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Subquantum Phase-Mixing and the Dynamic Emergence of the Born Rule during Primordial Inflation

M. Toufik Naser

Published Sep 15, 2026• 0 citations
Publisher

Abstract

Abstract We investigate the dynamical stability and relaxation of subquantum non-equilibrium states within the framework of De Broglie Bohm field theory during primordial inflation. By evaluating the quantum Hamilton-Jacobi formalism on a covariant Friedmann-Lemaître Robertson-Walker (FLRW) background, I demonstrate that non-linearities intrinsic to the field-theoretic quantum potential generate microscopic phase-space chaos. I solve the Jacobi deviation equation for coupled Fourier modes and extract the maximal Lyapunov exponent spectrum. I find that sub-horizon modes exhibit rapid exponential trajectory divergence driving coarse-grained subquantum entropy toward zero via Valentini's subquantum H-theorem. This dynamic phase-mixing enforces the Born rule well before horizon crossing, rendering anti-relaxation statistically prohibited for generic inflationary potentials. Finally, I map residual non-equilibrium corrections onto the Cosmic Microwave Background (CMB) power spectrum at low multipoles establishing current observational bounds from Planck 2018 data and providing forecasts for B-mode polarization measurements from LiteBIRD and CMB-S4.

DOI
10.21203/rs.3.rs-11025114/v1

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Abstract We investigate the dynamical stability and relaxation of subquantum non-equilibrium states within the framework of De Broglie Bohm field theory during primordial inflation. By evaluating the quantum Hamilton-Jacobi formalism on a covariant Friedmann-Lemaître Robertson-Walker (FLRW) background, I demonstrate that non-linearities intrinsic to the field-theoretic quantum potential generate microscopic phase-space chaos. I solve the Jacobi deviation equation for coupled Fourier modes and extract the maximal Lyapunov exponent spectrum. I find that sub-horizon modes exhibit rapid exponential trajectory divergence driving coarse-grained subquantum entropy toward zero via Valentini's subquantum H-theorem. This dynamic phase-mixing enforces the Born rule well before horizon crossing, rendering anti-relaxation statistically prohibited for generic inflationary potentials. Finally, I map residual non-equilibrium corrections onto the Cosmic Microwave Background (CMB) power spectrum at low multipoles establishing current observational bounds from Planck 2018 data and providing forecasts for B-mode polarization measurements from LiteBIRD and CMB-S4.
We present arguments suggesting that deviations from the Born probability rule could be generated for trans-Planckian field modes during inflation. Such deviations are theoretically possible in the de Broglie-Bohm pilot-wave formulation of quantum mechanics, according to which the Born rule describes a state of statistical equilibrium. We suggest that a stable equilibrium state can exist only in restricted conditions: on a classical background spacetime that is globally hyperbolic or in a mild quantum-gravity regime in which there is an effective Schrödinger equation with a well-defined time parameter. These arguments suggest that quantum equilibrium will be unstable at the Planck scale. We construct a model in which quantum nonequilibrium is generated by a time-dependent regulator for pilot-wave dynamics, where the regulator is introduced to eliminate phase singularities. Applying our model to trans-Planckian modes that exit the Planck radius, we calculate the corrected primordial power spectrum and show that it displays a power excess (above a critical wavenumber). We briefly consider how our proposals could be tested by measurements of the cosmic microwave background.
We illustrate through explicit numerical calculations how the Born rule probability densities of non-relativistic quantum mechanics emerge naturally from the particle dynamics of de Broglie–Bohm pilot-wave theory. The time evolution of a particle distribution initially not equal to the absolute square of the wave function is calculated for a particle in a two-dimensional infinite potential square well. Under the de Broglie–Bohm ontology, the box contains an objectively existing ‘pilot wave’ which guides the electron trajectory, and this is represented mathematically by a Schrödinger wave function composed of a finite out-of-phase superposition of M energy eigenstates (with M ranging from 4 to 64). The electron density distributions are found to evolve naturally into the Born rule ones and stay there; in analogy with the classical case this represents a decay to ‘quantum equilibrium’. The proximity to equilibrium is characterized by the coarse-grained subquantum H -function which is found to decrease roughly exponentially towards zero over the course of time. The time scale τ for this relaxation is calculated for various values of M and the coarse-graining length ε . Its dependence on M is found to disagree with an earlier theoretical prediction. A power law, τ ∝ M −1 , is found to be fairly robust for all coarse-graining lengths and, although a weak dependence of τ on ε is observed, it does not appear to follow any straightforward scaling. A theoretical analysis is presented to explain these results. This improvement in our understanding of time scales for relaxation to quantum equilibrium is likely to be of use in the development of models of relaxation in the early Universe, with a view to constraining possible violations of the Born rule in inflationary cosmology.
