Sofia D. Wechsler's 'The Wave Function Cannot Be a Real Wave—Then, Can We Speak of an Ontology of Particles?' (JQIS 12(4), 99–110, 2022; arXiv:2210.03530) argues that the wave function cannot be an ontic wave and then examines the remaining particle-ontology option: particles that always have definite positions but undergo random, discontinuous jumps with instantaneous position statistics given by the squared wave function (Gao's random discontinuous motion, RDM). Under special relativity this interpretation breaks entanglement correlations for time-separated detections unless a preferred-frame 'freezing' mechanism is introduced, and a Hardy-type interferometric variant is presented in which frame-dependent retrodictions force joint clicks of two detectors that standard quantum mechanics forbids — identified by the author as the one problem that seems to have no solution. The open problem is to construct, or prove impossible, a relativistically covariant dynamics for such particle-jump ontologies that reproduces standard quantum joint statistics for entangled systems in all inertial frames without a physically preferred frame, without classical self-interaction between branches of one particle's state, and without Hardy-type contradictions.

Background

Quantum mechanics predicts the statistics of detections but does not by itself fix what physically exists inside an apparatus. The PBR theorem (Pusey, Barrett and Rudolph, Nature Physics 8, 475 (2012)) showed — under an assumption of preparation independence — that the quantum state cannot be a merely statistical summary of a deeper, non-quantum state, which sharpened the old question: is the wave function a real wave, or only a calculational tool, with something else (particles?) constituting the reality in the apparatus?

Wechsler (2022) first argues the wave function cannot be a real wave: for two particles entangled in their paths, the collapse postulate (the rule that a detection updates the state of the pair instantly) together with the relativity of simultaneity lets different inertial frames assign incompatible pre-detection wave functions to the same pair, contradicting the idea of an ontic wave. She then turns to particle ontologies. The de Broglie–Bohm interpretation (particles guided by a real wave along continuous trajectories) is set aside on the strength of her separate nonrelativistic argument against continuous trajectories (JQIS 12(2), 2022). The remaining option is Shan Gao's random discontinuous motion (RDM): each particle always has a definite position, but jumps discontinuously and randomly from place to place; at every instant the set of occupied positions is distributed with density ∣ψ∣2|\psi|^2; an entangled state corresponds to strictly correlated simultaneous jumps of the two particles; measurement collapses the allowed region. Gao's original motivation — a charged particle spread over two wave-packets would self-repel and destroy the wave function — is refuted by Wechsler using second quantization (the two packets are mutually exclusive occupation branches, so no self-interaction arises), but the interpretation itself is then examined on its own terms.

The conflict with special relativity is the crux. Because the jumps are nonlocal (spacelike), a Lorentz boost can reverse the time order of successive stays or make two of them simultaneous, so in some inertial frames a single particle appears at two places at once, or disappears from space during a collapse. Gao's book (The Meaning of the Wave Function, Cambridge University Press, 2017, ch. 9) analyzed these distortions, concluded they are in principle unmeasurable, but conceded that RDM and its collapse evolution single out a preferred Lorentz frame — adopting a preferred-frame (absolute-simultaneity) route rather than constructing a covariant dynamics. Wechsler sharpens the problem in two ways. First, when the two members of an entangled pair are detected at different times, the correlated-jump rule violates the observed entanglement correlations; a 'freezing' rule (once one particle is detected in some frame, the partner stops jumping until detected) restores agreement with experiment but is defined in one preferred frame. Second, in a variant of Hardy's paradox (Hardy, Phys. Rev. Lett. 68, 2981 (1992) — an interferometric two-particle argument showing that local-realist reasoning conflicts with quantum statistics), she analyzes the path-entangled two-particle state (i∣u+⟩∣v−⟩+∣v+⟩∣v−⟩+i∣v+⟩∣u−⟩)/3(i|u^+\rangle|v^-\rangle+|v^+\rangle|v^-\rangle+i|v^+\rangle|u^-\rangle)/\sqrt{3} in an apparatus in which each particle passes two successive beam splitters BS±^\pm and BS'±^\pm and phase shifters: the particles travel on paths u±,v±u^\pm,v^\pm before the first beam splitters, on intermediate paths c±,d±c^\pm,d^\pm between the two beam-splitter stages, and on output paths u′±,v′±u'^\pm,v'^\pm feeding the detectors U±^\pm,V±^\pm. The lab-frame state after the first beam splitters (her eq. (10), with coefficients (−3,i,i,−1)/12(-3,i,i,-1)/\sqrt{12}) contains the component −∣d+⟩∣d−⟩/12-|d^+\rangle|d^-\rangle/\sqrt{12}: under the RDM reading, prior to meeting the beam splitters BS'±^\pm the two particles travel simultaneously on the intermediate paths d+d^+ and d−d^- with probability 1/12. In a boosted frame S' in which detector U+ clicks before the wave packets of p− have reached BS′−, the partially transformed state (eq. (13)) implies that whenever p− travels along d−d^-, p+ must have clicked U+; in the mirrored boosted frame S'', whenever p+ travels along d+d^+, p− must have clicked U− (eq. (14)). Composing these frame-dependent retrodictions with the joint occupation of d+d^+ and d−d^- implies that both U+ and U− click jointly — yet the lab-frame final wave function (eq. (12)) contains no ∣u′+⟩∣u′−⟩|u'^+\rangle|u'^-\rangle component, and quantum mechanics forbids the double click. This is claimed to be an empirically accessible contradiction between RDM-plus-relativity and the quantum predictions.

