Effective relational dynamics of a nonintegrable cosmological model

TL;DR

This work develops and applies an effective semiclassical framework for relational quantum dynamics to a nonintegrable closed FRW cosmology with a minimally coupled massive scalar field. By representing states through expectation values and Weyl-ordered moments and by switching between clock variables (α and φ) via explicit gauge transformations, the authors obtain locally valid relational evolution and quantify quantum backreaction up to leading order in ħ. They show that relational evolution is possible only for sufficiently semiclassical states and typically breaks down near the region of maximal expansion due to defocusing of classical trajectories and chaotic scattering, which mix internal time directions and undermine clock reliability. The findings highlight the intrinsic fragility of relational observables in generic nonintegrable cosmologies and point to the necessity of transient, clock-dependent descriptions of quantum evolution, with potential implications for how quantum gravity might meaningfully address the problem of time in realistic settings.

Abstract

We apply the effective approach to evaluating semiclassical relational dynamics to the closed Friedman--Robertson--Walker cosmological model filled with a minimally coupled massive scalar field. This model is interesting for studying relational dynamics in a more general setting because (i) it features a nontrivial coupling of the relational clock to the evolving degrees of freedom, (ii) no temporally global clock variable exists, and (iii) it is nonintegrable which is typical for generic dynamical systems. The effective approach is especially well geared for addressing the concept of relational evolution in this context, since it enables one to switch between different clocks and yields a consistent (temporally) local time evolution with transient observables so long as semiclassicality holds. We provide evidence that relational evolution in this model universe, while possible for sufficiently semiclassical states, generically breaks down in the region of maximal expansion. This is rooted in a defocusing of classical trajectories, which leads to a rapid spreading of states that are initially sharply peaked and to a mixing of internal time directions in this region. These results are qualitatively compared to previous work on this model, revisiting conceptual issues that have been raised earlier in the literature.

Figures (8)

• Figure 1: Two typical classical solutions to the closed FRW spacetime---both $\phi$ and $a$ generically fail to be globally valid internal clock functions in this model. Here we used $\alpha=\ln(a)$ as appropriate for the canonical discussion following (\ref{['hamcon2']}, \ref{['ham2']}). Insets (a) and (c) show extended segments of (both the expanding and recontracting branches of) relational evolution up to the point of maximal expansion, $\alpha_{max}=\ln(a_{max})$. The (new) scale factor $\alpha$ oscillates between points of regular (nonglobal) maxima $\alpha_{max,k}=\ln(a_{max,k})$ and (nonglobal) minima $\alpha_{min,k}=\ln(a_{min,k})$; Inset (b) shows a close--up of the same configuration space trajectory as (a) near $\alpha_{max}$, displaying the nonglobal extrema in a greater detail, while inset (d) depicts a close--up on an intermediate section of the trajectory in (c).
• Figure 2: Evolution of the canonical variables governed by (\ref{['ham2']}) for a rather benign classical solution. Notice how $\alpha$ features quasi--turning points close to the turning points of $\phi$ (also manifested in $p_\alpha$ having a local minimum close to the zeros of $p_\phi$).
• Figure 3: Defocusing of nearby trajectories, caustics develop along the extrema of $\phi$ (see also page2).
• Figure 4: (a) Classical trajectory (dotted) and patched up effective trajectory: $\alpha$--gauge (solid), $\phi$--gauge (dashed). (b) Moments in $\alpha$--gauge on the incoming branch: $(\Delta \phi)^2$ (thick, dashed), $(\Delta p_{\phi})^2$ (thin, dashed), $\Delta(\phi p_{\phi})$ (solid). $\alpha_{Q_1}$ is the quasi--turning point of $\alpha$ on the incoming branch, where the clock becomes 'slow' (see discussion and Fig. \ref{['fig:3a']}).
• Figure 5: (a) Moments in $\phi$--gauge: $(\Delta \alpha)^2$ (thick, dashed), $(\Delta p_{\alpha})^2$ (thin, dashed), $\Delta(\alpha p_{\alpha})$ (solid). (b) Moments in $\alpha$--gauge on the outgoing branch: $(\Delta \phi)^2$ (thick, dashed), $(\Delta p_{\phi})^2$ (thin, dashed), $\Delta(\phi p_{\phi})$ (solid). $\alpha_{Q_2}$ is the quasi--turning point of $\alpha$ on the outgoing branch, where the clock becomes 'slow' (see discussion and Fig. \ref{['fig:3b']}).

Theorems & Definitions (1)

• Theorem 4.1