[Paper Review] On some toy quantum cosmology

Robert Carroll|arXiv (Cornell University)|Oct 8, 2010

Cosmology and Gravitation Theories9 references3 citations

TL;DR

This paper proposes a toy quantum cosmology model where the quantum potential from de Broglie-Bohm mechanics is linked to gravity via Weyl integrable spacetime (WIST), embedding quantum effects into cosmological dynamics. Using a modified action with a conformal factor β² = m²exp(ℚ), it derives Friedmann-like equations where the quantum potential ℚ acts as a dynamical field influencing dark energy and curvature, suggesting a deep connection between quantum mechanics and gravity through geometric and thermodynamic principles.

ABSTRACT

Some connections of the quantum potential to gravitation are discussed.

Motivation & Objective

• To explore the connection between quantum mechanics and gravity through the lens of Weyl integrable spacetime (WIST), motivated by de Broglie-Bohm mechanics and quantum trajectories.
• To model a toy universe where quantum effects, represented by the quantum potential ℚ, influence cosmological evolution via a conformal factor β² = m²exp(ℚ).
• To derive modified Friedmann equations from a Weyl-Dirac action with a dynamical conformal factor, showing how quantum potential ℚ can generate effects akin to dark energy and curvature.
• To investigate the role of the cosmological constant Λ and scalar potential V in coupling quantum and gravitational dynamics through the conformal factor Φ = exp(−ℚ).
• To establish a bridge between thermodynamic principles (e.g., entropy, Fisher information) and quantum gravity by linking the quantum potential to entropy functionals in the WDW framework.

Proposed method

• Adopts a Weyl integrable spacetime (WIST) model with Weyl vector wλ = −2∂λlog(β), leading to a modified gravitational action (1.2) involving β², Ricci scalar R, and a conformal factor Φ = exp(−ℚ).
• Derives the action (1.2) from a Weyl-Dirac theory with arbitrary σ, showing cancellation of terms involving wλ and βλ, resulting in a simplified action with (σ+6)(∂β)² and 2Λβ⁴ terms.
• Applies the FRW metric (1.3) to the model, assuming ℚ = ℚ(a(t)) so that the quantum potential depends only on the scale factor, enabling cosmological dynamics.
• Transforms the action into a scalar-tensor form by identifying Φ = exp(−ℚ) = β²/m², linking the quantum potential to the conformal factor in a scalar-tensor theory with ω = −(1/4)(σ+6).
• Derives modified Friedmann equations (2.16) and (2.17) from the action, incorporating ρᴹ, Pᴹ, and potential V(Φ), and expresses them in terms of ℚ and its time derivatives.
• Connects the quantum potential ℚ to thermodynamic entropy via Fisher information and the WDW framework, introducing entropy functionals involving δP¹ᐟ²/δhᵢⱼ and Ricci flow-inspired functionals.

Experimental results

Research questions

• RQ1 How can the quantum potential ℚ from de Broglie-Bohm mechanics be geometrically embedded into a cosmological framework via Weyl geometry?
• RQ2 What cosmological dynamics emerge when the quantum potential ℚ is treated as a dynamical field coupled to gravity through the conformal factor Φ = exp(−ℚ)?
• RQ3 Can the interplay between the quantum potential ℚ and the cosmological constant Λ generate effective dark energy or curvature effects in a WIST model?
• RQ4 How do the modified Friedmann equations derived from the Weyl-Dirac action reflect the influence of quantum effects on large-scale structure and spacetime evolution?
• RQ5 What is the role of thermodynamic entropy and information-theoretic functionals (e.g., Fisher information, Perelman entropy) in connecting quantum mechanics and gravity in this model?

Key findings

• The action (1.2) reduces to I = ∫√−g d⁴x [−β²R + (σ+6)(∂β)² + 2Λβ⁴], showing that the quantum potential ℚ = ℏ²□|Ψ|/(m²|Ψ|) is encoded in β² = m²exp(ℚ).
• The conformal factor Φ = exp(−ℚ) = β²/m² is identified with the scalar field in a scalar-tensor theory, with ω = −(1/4)(σ+6), linking quantum mechanics to gravity via Weyl geometry.
• Modified Friedmann equations (2.16) and (2.17) are derived, showing that ℚ and its time derivatives govern the dynamics of Hubble parameter H and scale factor a(t).
• When matter Lagrangian ℒᴹ = 0, the equations reduce to (2.18), expressing ℚ's dynamics through ℚ̇² − ℚ̈ and Hubble term 3Hℚ̇, with Λ and V(Φ) contributing to curvature and potential energy.
• For a quadratic potential V(Φ) = cΦ² = cexp(−2ℚ), the term Φ(dV/dΦ) − 2V vanishes, simplifying the dynamics and suggesting a stable quantum-gravitational configuration.
• The quantum potential ℚ is shown to generate an entropy functional ∫𝒟h Pℚ dV in the WDW framework, linking quantum fluctuations to gravitational entropy and thermodynamic principles.