Gell-Mann and Hartle [Phys. Rev. A 89, 052125 (2014)] gave a formal characterization of the quasiclassical realms of our quantum universe: sets of alternative coarse-grained histories defined by adaptive branch-dependent coarse grainings built on a narrative framework of quasiclassical variables, decohering strongly through records stored in an environment constructed from the coarse graining itself. The paper, however, provided only the formal machinery: no concrete realistic dynamical model was shown to satisfy the combined conditions, the Hilbert-space factorization underlying the construction is obstructed in finite dimensions, no dynamical criterion or rate for record formation was given, and the stronger condition (Eq. 7.9, labeled 'Too Strong') that would make branch density matrices diagonal was argued only in general terms rather than determined for any concrete model. This problem asks for the physics behind the formalism: construct and verify the full characterization in a concrete realistic many-body or cosmological model, resolving the factorization obstruction, deriving record-formation criteria and timescales, and settling whether and when branch density matrices become diagonal. No prior work has carried a concrete realistic model through the full construction.

Background

In the decoherent (consistent) histories formalism, quantum theory is applied to a closed system such as the universe: a 'realm' is a set of alternative coarse-grained histories, each history being a sequence of exhaustive sets of orthogonal projection operators applied to an initial state vector Ψ^0\hat{\Psi}^0 with unitary evolution in between. Coarse graining means replacing the fully fine-grained set of alternatives at each time by a partition into groups; the groups define the projections P^αkk\hat{P}_{\alpha_k}^{k}. A history's class operator chains these projections, and its branch state vector is the result of applying the chain to Ψ^0\hat{\Psi}^0. The weakest standard consistency requirement is medium decoherence: the branch state vectors of different histories are mutually orthogonal, so the probabilities defined by the decoherence functional (the matrix of inner products of branch state vectors) obey the ordinary sum rules.

Gell-Mann and Hartle (2014) sharpened three ingredients of this picture. First, branch-dependent adaptive coarse grainings: the projections at time tkt_k may carry upper indices recording which earlier alternatives occurred, P^αkkαk−1⋯α1\hat{P}_{\alpha_k}^{k\alpha_{k-1}\cdots\alpha_1}, and even the times may depend on the preceding alternatives, so that the graining follows features of interest (e.g., a moving object) and avoids refining low-probability branches — aiming at graining that is as refined as decoherence permits. Second, narrative coarse grainings: the Schrödinger-picture projections at different times commute, which guarantees a common operator framework so that the histories 'tell a story' about the same variables; quasiclassical variables are averages of approximately conserved densities (energy, momentum, number) over spatial volumes, and a quasiclassical realm is a decoherent set of histories of such variables exhibiting, with high probability, patterns of correlation described by closed deterministic equations. Third, strong decoherence: when the commuting projections at a time define a system-environment tensor-product split H=Hs⊗He\mathcal{H}=\mathcal{H}^s\otimes\mathcal{H}^e with each projection acting only on the system factor, each branch state vector expands as Ψ^kαk−1⋯α1=∑rvr⊗zr\hat{\Psi}^{k\alpha_{k-1}\cdots\alpha_1}=\sum_r v_r\otimes z_r in system basis vectors vrv_r and environment vectors zrz_r, and strong decoherence requires the environment overlaps (z,z)e(z,z)^e (inner products taken in the environment factor He\mathcal{H}^e) to be proportional to Kronecker deltas over all past-alternative indices. Strong decoherence implies medium decoherence, implies orthogonality after insertion of any system observable, guarantees the existence of present records (environment projectors R^\hat{R} acting on the full present state Ψ^\hat{\Psi}, with R^Ψ^=Ψ^kαk−1⋯α1\hat{R}\hat{\Psi}=\hat{\Psi}^{k\alpha_{k-1}\cdots\alpha_1} correlating with past alternatives), ensures permanence of the past under future refinement, and yields a branchwise decomposition of system expectation values via branch density matrices ρkαk−1⋯α1s\rho^{k\alpha_{k-1}\cdots\alpha_1 s} obtained by tracing each branch state vector over the environment.

The construction rests on a mathematical caveat the authors themselves flag: the factorization of Hilbert space that makes each projection act as P^α=P^αs⊗Ie\hat{P}_\alpha=\hat{P}_\alpha^{s}\otimes I^{e} requires the projected subspaces to be infinite-dimensional, and in finite dimensions the dimension identity dim⁡(P^α)=dim⁡(P^αs) dim⁡(Ie)\dim(\hat{P}_\alpha)=\dim(\hat{P}_\alpha^{s})\,\dim(I^{e}) obstructs the factorization (a point credited to Brun and Hartle's 1999 study of the quantum harmonic chain). The paper also argues, though only in general terms, that records 'are not available instantaneously but generally take some time to form' (the analogous record construction attempted at the last time step of a history fails because orthogonality in the last alternative index is then not guaranteed), and that diagonality of the branch density matrices is not a consequence of strong or medium decoherence: it would require the stronger condition that the environment overlaps be proportional to Kronecker deltas over all indices including the present one (Eq. (7.9), which the authors label 'Too Strong' and argue would imply a Schmidt decomposition of the branch expansion, risking conflict with the quasiclassicality of the grained variables). Everything is formulated at the level of definitions and short proofs; no realistic dynamical system — the paper's own motivating examples are dust grains scattering cosmic microwave background photons, planet formation, and cosmological structure formation — is carried through the machinery.

Problem Statement

The Gell-Mann-Hartle characterization of quasiclassical realms — strongly decoherent sets of alternative histories defined by adaptive branch-dependent coarse grainings built on a narrative framework of quasiclassical variables — is currently a purely formal structure. The problem is to give it demonstrated physical content in a concrete, dynamical model of a realistic system (many-body or cosmological, in the spirit of the source's own examples such as a dust grain scattering ambient photons or cosmological density fluctuations), which jointly requires: (1) explicitly constructing the adaptive branch-dependent narrative coarse graining for the model and verifying the strong decoherence condition, i.e., that in the expansion of each branch state vector into system basis vectors and environment vectors, Ψ^kαk−1⋯α1=∑rvr⊗zr\hat{\Psi}^{k\alpha_{k-1}\cdots\alpha_1}=\sum_r v_r\otimes z_r, the environment overlaps (z,z)e(z,z)^e are proportional to Kronecker deltas over the past-alternative indices; (2) resolving the status of the system-environment Hilbert-space factorization on which this very definition rests — the construction requires commuting projections onto infinite-dimensional subspaces and is obstructed in finite dimensions by the dimension identity dim⁡(P^α)=dim⁡(P^αs) dim⁡(Ie)\dim(\hat{P}_\alpha)=\dim(\hat{P}_\alpha^{s})\,\dim(I^{e}), so a finite- or finite-approximation treatment needs exact conditions under which a (possibly approximate or dilated) split exists; (3) deriving dynamical criteria and rates for record formation — the framework guarantees records exist under strong decoherence and that they take time to form, but gives no criterion for how long, and the model analysis must supply this timescale; and (4) determining whether the stronger condition of Eq. (7.9) — environment overlaps proportional to Kronecker deltas over all indices, making the branch density matrices diagonal — is ever realized in realistic quasiclassical realms, and what phenomenological consequences such additional diagonality would have. Items (2) through (4) are not independent add-ons but the prerequisites, consequences, and natural strengthening of the construction in item (1), and must be realized jointly within one worked model or model class. The problem is answered when at least one concrete realistic model is carried through the full construction with the environment split justified and the diagonality question settled, and with the strong-decoherence condition either verified together with record-formation timescales derived, or else its failure quantified (in which case item (3) is not required).

References

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