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Klein-Gordon flow on FLRW spacetimes

Edwin Beggs, Shahn Majid

TL;DR

The paper develops a generally covariant quantum-mechanical framework (KGQM) for Klein-Gordon fields on FLRW spacetimes, expressing evolution with an external parameter $s$ and enabling separation of variables into temporal and spatial parts. For $\kappa>0$ it uncovers a discrete cosmological-atom spectrum from polar-separable spatial modes combined with temporal stationary states, while for all $\kappa$, temporal quantum mechanics emerges with the expansion factor $a(t)$ acting as a potential $1/a(t)^2$ in a 1D problem along the time axis. It further analyzes operator geodesic equations and Ehrenfest theorems in this setting, and develops a coordinate-time interpretation where temporal modes $F^{\pm}_{\omega}(t)$ can exhibit oscillatory or exponential behavior depending on the Hubble parameter $H$, with inflation driving non-oscillatory dynamics and a washout of spatial observables at late times. The work also demonstrates how transitions in $a(t)$ generate reflected temporal modes and outlines extensions to non-factorised states and density matrices, pointing to rich connections between quantum mechanics, cosmology, and quantum gravity-inspired formalisms. Overall, it provides a covariant, separation-of-variables toolkit for exploring quantum dynamics on expanding spacetimes and introduces novel phenomena tied to inflation and temporal quantum mechanics with potential cosmological applications.

Abstract

We study a new approach to generally covariant quantum mechanics applied in the case of an FLRW cosmological background. For positive spatial curvature we find a discrete series of solutions of the Klein-Gordon equation that can reasonably be called gravitationally bound `cosmological atom' states. For all cases of curvature, these modes, as well as more conventional atomic spatial modes bound by an external potential, extend to solutions of the Klein-Gordon equations viewed as stationary modes of Klein-Gordon quantum mechanics where wavefunctions are over spacetime and evolution is with respect to an external `geodesic time' parameter $s$. For general nonstationary states with fixed spatial eigenvector, the theory reduces to a novel 1-dimensional quantum system on the time $t$ axis with potential $1/a(t)^2$, where $a(t)$ is the Friedmann expansion factor. Its behaviour, and hence the evolution of spatial states, changes critically when the Hubble constant exceeds $2/3$ of the particle mass, as typically occurs during inflation. We also find washout of the evolution of spatial observables at late times and a backward-traveling reflected mode generated when the value of $H$ transitions to a larger value.

Klein-Gordon flow on FLRW spacetimes

TL;DR

The paper develops a generally covariant quantum-mechanical framework (KGQM) for Klein-Gordon fields on FLRW spacetimes, expressing evolution with an external parameter and enabling separation of variables into temporal and spatial parts. For it uncovers a discrete cosmological-atom spectrum from polar-separable spatial modes combined with temporal stationary states, while for all , temporal quantum mechanics emerges with the expansion factor acting as a potential in a 1D problem along the time axis. It further analyzes operator geodesic equations and Ehrenfest theorems in this setting, and develops a coordinate-time interpretation where temporal modes can exhibit oscillatory or exponential behavior depending on the Hubble parameter , with inflation driving non-oscillatory dynamics and a washout of spatial observables at late times. The work also demonstrates how transitions in generate reflected temporal modes and outlines extensions to non-factorised states and density matrices, pointing to rich connections between quantum mechanics, cosmology, and quantum gravity-inspired formalisms. Overall, it provides a covariant, separation-of-variables toolkit for exploring quantum dynamics on expanding spacetimes and introduces novel phenomena tied to inflation and temporal quantum mechanics with potential cosmological applications.

Abstract

We study a new approach to generally covariant quantum mechanics applied in the case of an FLRW cosmological background. For positive spatial curvature we find a discrete series of solutions of the Klein-Gordon equation that can reasonably be called gravitationally bound `cosmological atom' states. For all cases of curvature, these modes, as well as more conventional atomic spatial modes bound by an external potential, extend to solutions of the Klein-Gordon equations viewed as stationary modes of Klein-Gordon quantum mechanics where wavefunctions are over spacetime and evolution is with respect to an external `geodesic time' parameter . For general nonstationary states with fixed spatial eigenvector, the theory reduces to a novel 1-dimensional quantum system on the time axis with potential , where is the Friedmann expansion factor. Its behaviour, and hence the evolution of spatial states, changes critically when the Hubble constant exceeds of the particle mass, as typically occurs during inflation. We also find washout of the evolution of spatial observables at late times and a backward-traveling reflected mode generated when the value of transitions to a larger value.

Paper Structure

This paper contains 10 sections, 1 theorem, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Recap of the FLRW metric
  3. KGQM in an FLRW background by separation of variables
  4. Polar-separable eigenfunctions of $\Delta$
  5. $\kappa>0$ spherical case.
  6. $\kappa=0$ spatially flat case.
  7. $\kappa<0$ hyperbolic case.
  8. Operator geodesic equation in an FLRW background
  9. FLRW quantum mechanics with respect to coordinate time
  10. Concluding remarks

Key Result

Lemma 3.1

$\Delta_t$ is essentially self-adjoint with respect to the $a^3{\rm d} t$ measure on the space of fields for which $[\bar{G}({\partial}_t F) a^3]=0$ across the endpoints (for example, any mix of Neumann and Dirichlet conditions at the two limits).

Figures (5)

  • Figure 1: (a) Stationary state $F^+_\omega(t)$ in the oscillatory regime, shown for $H=1, \omega= 4$ and (b) temporal QM evolution under $s$ of an initial Gaussian $F(0,t)$ centred at $t=25$, shown for $H=0.1$. Both parts are for $a_0=1, \mu=0.5, \nu=10$.
  • Figure 2: Radial sector for (a) $\kappa>0$ with 'cosmological atom'-like modes $\psi_{n,l}(r)$ for $n=5$ and $n=6$ with allowed values of $l$ in each case. (b) $\kappa<0$ typical form of oscillatory decaying solutions $\psi_{\nu,l}$.
  • Figure 3: (a) Expectation value ${\langle}t{\rangle}$ for the Gaussian evolution in Figure \ref{['figF']} (b) $\mu {{\rm d} {\langle}P_t{\rangle}\over{\rm d} s}$ computed from the RHS of (\ref{['dvevPds']}) for the same solution. (c) The classical entropy increases during the evolution.
  • Figure 4: Expected value ${\langle} r {\rangle}(t)$ in KG solution/KGQM stationary state $\psi=F^{-,8}_{\omega}(t)\psi_{3,0}(r)+ F^{-,24}_\omega(t)\psi_{5,0}(r)$ evolving a sum of $l=0$ cosmological atom states for levels $n=3,5$. The dashed line is the corresponding behaviour in regular quantum mechanics oscillating between the expected value in states $\psi_{3,0}\pm\psi_{5,0}$. Plots are for $\kappa=a_0=1$, $H=0.08$, $\omega=4$.
  • Figure 5: Solution for $F_\omega(t)$ covering an exponential period $t\in (2,2.2)$ where the Hubble constant jumps to above the bound (\ref{['mH']}) for the oscillatory regime (as could happen during inflation). The imaginary part looks broadly similar. The plot is for $a_0=\nu=1$, $\omega=4$, $H_1=1$, $H_2=2.7$.

Theorems & Definitions (2)

  • Lemma 3.1
  • Example 3.2