[Paper Review] Remarks on gravity, entropy, and information

Robert Carroll|ArXiv.org|Feb 5, 2006

Cosmology and Gravitation Theories13 references3 citations

TL;DR

This paper explores deep connections between gravity, entropy, and information using de Donder-Weyl covariant field theory and the Wheeler-DeWitt (WDW) equation. It derives a covariant Bohmian formulation of quantum fields, reformulates the WDW equation in terms of extrinsic curvature and multifingered time, and shows that the Hamiltonian constraint in modified ADM formalism leads to a dynamical time rate, with key results linking differential entropy and Fisher information to spacetime geometry and quantum gravity constraints.

ABSTRACT

This is a partially survey collection of material on gravity, entropy, and information with some new heuristic results related to the WDW equation.

Motivation & Objective

• To investigate the interplay between gravity, entropy, and information using covariant quantum field theory and the WDW equation.
• To derive a covariant Bohmian formulation of quantum fields from the de Donder-Weyl formalism, suggesting a potential bridge between quantum mechanics and general relativity.
• To reformulate the WDW equation in terms of extrinsic curvature and multifingered time, emphasizing the role of time as a dynamical variable.
• To examine the implications of modifying the ADM action by introducing a slicing density α, leading to a scalar-weighted Hamiltonian density and improved constraint behavior.
• To analyze the role of differential entropy and Fisher information in the context of spacetime geometry and quantum gravity constraints.

Proposed method

• Uses the de Donder-Weyl (dDW) formalism to treat space and time on equal footing, enabling manifest covariance in field equations.
• Derives the functional Hamilton-Jacobi (HJ) equation from the dDW HJ equation restricted to a Cauchy surface Σ, linking dDW eikonal functions to canonical wave functionals.
• Applies the dDW formalism to scalar field theories, deriving the WDW equation via the relation ∂ₜS = ∫ d³x ∂ₜSᵗ|Σ and δS/δy(x) = ∂ᵧSᵗ|Σ.
• Introduces a modified ADM action with a slicing density α, leading to a Hamiltonian density ṠH = h¹ᐟ²H of scalar weight 2, which avoids dependence on the Hamiltonian density in equations of motion.
• Derives equations of motion from the modified action, showing that δS/δπⁱʲ = 2N(πⁱʲ − hⁱʲπ/2) and δS/δhᵢⱼ = −α times a combination of Ricci and curvature terms.
• Demonstrates that the equations of motion derived from the modified action are equivalent to the standard 3+1 equations only when ⁴Rᵢⱼ = 0, and that the constraint structure is preserved even when constraints are violated.

Experimental results

Research questions

• RQ1 How can the de Donder-Weyl formalism be used to derive a covariant Bohmian interpretation of quantum field theory?
• RQ2 What is the role of extrinsic curvature in defining time evolution in canonical quantum gravity?
• RQ3 How does modifying the ADM action via a slicing density α affect the Hamiltonian constraint and the dynamics of the WDW equation?
• RQ4 In what way do differential entropy and Fisher information emerge from the structure of the WDW equation and spacetime geometry?
• RQ5 Why is the modified Hamiltonian density ṠH = h¹ᐟ²H better behaved than the standard ADM Hamiltonian when constraints are violated?

Key findings

• The functional HJ equation is derived as a consequence of the dDW HJ equation restricted to a Cauchy surface, under the assumption that the HJ eikonal functional S is related to dDW eikonal functions Sᵘ.
• The modified ADM action with slicing density α leads to a Hamiltonian density ṠH = h¹ᐟ²H of scalar weight 2, which is independent of the Hamiltonian density in the equations of motion.
• The equations of motion derived from the modified action are equivalent to the standard 3+1 equations only when ⁴Rᵢⱼ = 0, indicating that the modified formalism is more robust under constraint violation.
• The variation of the modified Hamiltonian density δṠH does not contain a term proportional to the Hamiltonian density, unlike the standard ADM case, which improves the behavior of the theory under constraint violation.
• The slicing density α and shift βⁱ are not varied in the action principle; instead, constraints are imposed on initial data and preserved dynamically, ensuring consistency.
• The paper shows that the condition ⁴Rᵢⱼ = 0 is equivalent to the vanishing of the combination Gᵢⱼ + hᵢⱼG⁰₀, which is independent of the Hamiltonian and momentum densities, highlighting a fundamental difference between Rᵢⱼ = 0 and Gᵢⱼ = 0 as equations of motion.