The neighboring literature shows partial progress on adjacent questions but not on this one. Tumulka's relativistic version of the Ghirardi–Rimini–Weber (GRW) collapse model (J. Stat. Phys. 125, 821 (2006)) is a Lorentz-invariant collapse theory whose primitive ontology is flashes (discrete point events) rather than persisting particles, and it covers only noninteracting particles. Bedingham's dynamical state reduction for an Einstein–Podolsky–Rosen (EPR) experiment (J. Phys. A 42, 465301 (2009)) is precisely the kind of collapse account Wechsler criticizes for being unable to respond to dynamically changed measurement settings. Jones, Guaita and Bassi (Phys. Rev. A 103, 042216 (2021)) proved a no-go for one particular route of relativizing GRW. Lienert and Tumulka's Born rule for arbitrary Cauchy surfaces (Lett. Math. Phys. 110, 753 (2020)) supplies a covariant multi-time probability framework for wave functions, with interacting systems still unresolved. Bedingham and Pearle (Phys. Rev. A 112, 042218, 2025) constructed a Poincaré-covariant continuous-spontaneous-localization (CSL) model with quantised time variables, but only for distinguishable particles. Levy and Hemmo (Found. Phys. 55, 73, 2025) proved that a Lorentz-invariant and contextual theory (such as quantum field theory) admits no deterministic hidden-variables model — genuinely stochastic hidden variables are required — which permits stochastic jump ontologies in principle but constructs none. No known result delivers a covariant dynamics for persistent particles with definite positions at every instant, and none evaluates Wechsler's freezing or Hardy-variant arguments.

Problem Statement

Does there exist a relativistically covariant mechanism — a complete dynamics — for a particle-based ontology whose particles undergo random, discontinuous ontic jumps in the sense of Gao's random discontinuous motion (each particle has a definite position at every instant; the set of occupied positions at an instant is distributed as ∣ψ∣2|\psi|^2; entangled states impose strictly correlated simultaneous jumps) — where these clauses fix the nonrelativistic ontology that is to be relativized, so that an admissible covariant dynamics may reformulate the jump rule with 'instant' and 'simultaneous' understood relative to arbitrary spacelike Cauchy hypersurfaces rather than to an absolute simultaneity relation — such that: (a) the correlations of entanglement are enforced so that time-separated and frame-separated detections yield the same joint statistics in all inertial frames, i.e. standard quantum joint-detection statistics — including in Hardy-type interferometric arrangements such as Wechsler's two-particle setup with state (i∣u+⟩∣v−⟩+∣v+⟩∣v−⟩+i∣v+⟩∣u−⟩)/3(i|u^+\rangle|v^-\rangle+|v^+\rangle|v^-\rangle+i|v^+\rangle|u^-\rangle)/\sqrt{3} — are exactly reproduced; and (b) no physically preferred frame or foliation, no classical self-interaction between the branches of a single particle's wave function, and no empirical contradiction with the quantum predictions (such as the joint U+–U– detector clicks that quantum mechanics forbids in Wechsler's Hardy-variant setup) is introduced? Alternatively, prove that no dynamics of this class satisfies (a) and (b) — the impossibility argument covering covariant reformulations of the jump rule on arbitrary spacelike hypersurfaces, not only foliation-dependent or otherwise restricted subclasses — so that some preferred-frame structure is unavoidable for particle-jump ontologies of this kind.

