Gell-Mann and Hartle [Phys. Rev. A 85, 062120 (2012)] reformulated decoherent histories quantum mechanics for a closed system by postulating one real fine-grained history, described in a preferred sum-over-histories variable set and embedded in an ensemble of imagined histories assigned extended probabilities (values possibly negative or exceeding one) through a fundamental distribution built from the quantum state and action. Standard quantum probabilities are recovered for coarse-grained sets whose histories are recorded. The authors explicitly acknowledge a bundle of unresolved issues: whether deeper principles can justify the fundamental distribution rather than postulating it; what restrictions on allowed records and coarse grainings, beyond the general recorded/decoherence condition, guarantee that the ensemble reliably describes the quasiclassical realms used by information-gathering systems (the typicality condition); how to quantify and control the approximations inherent in the record condition in realistic models; how to delineate regimes where fine-grained histories obtain zero or negligible extended probability (Dowker's spin examples); and how to extend the formulation beyond a fixed spacetime background to fully quantum-gravitational contexts. No resolution of any of these issues is known in the literature to date. The problem is open.

Background

For a closed quantum system — ultimately the whole universe — the decoherent (consistent) histories (DH) formulation of Griffiths, Omnès, Gell-Mann and Hartle assigns probabilities not to instantaneous states but to sets of alternative coarse-grained histories ('realms'); interference between histories generally prevents assigning ordinary probabilities to completely fine-grained ones. A 'quasiclassical realm' is a decoherent set of coarse-grained histories defined by quasiclassical variables (suitably averaged, approximately conserved quantities such as local energy and number density) whose individual histories exhibit patterns of correlation described by classical equations of motion; 'IGUS' abbreviates information gathering and utilizing system, the framework's replacement for human observers. A long-standing difficulty of the DH approach is that it admits many mutually incompatible decoherent realms, so that unadorned DH supplies no unified notion of what is 'real' across realms. Gell-Mann and Hartle (2012) addressed this by postulating that there is one real fine-grained history qreal(t)q_{\mathrm{real}}(t), specified in a preferred complete set of sum-over-histories variables (particle paths or field configurations), and by embedding it in an ensemble of comparable imagined histories — analogous to the ensemble of statistical mechanics — assigned extended probabilities w[q(t)]w[q(t)], i.e., signed quantities that may be negative or greater than one and are read as 'don't bet — not settleable' when non-standard. The fundamental distribution is defined directly from the state and action, w[q(t)]=Re[Ψ∗(qf,tf) eiS[q(t)]/ℏ Ψ(q0,t0)]w[q(t)]=\mathrm{Re}\left[\Psi^{*}(q_f,t_f)\,e^{iS[q(t)]/\hbar}\,\Psi(q_0,t_0)\right], and every coarse graining (partition of the fine-grained histories into classes cαc_\alpha) receives the summed extended probability p(α)=∫cαDq w[q(t)]p(\alpha)=\int_{c_\alpha}Dq\,w[q(t)], which satisfies the probability sum rules exactly. Standard quantum mechanics is recovered for sets of histories that are recorded: a set of chain (class-operator) histories CαC_\alpha is recorded if there are orthogonal projections RαR_\alpha at a time after the last alternative such that Rα∣Ψ⟩≈Cα∣Ψ⟩R_\alpha|\Psi\rangle\approx C_\alpha|\Psi\rangle (Eq. (3.1) of the source); then p(α)≈⟨Ψ∣Rα∣Ψ⟩≈∥Cα∣Ψ⟩∥2p(\alpha)\approx\langle\Psi|R_\alpha|\Psi\rangle\approx\|C_\alpha|\Psi\rangle\|^2 and the set decoheres, yielding the Born-rule probabilities of DH. Three gaps survive this construction. First, whether the specific form of w[q(t)]w[q(t)] follows from deeper principles is left open by the authors. Second, the record condition (3.1) is very general, and in both classical statistical mechanics and this quantum ensemble one can construct pathological coarse grainings (e.g., partitioning on whether the initial configuration lies in a tiny region around its actual value) for which the real history is assigned negligible or even zero probability; the real coarse-grained history rr is then atypical, violating the typicality condition −log⁡p(r)≪−∑αp(α)log⁡p(α)=S-\log p(r)\ll-\sum_\alpha p(\alpha)\log p(\alpha)=S (Eq. (5.1)). Fay Dowker exhibited simple spin models in which, for every fine-grained history, some coarse graining assigns it probability zero, and the allowed records and coarse grainings must therefore be restricted beyond (3.1) if the ensemble is to reliably describe the quasiclassical realms exploited by IGUSs — but the authors give no explicit restriction. Third, the approximation sign in (3.1) is quantified only through crude estimates (overlap magnitudes of order 10−102010^{-10^{20}} for quasiclassical branches) in the predecessor treatment (Hartle 2008), and the whole formulation works in a fixed, non-fluctuating background spacetime, while its motivating application is quantum cosmology where spacetime geometry itself fluctuates.