We extend de Broglie--Bohm (dBB) pilot-wave theory by introducing a complex source field into the Schrödinger equation and examining its effects on quantum equilibrium and nonequilibrium dynamics. In dBB theory, the physical particle distribution P, rather than the Born density ∣ψ∣2|ψ|^2∣ψ∣2, carries the ensemble probability, so a source term may modify the pilot wave without violating conservation of total particle probability. We derive the source-modified Hamilton--Jacobi and continuity equations, the transport equation for the nonequilibrium ratio f=P/∣ψ∣2f=P/|ψ|^2f=P/∣ψ∣2, and an exact entropy-production formula. Three applications follow. First, suitably designed sources can drive exponential relaxation toward quantum equilibrium. Second, the entropy-production rate admits a Prigogine-type bilinear form, providing a basis for a subquantum thermodynamics with entropy-producing and entropy-extracting regimes. Third, a tuned source can exactly cancel the Bohmian quantum potential, yielding classical particle trajectories while the guiding wave retains nontrivial structure and the nonequilibrium ratio remains conserved. The resulting framework provides a unified setting for studying source-driven quantum nonequilibrium, entropy exchange, and classicalization, and motivates further investigation of the physical and ontological status of the source field.
The quantum cosmology of the flat Friedmann-Lema\^{\i}tre-Robertson-Walker universe, filled with a scalar field, is considered in the de Broglie--Bohm (dBB) interpretation framework. A stiff-matter quantum bounce solution is obtained. The bouncing and subsequent pre-inflationary and inflationary dynamics are studied in details. We consider some representative primordial inflation models as examples, for which analytical expressions characterizing the dynamical quantities can be explicitly derived. The dependence of the inflationary dynamics on the quantum bounce parameters is then analyzed. The parameters emerging from our description are constrained by requiring the produced dynamics to be in accordance with some key cosmological quantities. The constraining conditions are also illustrated through regions of parameter space in terms of the bounce quantities.
We study incomplete relaxation to quantum equilibrium at long wavelengths, during a preinflationary phase, as a possible explanation for the reported large-scale anomalies in the cosmic microwave background. Our scenario makes use of the de Broglie--Bohm pilot-wave formulation of quantum theory, in which the Born probability rule has a dynamical origin. The large-scale power deficit could arise from incomplete relaxation for the amplitudes of the primordial perturbations. We show, by numerical simulations for a spectator scalar field, that if the preinflationary era is radiation dominated then the deficit in the emerging power spectrum will have a characteristic shape (an inverse-tangent dependence on wave number kkk, with oscillations). It is found that our scenario is able to produce a power deficit in the observed region and of the observed (approximate) magnitude for an appropriate choice of cosmological parameters. We also discuss the large-scale anisotropy, which might arise from incomplete relaxation for the phases of the primordial perturbations. We present numerical simulations for phase relaxation, and we show how to define characteristic scales for amplitude and phase nonequilibrium. The extent to which the data might support our scenario is left as a question for future work. Our results suggest that we have a potentially viable model that might explain two apparently independent cosmic anomalies by means of a single mechanism.
We show that inflationary cosmology may be used to test the statistical predictions of quantum theory at very short distances and at very early times. Hidden-variables theories, such as the pilot-wave theory of de Broglie and Bohm, allow the existence of vacuum states with nonstandard field fluctuations (``quantum nonequilibrium''). We show that inflationary expansion can transfer microscopic nonequilibrium to macroscopic scales, resulting in anomalous power spectra for the cosmic microwave background. The conclusions depend only weakly on the details of the de Broglie-Bohm dynamics. We discuss, in particular, the nonequilibrium breaking of scale invariance for the primordial (scalar) power spectrum. We also show how nonequilibrium can generate primordial perturbations with nonrandom phases and intermode correlations (primordial non-Gaussianity). We address the possibility of a low-power anomaly at large angular scales, and show how it might arise from a nonequilibrium suppression of quantum noise. Recent observations are used to set an approximate bound on violations of quantum theory in the early Universe.