References

  1. Sofia D. Wechsler, 'The Wave Function Cannot Be a Real Wave—Then, Can We Speak of an Ontology of Particles?', Journal of Quantum Information Science 12(4), 99–110 (2022), DOI 10.4236/jqis.2022.124009, https://doi.org/10.4236/jqis.2022.124009; arXiv:2210.03530, https://arxiv.org/abs/2210.03530
  2. Sofia D. Wechsler, 'About the Nature of the Quantum System—An Examination of the Random Discontinuous Motion Interpretation of the Quantum Mechanics', Journal of Quantum Information Science 11(3), 99–111 (2021), DOI 10.4236/jqis.2021.113008, https://doi.org/10.4236/jqis.2021.113008; arXiv:2107.05633, https://arxiv.org/abs/2107.05633
  3. Sofia D. Wechsler, 'A Non-Relativistic Argument against Continuous Trajectories of Particles', Journal of Quantum Information Science 12(2), 29–36 (2022), DOI 10.4236/jqis.2022.122004, https://doi.org/10.4236/jqis.2022.122004; arXiv:2204.02184, https://arxiv.org/abs/2204.02184
  4. Sofia Wechsler, 'What in Fact Proves the Violation of the Bell-Type Inequalities?', Journal of Quantum Information Science 13(1), 24–32 (2023), DOI 10.4236/jqis.2023.131002, https://doi.org/10.4236/jqis.2023.131002; arXiv:2303.08821, https://arxiv.org/abs/2303.08821
  5. Shan Gao, The Meaning of the Wave Function: In Search of the Ontology of Quantum Mechanics, Cambridge University Press (2017), DOI 10.1017/9781316407479, https://doi.org/10.1017/9781316407479; book preprint arXiv:1611.02738, https://arxiv.org/abs/1611.02738
  6. Shan Gao, 'A Puzzle for the Field Ontologists', Foundations of Physics 50, 1541–1553 (2020), DOI 10.1007/s10701-020-00390-0, https://doi.org/10.1007/s10701-020-00390-0
  7. Shan Gao, 'Can the Ontology of Bohmian Mechanics Consists Only in Particles? The PBR Theorem Says No', Foundations of Physics 53, article 91 (2023), DOI 10.1007/s10701-023-00731-9, https://doi.org/10.1007/s10701-023-00731-9; preprint arXiv:2501.03275, https://arxiv.org/abs/2501.03275
  8. Matthew F. Pusey, Jonathan Barrett, Terry Rudolph, 'On the reality of the quantum state', Nature Physics 8, 475–478 (2012), DOI 10.1038/nphys2309, https://doi.org/10.1038/nphys2309
  9. Lucien Hardy, 'Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories', Physical Review Letters 68, 2981–2984 (1992), DOI 10.1103/PhysRevLett.68.2981, https://doi.org/10.1103/PhysRevLett.68.2981
  10. Roderich Tumulka, 'A Relativistic Version of the Ghirardi–Rimini–Weber Model', Journal of Statistical Physics 125, 821–840 (2006), DOI 10.1007/s10955-006-9227-3, https://doi.org/10.1007/s10955-006-9227-3
  11. Daniel J. Bedingham, 'Dynamical state reduction in an EPR experiment', Journal of Physics A: Mathematical and Theoretical 42, 465301 (2009), DOI 10.1088/1751-8113-42-46/465301, https://doi.org/10.1088/1751-8113-42-46/465301
  12. C. Jones, T. Guaita, A. Bassi, 'Impossibility of extending the Ghirardi-Rimini-Weber model to relativistic particles', Physical Review A 103, 042216 (2021), DOI 10.1103/PhysRevA.103.042216, https://doi.org/10.1103/PhysRevA.103.042216
  13. Matthias Lienert, 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
  14. Daniel J. Bedingham, Philip Pearle, 'Relativistic collapse model with quantized time variables', Physical Review A 112, 042218 (2025), DOI 10.1103/j4lc-s4pw, https://doi.org/10.1103/j4lc-s4pw; arXiv:2506.07959, https://arxiv.org/abs/2506.07959
  15. Avi Levy, Meir Hemmo, 'Locality and Probability in Relativistic Quantum Theories and Hidden Variables Quantum Theories', Foundations of Physics 55, article 73 (2025), DOI 10.1007/s10701-025-00885-8, https://doi.org/10.1007/s10701-025-00885-8; arXiv:2503.12967, https://arxiv.org/abs/2503.12967