Arrows of time in the bouncing universes of the no-boundary quantum state: validity beyond the semiclassical minisuperspace derivation

The no-boundary wave function (NBWF) — Hartle and Hawking's proposal for the quantum state of the universe — predicts, in a semiclassical minisuperspace treatment, that its classical bouncing histories have fluctuations in a low-excitation state at the bounce, so that the fluctuation arrow of time is bidirectional (fluctuations grow away from the bounce in both directions of time), with the electromagnetic, thermodynamic, and psychological arrows aligned with it; in non-bouncing (recollapsing) histories the arrow instead points in a single direction across the whole spacetime. Hartle and Hertog's 2012 derivation of these results acknowledged that they rest on unverified approximations: classical histories are extrapolated through the bounce, where the wave function's classicality conditions fail; whether histories extend quantum mechanically through singularities is open; backreaction and nonlinear fluctuations are neglected; and the very-large-scale inhomogeneities of eternal-inflation regimes complicate quantitative predictions. Later work by the same program replaced classical extrapolation through the bounce with computed quantum transitions between classical histories, treated nonlinear backreaction exactly in a solvable Bianchi IX model (a homogeneous but anisotropic universe with two independent scale factors), and computed quantum transitions through singularities — but all within saddle-point minisuperspace. Whether the arrows-of-time predictions of the no-boundary state for bouncing universes hold beyond these approximations, in the full quantum theory, remains open.

Background

The visible universe exhibits several time asymmetries, called arrows of time: the fluctuation arrow (the growth of deviations from homogeneity into nucleated bubbles, galaxies, stars, planets, and biota), the retardation of electromagnetic radiation, the thermodynamic arrow (the tendency of presently isolated systems to evolve toward equilibrium), and the psychological arrow (our distinction between past, present, and future). Since the dynamical laws are time neutral — CPT invariant in quantum field theory — these asymmetries are standardly attributed to the state of the universe rather than to the laws, an idea going back to Boltzmann's 1897 observation that the second law of thermodynamics follows if the present state evolved from an improbable special state. A final theory of the universe therefore needs, besides dynamics, a theory of its quantum state, and arrows of time should be derived as predictions of that state for the ensemble of coarse-grained classical histories it allows.

The no-boundary wave function (NBWF) of Hartle and Hawking is a leading candidate for such a theory of the state. It is defined by a sum over regular four-dimensional geometries and fields on a four-disk with a single boundary three-sphere on which the wave function's arguments are specified, weighted by the exponential of minus the Euclidean action. In the semiclassical (steepest-descent) approximation the wave function is a sum of terms of the form $\exp[(-I_R+iS)/\hbar]$, one for each complex extremum of the action — a 'fuzzy instanton' — that is regular on the disk; $I_R$ and $S$ are the real and imaginary parts of its action. Where the classicality conditions hold — the imaginary part $S$ varies rapidly compared to the real part $I_R$ across superspace (the configuration space of three-geometries and fields) — the NBWF predicts classical Lorentzian histories as the integral curves of $S$, with probabilities proportional to $\exp(-2I_R/\hbar)$. Hartle and Hertog (2012) worked in a minisuperspace truncation (spatially closed, homogeneous, isotropic backgrounds with a massive scalar field and a positive cosmological constant, plus linear fluctuations), where the fuzzy instantons are labeled by the scalar field value $\phi_0$ at the regular 'South Pole' of the disk: the state predicts classicality only for $\phi_0\gtrsim 1.27$ (in Planck units), inflation on every classical history, a non-bouncing class ($1.27\lesssim\phi_0\lesssim 1.55$) with big bang and big crunch singularities, and a bouncing class ($\phi_0\gtrsim 1.55$) whose histories contract to a minimum radius and re-expand.

The paper's subject is the arrows of time of these histories. Building on Hawking, Laflamme and Lyons (1993), who found that in recollapsing NBWF universes the fluctuation arrow points in one direction over the whole spacetime, Hartle and Hertog showed that regularity of the fuzzy instantons forces fluctuations to vanish at the South Pole, so each Lorentzian history has exactly one range of three-surfaces where fluctuations are small — near one singularity for non-bouncing histories, at the bounce for bouncing ones. The fluctuation arrow is therefore bidirectional in bouncing universes: fluctuations are small at the bounce and grow away from it in both directions of time, and the electromagnetic, thermodynamic, and psychological arrows are argued to be aligned with the fluctuation arrow on each side. Observers on one side of a bounce can neither influence nor be causally influenced by events on the other side, since the arrow structure itself shields them; the period 'before the big bang' is in this sense observationally irrelevant — though, in the authors' words, whether history can be extended quantum mechanically through classical singularities is still an open question.

