In the decoherent-histories formulation of the quantum mechanics of closed systems, Gell-Mann and Hartle (Phys. Rev. A 76, 022104 (2007)) characterized the quasiclassical realm underlying everyday physics — decoherent, narrative coarse grainings by hydrodynamic conserved-field variables — and explicitly left open how such realms are selected and whether they are unique. The problem asks for a complete, principled characterization of the maximally refined quasiclassical realms emergent from the Hamiltonian and initial quantum state alone: (1) an operational narrative-realm selection rule that discriminates among the many decoherent coarse grainings and excludes trivial realms; (2) a determination of whether incompatible maximal quasiclassical realms with high predictability and utility exist, or whether the quasiclassical family is essentially unique; and (3) quantitative characterization of decoherence times, admissible coarse-graining scales, and noise at near-maximal refinement for realistic quasiclassical variables. All three components remain open: partial progress exists, but no published work supplies a completed selection principle, and the uniqueness question is restated as open by Hartle and Gell-Mann (2021).

Background

To describe the universe as a whole one needs a quantum mechanics of closed systems, with no external observer or measuring apparatus. The decoherent (or consistent) histories formulation provides one: a history is a sequence of mutually exclusive alternatives at a succession of times, each alternative represented by a projection operator; the chain operator CαC_\alpha is the product of the projections defining history α\alpha (in time order, Heisenberg picture). A set of alternative histories — the authors call it a realm — can be assigned probabilities only when it decoheres, i.e. when the off-diagonal elements of the decoherence functional D(α,β)=Tr(Cα ρ Cβ†)D(\alpha,\beta)=\mathrm{Tr}(C_\alpha\,\rho\,C_\beta^\dagger) vanish (medium decoherence), where ρ\rho is the initial state of the closed system. There is no non-trivial probabilistic description at the fine-grained level: decohering perfectly fine-grained sets of histories are necessarily trivial, for example sets whose only non-zero histories never change between successive times (the source paper devotes an appendix to this), so some coarse graining — grouping alternatives and sacrificing information — is inescapable for probability.

Different coarse grainings define different realms. Two realms are compatible when both are coarse grainings of one common finer-grained decoherent set; otherwise they are incompatible, and probabilities may not be combined across them. Quantum mechanics by itself prefers no realm over another. Observers, which the authors call information gathering and utilizing systems (IGUSes), use a tiny subset: the quasiclassical realms, whose histories are narratives — sequences of projections at successive times related by a rule, so that the histories follow the same or related quantities through time rather than presenting unrelated scraps — and whose variables are quasiclassical: spatial averages over small volumes of densities of conserved quantities (energy, momentum) and approximately conserved quantities (e.g. baryon number), i.e. the variables of classical hydrodynamics such as the Navier–Stokes equations. Approximate conservation yields approximate decoherence, high predictability through closed effective equations of motion, and local equilibrium; utility — the usefulness of a realm's histories for prediction and information gathering — is, alongside predictability, the source's criterion for realistic realms. Maximally refined quasiclassical realms — as fine-grained as decoherence and classicality permit — are, on the authors' view, features of the universe emergent from the Hamiltonian HH, the initial quantum state ∣Ψ⟩|\Psi\rangle, and the long sequence of chance events, not choices of the observer. The source paper develops this picture in a simplified model universe in a box — a large finite region of nearly flat spacetime whose fields act on bulk matter through short-range forces — explicitly setting aside gravitational and cosmological complications.

The same coarse graining connects to thermodynamics: for a coarse-grained description specified by expected values ⟨Am⟩\langle A_m\rangle, the appropriate entropy is the maximum (von Neumann) entropy S=−Tr(ρ~log⁡ρ~)S=-\mathrm{Tr}(\tilde\rho\log\tilde\rho) over effective density matrices ρ~\tilde\rho consistent with those expectations (the Jaynes maximum-entropy construction), which for quasiclassical variables reproduces the usual thermodynamic entropy of statistical mechanics, low for the early universe and increasing since.

Two lines of prior work frame the open problem. First, J. J. Halliwell (Phys. Rev. Lett. 83, 2481 (1999)) demonstrated decoherence and small fluctuations for hydrodynamic histories of NN nonrelativistic particles; the source paper calls this result 'the best we have today' but stresses that its coarse graining is likely far coarser than maximal, while much more quantitative oscillator-model studies of classicality (decoherence times, noise) are less realistic. Second, F. Dowker and A. Kent (J. Stat. Phys. 82, 1575 (1996)) argued that the histories formulation is incomplete without a fundamental set-selection principle singling out the quasiclassical sets, and A. Kent (Phys. Rev. Lett. 78, 2874 (1997)) showed that different consistent sets can license contrary inferences. The 2007 paper responds that a suitable rule defining narrative realms is being sought 'using ideas about simplicity, complexity, and logical depth' — algorithmic information content (the length of the shortest program computing a description) and logical depth (a measure of the computational work embodied in a description) — and, in a footnote, explicitly leaves open 'whether there are incompatible maximal realms that also exhibit high levels of predictability and utility'. Three later references might appear to close this gap but do not: Wallden's set-selection suggestion (ref 5) does not single out a unique set; the numerical demonstration of Strasberg, Reinhard and Schindler (ref 6) establishes coherence suppression for slow, coarse observables of a random-matrix model, not a completed operational criterion for maximally refined quasiclassical realms; and the algorithmic search of Adil et al. (ref 14) discovers tensor-product quasiclassical subsystems rather than the hydrodynamic narrative realms at issue.

