In the decoherent (consistent) histories formulation of quantum mechanics for a closed system such as the universe, decoherence alone admits a vast multiplicity of incompatible sets of alternative histories, so the quasiclassical realm of familiar experience cannot be identified without an additional selection principle. Gell-Mann and Hartle organized the strongly decohering histories into 'full' sets falling into equivalence classes labeled by Hilbert-space bases, and posed — but did not answer — the question of which quantitative, dynamics-informed measure of classicality, optimized over these sets, picks out the quasiclassical realm(s) from the Hamiltonian and density matrix alone. More than three decades of subsequent work, including the authors' own augmented-entropy proposal, have produced only candidate measures with known pathologies and no completed selection procedure, no implementation in any concrete model, and no determination of whether the quasiclassical realm is essentially unique. The problem is to complete this selection program: define the measure with its pathology-excluding restrictions, make it well defined for mixed density matrices where medium decoherence no longer implies strong decoherence, carry out the optimization/selection procedure from which the quasiclassical realm(s) emerge from (H,ρ)(H, \rho) alone, and settle the uniqueness question.

Background

The decoherent (or consistent) histories formulation treats quantum mechanics as a theory of a closed system — ideally the universe — characterized by a Hamiltonian HH and a density matrix ρ\rho (the initial condition, in the Heisenberg picture), with observers and apparatus inside the system. An exhaustive set of alternative coarse-grained histories is specified by sequences of projection operators Pαk(tk)P_{\alpha_k}(t_k) onto ranges of values of observables at times t1<⋯<tnt_1 < \cdots < t_n; an individual history α=(α1,…,αn)\alpha = (\alpha_1,\ldots,\alpha_n) corresponds to the chain operator Cα=Pαn(tn)⋯Pα1(t1)C_\alpha = P_{\alpha_n}(t_n)\cdots P_{\alpha_1}(t_1). Probabilities p(α)=Tr(CαρCα†)p(\alpha) = \mathrm{Tr}(C_\alpha \rho C_\alpha^\dagger) obey the usual sum rules only when the set decoheres, i.e. when the decoherence functional D(α′,α)=Tr(Cα′ρCα†)D(\alpha',\alpha) = \mathrm{Tr}(C_{\alpha'} \rho C_\alpha^\dagger) is negligible off-diagonal (vanishing real part is 'weak' decoherence; vanishing of the whole functional is 'medium' decoherence, the standard condition).

Gell-Mann and Hartle's 1990 paper 'Alternative Decohering Histories in Quantum Mechanics' (a conference-proceedings article posted to arXiv only in 2019) introduced two refinements of this framework. First, strong decoherence: the condition Cαρ=RαρC_\alpha \rho = R_\alpha \rho with mutually orthogonal projection operators RαR_\alpha, which they identified with the existence of generalized records — present data perfectly correlated with past histories, so that 'what happened' can be reconstructed without assuming an observer. For pure ρ\rho, medium decoherence implies strong decoherence, but for mixed ρ\rho it generally does not (their example: the completely indifferent initial state ρind=1/Tr(1)\rho_{\mathrm{ind}} = 1/\mathrm{Tr}(1), for which strong decoherence forces all projections at different times to commute). Second, full sets: a strongly decohering set can be fine-grained until the record projections RαR_\alpha form a complete set of one-dimensional projections defining a basis of Hilbert space; such full sets fall into equivalence classes sharing that basis. This was meant to replace the older notion of maximal decoherent sets, which contain vast trivial redundancy.

The physical target is the quasiclassical realm: a decohering set of histories whose probabilities are concentrated on histories exhibiting correlations governed by deterministic classical equations (Einstein's, Navier–Stokes', Maxwell's), expected to be defined by hydrodynamic variables — averages over suitable spatial volumes of densities of conserved quantities such as energy and momentum. The ambition is to show that the quasiclassical realm is an emergent feature of (H,ρ)(H, \rho), not something posited from outside quantum mechanics. The obstacle is that decoherence is wildly non-unique: most full sets bear no resemblance to classical physics, and different inequivalent quasiclassical realms may coexist. The 1990 paper therefore sought a quantitative measure of classicality that references both ρ\rho and HH (the dynamics must enter, otherwise one state vector is as good as another), to be maximized or minimized over full sets and their equivalence classes. It offered two candidates with complementary defects: S^\hat S, the entropy of formal probabilities built from the Schrödinger-picture chains and the indifferent density matrix ρind\rho_{\mathrm{ind}}, which rewards time-translation correlation of the projections but ignores the actual ρ\rho; and S(ρ~)S(\tilde\rho), the maximum of −Tr(ρ~log⁡ρ~)-\mathrm{Tr}(\tilde\rho \log \tilde\rho) over density matrices ρ~\tilde\rho preserving the histories' decoherence functional, which rewards determinism and stretched-out histories but is indifferent to ρ\rho-independent redundant refinements of the histories (and actually favors those connected with ρ\rho) and does not discriminate among sets related by reassignment of the times tkt_k or by unitary transformations of Hilbert space that preserve ρ\rho — trivial variants that merely stretch the same record information out in time in uninteresting ways.