The derivation rests on acknowledged approximations, stated as such in the paper: (i) the saddle-point (fuzzy-instanton) approximation and the extrapolation of classical histories by the classical equations of motion through regions — notably the bounce — where the classicality conditions fail, since $S$ varies slowly there; the authors call this extrapolation 'an assumption which can in principle be checked in the full quantum mechanical theory'; (ii) the unresolved question of quantum continuation through classical singularities; (iii) the difficulty of quantitatively estimating our time-distance from the small-fluctuation regime at the bounce in the presence of the very-large-scale inhomogeneities that the NBWF predicts in regimes of eternal inflation (regimes where fluctuations leave the horizon while the scalar field value is high enough that quantum fluctuations dominate the classical roll); and (iv) the minisuperspace truncation itself, with linearly treated fluctuations and assumed-small backreaction (the feedback of the fluctuations on the background geometry). A later literature has taken up precisely these questions — quantum transitions replacing classical extrapolation through the bounce and through singularities, exact treatments of nonlinear backreaction in solvable models, contested definitions of the no-boundary path integral itself, and ongoing work on classicality in bouncing no-boundary histories. What has not been achieved is the check 'in the full quantum theory' that the authors called for: a treatment simultaneously free of the saddle-point, minisuperspace, linear, and small-backreaction approximations, from which the arrows-of-time conclusions would follow.

Problem Statement

Hartle and Hertog (2012) derived — within the saddle-point (fuzzy-instanton) approximation, a minisuperspace truncation with linear fluctuations, and neglected backreaction — the prediction that the classical bouncing histories of the no-boundary quantum state have small fluctuations at the bounce, a bidirectional fluctuation arrow of time (fluctuations growing away from the bounce in both directions), and electromagnetic, thermodynamic, and psychological arrows aligned with the fluctuation arrow on each side. The problem is to establish — or refute — these arrows-of-time predictions of the no-boundary state for its bouncing universes in the quantum theory, beyond the approximations in which they were derived. The components below are the gaps the source paper itself identifies as unestablished; they serve one overarching objective, namely extending and rigorously justifying the 2012 conclusions:

(i) History extension through the bounce: what replaces, at the quantum level, the extrapolation of classical histories by the classical equations of motion through the regime where the classicality conditions fail (where the imaginary part $S$ of the action varies slowly compared to the real part $I_R$)? The quantum-transition treatment of Hartle and Hertog (2015) and Bramberger, Hertog, Lehners and Vreys (2017) — in which the state supplies branching probabilities between classical histories on either side of the bounce, rather than a deterministic classical continuation — was carried out within saddle-point minisuperspace; the question is whether the bidirectional arrow and the arrow alignments survive in a treatment going beyond these approximations.

(ii) Passage through singularities: whether and how history can be continued quantum mechanically through classically singular regimes, beyond the minisuperspace transition amplitudes computed to date.

(iii) Robustness to the neglected degrees of freedom: whether the small-fluctuations-at-the-bounce prediction and the arrow conclusions survive nonlinear backreaction (beyond exactly solvable models) and the truncation to homogeneous, isotropic backgrounds with linear perturbations.

(iv) Predictive control: whether quantitative predictions — such as measures of how far in time our present surface is from the small-fluctuation regime at the bounce, or probabilities for observables — can be controlled in the presence of the very-large-scale stochastic inhomogeneities that the no-boundary state predicts in regimes of eternal inflation. The source states that these inhomogeneities 'do not alter our finding that the fluctuation arrow of time reverses at the bounce', so this component concerns quantitative prediction rather than the arrow conclusion itself. Components (i)-(iii) are the required objective, since each bears on the arrow conclusions; component (iv) is recorded as context and is not itself a required deliverable.

A solution may take the form of a controlled derivation confirming (possibly in modified form, e.g. with branching replacing deterministic continuation) or refuting the bidirectional-arrow and arrow-alignment conclusions beyond these approximations; or of a worked counterexample, likewise within the no-boundary framework and beyond these approximations, showing that the arrow predictions themselves fail. A demonstration that a derivation step fails only at the saddle-point minisuperspace level would not settle the question: the arrow conclusion holds within the saddle-point minisuperspace treatment, and whether it survives beyond these approximations is exactly what is open.

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