In time-neutral generalized (decoherent-histories) quantum mechanics, Hartle (2013) showed by examples that quantum-mechanical and thermodynamic arrows of time can emerge from cosmological boundary conditions, that they may be local — pointing in different directions in different spacetime regions, as in a bouncing universe — and that some boundary conditions yield no well-defined arrow at all. The open problem is to convert these examples into a general theory: necessary and sufficient conditions on the boundary data (an initial/final pair of density matrices, or a single density matrix at a bounce) that determine whether local arrows emerge, whether the quantum and thermodynamic arrows align, or whether no arrow is defined. The same program includes characterizing the physics of two-boundary situations with nontrivial initial and final density matrices and quantifying what can be inferred across a bounce from a mixed bounce state. Two broader directions that the paper discusses without posing them as open problems are extending the construction from fixed classical backgrounds to quantum spacetime and delimiting where the emergent local notions of quantum mechanics (states on spacelike surfaces, unitary evolution, reduction) are valid. A 2020 sequel by Hartle proved necessary decoherence constraints on two-boundary data — a pure final state admits no nontrivial decoherent set of histories, and the rank of the final density matrix bounds how many histories can decohere — which partially advances the two-boundary component, but the general classification and the remaining components survive as an open core.

Background

The familiar quantum mechanics of laboratory measurements already contains an arrow of time: von Neumann's second evolution law, the reduction of the state on measurement, operates in only one time direction, while the Schrödinger equation is reversible. For the universe as a whole — a closed system with no outside observer — such a built-in arrow is not acceptable, and one instead uses the decoherent (consistent) histories formulation, in which quantum theory assigns probabilities to coarse-grained histories of a closed system. A set of alternative histories, represented by chains of projection operators CαC_\alpha, can be assigned probabilities only when quantum interference between its members is negligible, i.e. when the set decoheres. In the time-neutral version of this formalism, developed by Gell-Mann and Hartle building on the two-time measurement formalism of Aharonov, Bergmann and Lebowitz, the theory is specified by a triple (H,ρi,ρf)(H,\rho_i,\rho_f): a time-neutral Hamiltonian and density matrices representing an initial and a final condition, entering symmetrically through p(α)=N Tr(ρfCαρiCα†)p(\alpha)=N\,\mathrm{Tr}(\rho_f C_\alpha \rho_i C_\alpha^\dagger) with N−1=Tr(ρfρi)N^{-1}=\mathrm{Tr}(\rho_f\rho_i), so that no arrow is built into the framework. More generally, generalized quantum theory is any framework specified by a decoherence functional D(α,α′)D(\alpha,\alpha') satisfying hermiticity, normalization, positivity, and the principle of superposition, with decoherent sets obeying D(α,α′)≈δαα′p(α)D(\alpha,\alpha')\approx\delta_{\alpha\alpha'}p(\alpha).

With the conventional pair of boundary conditions — a special low-entropy initial ρi\rho_i and a final condition of indifference ρf∝I\rho_f\propto I — both a thermodynamic arrow (growth of the usual coarse-grained entropy) and a quantum-mechanical arrow emerge codirectionally. Hartle (2013) examined what other boundary data produce. When both ρi\rho_i and ρf\rho_f are nontrivial, there is generally no clear definition of a global thermodynamic or quantum-mechanical arrow: the entropy may rise and then fall, and there is no single state at a moment of time from which either future or past could be predicted. For bouncing universes — motivated by the no-boundary wave function of Hartle and Hawking (the proposal that the quantum state of the universe is given by a path integral over compact geometries, with no boundary in the past) and by the Hartle–Hertog result that in the no-boundary state matter fluctuations are small at the bounce rather than at the large-volume ends — he posited a generalized quantum theory with a single density matrix ρ0\rho_0 at the bounce and a decoherence functional D(β,α;β′,α′)=Tr(CβBρ0 CαA†Cα′Aρ0 Cβ′B†)D(\beta,\alpha;\beta',\alpha')=\mathrm{Tr}(C^B_\beta\sqrt{\rho_0}\,C^A_{\alpha}{}^\dagger C^A_{\alpha'}\sqrt{\rho_0}\,C^B_{\beta'}{}^\dagger) over chain operators on the two sides. Each side then has coincident quantum and thermodynamic arrows pointing away from the bounce, so arrows are local and can be consistent with global time symmetry; for a pure ρ0\rho_0 the joint probabilities factor exactly across the bounce, p(β,α)=pB(β)pA(α)p(\beta,\alpha)=p_B(\beta)p_A(\alpha), and the far side is uncorrelated with ours. In all of the paper's examples where both arrows were locally available they pointed the same way — but, as the paper itself notes, a few examples do not make a general result.

Problem Statement

Develop the general theory, of which Hartle (2013) supplied only examples, of how arrow-of-time structure follows from boundary conditions in time-neutral quantum mechanics of closed systems. The common objective is a classification of boundary data by their arrow structure, with the following jointly required components: (a) necessary and sufficient conditions on the boundary data — the pair (ρi,ρf)(\rho_i,\rho_f) for an initial/final formulation, or the bounce density matrix ρ0\rho_0 of the bounce decoherence functional — for (i) a well-defined local quantum-mechanical arrow to emerge, (ii) the local quantum and thermodynamic arrows to be codirectional, and (iii) no well-defined quantum arrow at all (the three outcomes are exhaustive, with (ii) the stronger case contained in (i) and (iii) the complement of (i)); the conditions should specify which properties of the data matter (pure versus mixed, rank, degree of entropic specialness relative to a condition of indifference). (b) A characterization of the physics of two-boundary situations in which both ρi\rho_i and ρf\rho_f are nontrivial — decoherence, predictability, and entropy evolution — in which no global arrow is currently defined, beyond the sharp necessary constraints already established, such as the rank bound on decoherent alternatives (Hartle 2020). (c) A quantification of cross-bounce inference for mixed ρ0\rho_0: how much information about histories on the far side of a bounce is contained in observations on one side — per the source, in the joint probabilities of the bounce decoherence functional — given that for pure ρ0\rho_0 the two sides are exactly uncorrelated. Two broader directions discussed by the source lie beyond the classification core: extending the time-neutral generalized quantum framework and the bounce decoherence-functional construction to quantum dynamical spacetime, and delimiting where the emergent local notions of quantum mechanics — states on spacelike surfaces, their unitary evolution, and reduction — are valid; the paper touches these only as motivation for the bounce model and as a speculative concluding remark, and they remain context for, rather than required components of, the classification.

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