The authors later (in 'Strong Decoherence', 1995) proposed a third candidate, the augmented entropy Σ=KU′(E^)+S\Sigma = K_{U'}(\hat E) + S: the sum of the effective complexity of the coarse-grained ensemble — the length of the shortest program (algorithmic information content, in the sense of Kolmogorov, Chaitin, and Solomonoff) describing its regularities — and the coarse-grained Shannon entropy of the histories' probabilities. Smaller Σ\Sigma should mean more classical. But they exhibited an explicit pathology: a set that projects at every time onto the state vector ∣Ψ⟩|\Psi\rangle and its orthogonal complement makes Σ\Sigma small without being remotely quasiclassical, so unspecified 'suitable restrictions' on the admissible operator sets are needed before the measure can define anything.

The problem this raises is a recognized open issue in the field, sometimes called the set-selection problem: Dowker and Kent's 1996 critique of the consistent-histories approach concluded that the formalism must be supplemented by some selection principle to yield unconditional predictions about which histories are classical. Several adjacent research programs approach classicality emergence from other directions without solving this one: environment-induced decoherence and quantum Darwinism (einselection of robust pointer states of open subsystems through environmental monitoring); quantum mereology (an in-principle algorithm of Carroll and Singh that selects a Hilbert-space tensor factorization whose subsystems have quasiclassical dynamics, by minimizing entanglement growth and internal spreading); and recent complexity-based definitions of wavefunction branches (decompositions of the state into components that are hard to interfere but easy to distinguish, proposed by Weingarten and by Taylor and McCulloch).

Problem Statement

Given a closed quantum system specified by its Hamiltonian HH and density matrix ρ\rho (in general mixed — the realistic case for the universe) that is expected to exhibit a quasiclassical realm, complete the quasiclassical-realm selection program of Gell-Mann and Hartle: formulate a precise, dynamics-informed measure of classicality for full sets of alternative strongly decohering histories, together with a completed — possibly multi-step — optimization or selection procedure over those full sets and over the equivalence classes, labeled by Hilbert-space bases, to which they belong, such that the quasiclassical realm(s) emerge as features of (H,ρ)(H, \rho) alone rather than by assumption. A complete answer should: (i) specify the restrictions on admissible sets of projection operators that exclude pathological strongly decohering sets — such as the set projecting at every time onto the state vector ∣Ψ⟩|\Psi\rangle and its orthogonal complement, which minimizes the augmented entropy Σ\Sigma without yielding quasiclassical behavior, and the trivial refinements that merely repeat projections at reassigned times or under ρ\rho-preserving unitaries — while remaining well defined for arbitrary mixed ρ\rho, where medium decoherence no longer implies strong decoherence and generalized records must be constructed rather than inferred; the restrictions must not themselves presuppose which variables or which sets are quasiclassical — in the program's own formulation the quasiclassical variables must 'emerge as a consequence of ρ\rho and HH and not have to be put in as an assumption', and the authors suggested that the pathology-excluding properties might be connected with locality — so that the quasiclassical realm is selected by the measure applied to (H,ρ)(H, \rho) rather than built into the admissible class; (ii) establish that the resulting criterion is well defined and precise, given that the algorithmic-information terms entering the candidate measures are uncomputable; and (iii) determine whether the quasiclassical realm so selected is essentially unique, or whether the same (H,ρ)(H, \rho) exhibits essentially inequivalent multiple quasiclassical realms — either conclusion settles the question — stating explicitly the notion of equivalence between quasiclassical realms on which the determination rests.

The question posed here is the measure and the selection procedure themselves; model demonstrations of the procedure in a nontrivial closed system — for example the quantum harmonic chain (a lattice of coupled quantum harmonic oscillators) or a minisuperspace quantum-cosmological model (a truncated quantum-gravity treatment retaining only finitely many homogeneous cosmological degrees of freedom) — are presented by the program's authors as a further test of the measure ('a major research effort'). Regimes where quantum gravity cannot be ignored (large fluctuations of the spacetime metric, sums over spacetime topology), and a standalone systematic classification of when medium decoherence fails to imply strong decoherence, lie outside the scope of the question posed here.

References

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