Problem Statement

Within the extended-probability ensemble (EPE) formulation of Gell-Mann and Hartle (Phys. Rev. A 85, 062120 (2012)) — one real fine-grained history qreal(t)q_{\mathrm{real}}(t) in a preferred sum-over-histories variable set, embedded in an ensemble with the fundamental distribution w[q(t)]=Re[Ψ∗(qf,tf)eiS[q(t)]/ℏΨ(q0,t0)]w[q(t)]=\mathrm{Re}[\Psi^{*}(q_f,t_f)e^{iS[q(t)]/\hbar}\Psi(q_0,t_0)] — establish the formulation's justification, reliability, and scope by resolving the five interlinked issues the authors themselves acknowledge as remaining: turning the EPE construction from a postulate-based framework into a principled and controlled description of a closed quantum system. These five issues are interlinked but each is separately resolvable: resolving any one of them resolves the corresponding part of the objective, does not require the others to be settled first, and leaves the remaining issues open. Specifically: (1) determine whether there exist deeper principles or dynamical derivations that fix the specific form of the fundamental distribution w[q(t)]w[q(t)] beyond postulating it from the state and action, or establish that no such derivation exists within quantum theory; (2) identify the precise restrictions on allowed records and coarse grainings, beyond the general recorded/decoherence condition Rα∣Ψ⟩≈Cα∣Ψ⟩R_\alpha|\Psi\rangle\approx C_\alpha|\Psi\rangle, under which the ensemble reliably describes the quasiclassical realms exploited by IGUSs — i.e., give operational, systematic criteria for selecting coarse grainings that satisfy the typicality condition −log⁡p(r)≪−∑αp(α)log⁡p(α)-\log p(r)\ll-\sum_\alpha p(\alpha)\log p(\alpha) in realistic situations, so that the real coarse-grained history is typical and pathological grainings (such as those in Dowker's spin examples, where every fine-grained history admits some coarse graining assigning it zero probability) are excluded; (3) quantify and control the approximations inherent in the record condition Rα∣Ψ⟩≈Cα∣Ψ⟩R_\alpha|\Psi\rangle\approx C_\alpha|\Psi\rangle — going beyond crude overlap estimates — and assess their sufficiency in realistic models, i.e., specify how approximately a set of histories must be recorded for its coarse-grained extended probabilities to be reliably positive, normalized, and empirically usable; (4) delineate, in concrete model examples (including spin models of the Dowker type), the regimes in which fine-grained histories obtain zero or negligible extended probability, and determine the empirical significance of such cases; and (5) extend the formulation beyond the assumed fixed background spacetime to contexts with gross quantum fluctuations of spacetime geometry, i.e., formulate the EPE construction for quantum spacetime without a fixed notion of time.

