[Paper Review] Entropy, geometry, and the quantum potential

Robert Carroll|ArXiv.org|Nov 20, 2005

Quantum Mechanics and Applications10 references3 citations

TL;DR

This paper proposes that the quantum potential in the Wheeler-DeWitt (WDW) equation emerges naturally from entropy and Fisher information principles when a complex wave function ψ = Rexp(iS/ℏ) is introduced into classical gravity. By interpreting R² as a probability density and linking momentum fluctuations to Fisher information, the quantum potential arises as an information-theoretic correction to the classical action, unifying quantum effects with geometric and thermodynamic structures in quantum gravity.

ABSTRACT

We sketch and emphasize the automatic emergence of a quantum potential Q in e.g. classical WDW type equations upon inserting a (Bohmian) complex wave function. The interpretation of Q in terms of momentum fluctuations via Fisher information and entropy ideas is discussed along with the essentially forced role of the amplitude squared as a probability density. We also review the constructions of Padmanabhan connecting entropy and the Einstein equations.

Motivation & Objective

• To demonstrate that the quantum potential Q arises automatically in WDW-type equations upon introducing a complex wave function ψ = Rexp(iS/ℏ).
• To interpret the quantum potential Q in terms of momentum fluctuations via Fisher information and entropy, grounding quantum behavior in statistical mechanics.
• To establish R² as a natural probability density in the context of geometric and information-theoretic structures.
• To connect the emergence of quantum effects in gravity with entropy functionals and the Einstein equations via information geometry.
• To unify quantum mechanics and general relativity by showing that the quantum potential arises from information-theoretic principles without requiring a separate matter Lagrangian.

Proposed method

• Formulate an entropy functional S using perturbations uₐ in a curved spacetime, with M and N tensors chosen to yield the Einstein equations upon extremization.
• Introduce a complex wave function ψ = Rexp(iS/ℏ) into the WDW equation, leading to the emergence of a quantum potential Q via the amplitude R.
• Apply Fisher information theory to relate momentum fluctuations to the quantum potential, with Q proportional to the Fisher information of the probability density R².
• Use the ADM formalism to derive the WDW equation from a Hamiltonian involving δS/δhᵢⱼ and field momenta πᵢⱼ, including stochastic fluctuations fᵢⱼ.
• Construct a Hamiltonian term Q = ∫Q P dV where P = R², showing that Q arises purely from the metric and probability density via information geometry.
• Establish a connection between the quantum potential and a complex velocity field in Weyl geometry, with Aᵤ ∝ ∂ᵤlog(P), leading to a Fisher-type metric.

Experimental results

Research questions

• RQ1 How does the quantum potential Q emerge from a complex wave function in a classical WDW-type equation?
• RQ2 What is the information-theoretic origin of the quantum potential in terms of Fisher information and entropy?
• RQ3 Can the probability density R² be derived from geometric and thermodynamic principles without postulating it?
• RQ4 How does the inclusion of momentum fluctuations fᵢⱼ via exact uncertainty relations lead to the quantum potential?
• RQ5 To what extent can the Einstein equations be derived from an entropy functional, and how does this relate to the WDW equation?

Key findings

• The quantum potential Q emerges automatically in the WDW equation upon introducing a complex wave function ψ = Rexp(iS/ℏ), without requiring additional postulates.
• The probability density is identified as P = R², which arises naturally from the requirement of consistency with Fisher information and momentum fluctuation constraints.
• The quantum potential is shown to be proportional to the Fisher information of the probability density P, specifically Q ∝ ∫ Dh ∫ dx N Gᵢⱼₖₗ (δP/δhᵢⱼ)(δP/δhₖₗ), linking quantum effects to information geometry.
• The WDW equation is derived from a Hamiltonian that includes a term Q = ∫Q P dV, where Q is constructed solely from the metric and P, implying that quantum behavior arises from geometric and statistical structure.
• The approach unifies quantum mechanics and gravity by showing that the quantum potential arises from information-theoretic principles, with no need for a separate matter Lagrangian.
• The formalism reproduces the standard rate equation for metric evolution (5.16) and extends it to include stochastic fluctuations fᵢⱼ, leading to a consistent quantum gravity framework via information geometry.