinger Equations with Particle Creation and Ultraviolet Cut-Off", Annales Henri Poincaré 22, 1887-1936 (2021). DOI: 10.1007/s00023-020-01009-w, https://doi.org/10.1007/s00023-020-01009-w (preprint arXiv:2001.05920, https://arxiv.org/abs/2001.05920)
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Matthias Lienert and Roderich Tumulka, "Multi-time version of the Landau-Peierls formulation of quantum electrodynamics", Annals of Physics 479, 170043 (2025). DOI: 10.1016/j.aop.2025.170043, https://doi.org/10.1016/j.aop.2025.170043 (preprint arXiv:2410.04535, https://arxiv.org/abs/2410.04535)
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Rashi Kaimal and Roderich Tumulka, "Scattering cross section formula derived from macroscopic model of detectors", Journal of Physics A: Mathematical and Theoretical 59, 295303 (2026). DOI: 10.1088/1751-8121/ae880a, https://doi.org/10.1088/1751-8121/ae880a (preprint arXiv:2601.01625, https://arxiv.org/abs/2601.01625)