References

  1. M. Gell-Mann and J. B. Hartle, 'Decoherent histories quantum mechanics with one real fine-grained history', Physical Review A 85, 062120 (2012). DOI: 10.1103/PhysRevA.85.062120; arXiv:1106.0767; https://doi.org/10.1103/PhysRevA.85.062120
  2. J. B. Hartle, 'Quantum mechanics with extended probabilities', Physical Review A 78, 012108 (2008). DOI: 10.1103/PhysRevA.78.012108; arXiv:0801.0688; https://doi.org/10.1103/PhysRevA.78.012108
  3. M. Gell-Mann and J. B. Hartle, 'Adaptive coarse graining, environment, strong decoherence, and quasiclassical realms', Physical Review A 89, 052125 (2014). DOI: 10.1103/PhysRevA.89.052125; arXiv:1312.7454; https://doi.org/10.1103/PhysRevA.89.052125
  4. J. B. Hartle, 'Decoherent Histories Quantum Mechanics Starting with Records of What Happens', arXiv preprint arXiv:1608.04145 (2016); https://arxiv.org/abs/1608.04145
  5. J. B. Hartle, 'What are the Realities', arXiv preprint arXiv:2112.10282 (2021); https://arxiv.org/abs/2112.10282
  6. A. Kent, 'Path Integrals and Reality', arXiv preprint arXiv:1305.6565 (2013); https://arxiv.org/abs/1305.6565
  7. E. Okon and D. Sudarsky, 'On the Consistency of the Consistent Histories Approach to Quantum Mechanics', Foundations of Physics 44, 19–33 (2014). DOI: 10.1007/s10701-013-9760-2; https://doi.org/10.1007/s10701-013-9760-2
  8. F. Dowker and Y. Ghazi-Tabatabai, 'The Kochen–Specker theorem revisited in quantum measure theory', Journal of Physics A: Mathematical and Theoretical 41, 105301 (2008). DOI: 10.1088/1751-8113-41/10/105301; arXiv:0711.0894; https://doi.org/10.1088/1751-8113/41/10/105301
  9. J. J. Halliwell and J. M. Yearsley, 'Negative probabilities, Fine's theorem, and linear positivity', Physical Review A 87, 022114 (2013). DOI: 10.1103/PhysRevA.87.022114; arXiv:1210.6870; https://doi.org/10.1103/PhysRevA.87.022114
  10. J. Steeger, 'One world is (probably) just as good as many', Synthese 200, 97 (2022). DOI: 10.1007/s11229-022-03499-z; https://doi.org/10.1007/s11229-022-03499-z
  11. P. Wallden, 'Distinguishing Initial State-Vectors from Each Other in Histories Formulations and the PBR Argument', Foundations of Physics 43, 1502–1525 (2013). DOI: 10.1007/s10701-013-9759-8; https://doi.org/10.1007/s10701-013-9759-8
  12. D. P. B. Schroeren, 'Decoherent Histories of Spin Networks', Foundations of Physics 43, 310–328 (2013). DOI: 10.1007/s10701-013-9698-4; https://doi.org/10.1007/s10701-013-9698-4
  13. H. J. D. Miller and J. Anders, 'Time-reversal symmetric work distributions for closed quantum dynamics in the histories framework', New Journal of Physics 19, 062001 (2017). DOI: 10.1088/1367-2630/aa703f; https://doi.org/10.1088/1367-2630/aa703f
  14. A. Adil, M. S. Rudolph, A. Arrasmith, Z. Holmes, A. Albrecht, and A. Sornborger, 'Search for classical subsystems in quantum worlds', Physical Review D 113, 103535 (2026). DOI: 10.1103/vj4x-96fk; arXiv:2403.10895; https://doi.org/10.1103/vj4x-96fk
  15. P. Solinas and S. Gherardini, 'Quantumness certification via non-demolition measurements', Advances in Physics: X 11, 2637832 (2026). DOI: 10.1080/23746149.2026.2637832; arXiv:2512.09734; https://doi.org/10.1080/23746149.2026.2637832
  16. M. Caprio and S. Mukherjee, 'Extended Probabilities and their Application to Statistical Inference', in Understanding Information and Its Role as a Tool, pp. 299–352 (World Scientific, 2025). DOI: 10.1142/9789811294921_0011; https://doi.org/10.1142/9789811294921_0011
  17. R. B. Griffiths, Consistent Quantum Theory, Cambridge University Press (2002). DOI: 10.1017/cbo9780511606052; https://doi.org/10.1017/cbo9780511606052