Problem Statement

Give a complete, principled characterization of the quasiclassical realm(s) of a closed quantum system specified by its Hamiltonian HH and initial quantum state ∣Ψ⟩|\Psi\rangle, as left open by Gell-Mann and Hartle, 'Quasiclassical coarse graining and thermodynamic entropy' (Phys. Rev. A 76, 022104 (2007)). The question has three jointly required components:

  1. A principled narrative-realm selection rule. A concrete, operational criterion that takes HH and ∣Ψ⟩|\Psi\rangle (plus stated admissibility restrictions on projection operators, e.g. locality) as input, discriminates among the many decoherent coarse grainings of a closed system, excludes trivial non-narrative realms (such as realms repeating the same projections at every time, in which nothing happens), and determines the maximally refined quasiclassical realms as emergent features of the universe rather than choices of the observer.

  2. Resolution of the uniqueness question. A determination of whether there exist incompatible maximal quasiclassical realms — mutually inequivalent maximally refined decoherent coarse grainings that each exhibit high levels of predictability and utility — or whether the family of maximally refined quasiclassical realms emergent from (H,∣Ψ⟩)(H,|\Psi\rangle) is essentially unique up to compatibility, together with any fundamental selection principle this entails. The source states this as: 'We leave open whether there are incompatible maximal realms that also exhibit high levels of predictability and utility.'

  3. Quantitative characterization at maximal refinement. For the model classes to which the rule applies, quantitative results for the selected maximally refined realms — decoherence times, the finest coarse-graining scales consistent with decoherence and classicality, and the effects of quantum noise on the followed quasiclassical variables — at a precision comparable to what oscillator models achieve, replacing the qualitative and semiclassical arguments available to the source.

These three components jointly answer one question: whether the everyday classical description of the universe is a derivable and essentially unique emergent feature of (H,∣Ψ⟩)(H,|\Psi\rangle) within decoherent-histories quantum mechanics. The uniqueness question concerns the maximally refined quasiclassical realms themselves and not a class of coarse grainings artificially restricted to make it decidable, and the quasiclassicality of part (3) concerns histories at near-maximal refinement, not only at coarse grainings far from it. The uniqueness in part (2) is a property of the maximally refined quasiclassical realms of (H,∣Ψ⟩)(H,|\Psi\rangle) themselves, not of any particular selection rule, and the predictability, utility, and oscillator-model precision invoked above are quantitative requirements to be stated for an explicit model class with stated parameters and tolerances (for instance a decoherence time or noise level reproduced to within a stated factor). The source's further acknowledged directions — generalization beyond the box model to quantum spacetime, gravitational clumping, black holes, proton decay, and accelerated expansion, and the possible role of non-additive q≠1q\neq1 entropies in metastable gravitational regimes — are neighboring program directions outside this problem's completion boundary.

References

  1. Murray Gell-Mann and James B. Hartle, 'Quasiclassical coarse graining and thermodynamic entropy', Physical Review A 76, 022104 (2007). DOI: 10.1103/PhysRevA.76.022104; arXiv:quant-ph/0609190. https://doi.org/10.1103/PhysRevA.76.022104
  2. Murray Gell-Mann and James 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://arxiv.org/abs/1312.7454
  3. Murray Gell-Mann and James 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://arxiv.org/abs/1106.0767
  4. James B. Hartle and Murray Gell-Mann, 'A measure of classicality', arXiv:2104.14465 (2021). https://arxiv.org/abs/2104.14465
  5. Petros Wallden, 'Contrary inferences in consistent histories and a set selection criterion', Foundations of Physics 44, 1195–1215 (2014). DOI: 10.1007/s10701-014-9837-6; arXiv:1402.3733. https://doi.org/10.1007/s10701-014-9837-6
  6. Philipp Strasberg, Teresa E. Reinhard and Joseph Schindler, 'First principles numerical demonstration of emergent decoherent histories', Physical Review X 14, 041027 (2024). DOI: 10.1103/PhysRevX.14.041027; arXiv:2304.10258. https://doi.org/10.1103/PhysRevX.14.041027
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  8. Adamantia Zampeli, Georgios E. Pavlou and Petros Wallden, 'Contrary inferences for classical histories within the consistent histories formulation of quantum theory', Foundations of Physics 56, article 3 (2026); published online 1 December 2025. DOI: 10.1007/s10701-025-00902-w; arXiv:2205.15893. https://doi.org/10.1007/s10701-025-00902-w
  9. C. Jess Riedel, 'Wavefunction branches demand a definition!', Quantum Views 9, 85 (2025). DOI: 10.22331/qv-2025-06-16-85; arXiv:2506.15663. https://doi.org/10.22331/qv-2025-06-16-85
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