[Paper Review] Relativistic Bohmian mechanics from scalar gravity

Benjamin Koch|ArXiv.org|Oct 15, 2008

Quantum Mechanics and Applications1 references3 citations

TL;DR

This paper derives the fundamental equations of relativistic de Broglie-Bohm (dBB) mechanics for a single particle from a scalar theory of curved spacetime, analogous to Nordström's theory. By matching the dBB quantum phase and pilot wave to geometric quantities in the scalar gravity model, the authors show that the continuity equation, Hamilton-Jacobi equation with quantum potential, and particle trajectory equations emerge naturally, establishing a duality between dBB mechanics and this geometric framework.

ABSTRACT

In this article we show that the fundamental equations of relativistic Bohmian mechanics for a single particle can be derived from a scalar theory of curved space-time.

Motivation & Objective

• To establish a geometric foundation for relativistic de Broglie-Bohm mechanics using a scalar theory of gravity.
• To resolve the ad hoc nature of postulating the quantum potential and three dBB equations by deriving them from a consistent geometric framework.
• To demonstrate that the dBB equations for a single particle arise as exact solutions in a scalar gravity model, not just in an approximation.
• To explore the possibility of a deeper duality between quantum mechanics and classical geometry, particularly in the context of the Klein-Gordon equation.
• To lay the groundwork for extending this duality to many-particle systems, interacting theories, and quantum field theory in future work.

Proposed method

• The paper uses a scalar theory of gravity with Ricci scalar curvature proportional to the stress-energy tensor trace (R = κT), and a vanishing Weyl curvature tensor (Cμναβ = 0), ensuring conformal flatness.
• It introduces a conformally rescaled Minkowski metric, gμν = φ²(x)ημν, where φ(x) is a scalar field related to the dBB pilot wave P(x).
• The Hamilton-Jacobi function SH in the stress-energy tensor is matched to the dBB quantum phase SQ via SH = SQ, ensuring consistency with the particle momentum postulate.
• The continuity equation (1) and quantum potential equation (2) in dBB theory are derived from the conservation of four-momentum ∇μpμ = 0 in the curved geometry.
• The particle trajectory equation (7) in dBB mechanics is recovered by expressing the geodesic equation in the conformally rescaled spacetime and matching it to the dBB equation of motion.
• Consistency between the derived dBB trajectory and the geodesic equation in curved spacetime is proven by showing that both equations yield the same dynamics under the matching conditions, with an effective scalar function f(x) derived from the geometry.

Experimental results

Research questions

• RQ1 Can the fundamental equations of relativistic de Broglie-Bohm mechanics be derived from a geometric scalar gravity model rather than being postulated?
• RQ2 What specific geometric conditions must be imposed on a scalar gravity theory to reproduce the dBB continuity equation, quantum potential, and trajectory equations?
• RQ3 Is there a duality between the dBB quantum phase and the Hamilton-Jacobi function in a curved spacetime framework?
• RQ4 How can the non-local quantum potential in dBB mechanics be geometrically interpreted via a scalar field in curved spacetime?
• RQ5 Can this geometric derivation be extended to many-particle systems, fermionic fields, or quantum field theory?

Key findings

• The continuity equation of dBB mechanics (1) is derived from the conservation of four-momentum ∇μpμ = 0 in the scalar gravity model, with the momentum expressed via the Hamilton-Jacobi function SH.
• The quantum potential equation (2) in dBB mechanics is reproduced by matching the Ricci scalar curvature R to the dBB quantum potential Q via the derived relation R = κT and the conformal factor φ.
• The particle trajectory equation (7) in dBB mechanics is recovered as the equation of motion in the conformally rescaled spacetime, with the proper time τ related to the affine parameter s via dτ/ds = φ⁻².
• The consistency between the dBB trajectory and the geodesic equation in curved spacetime is proven by showing that both yield the same dynamics when the matching conditions are applied, with an effective scalar function f(x) = -3/2 (∂aφ²)(∂aSH)/(M³φ⁴).
• The duality between dBB mechanics and the scalar gravity model is established via three matching conditions: φ ∝ P, SH ∝ SQ, and the coupling constant must be negative for consistency with real-valued phases.
• The non-relativistic limit of the derived equations reproduces the standard non-relativistic dBB theory, including the Schrödinger equation and the original de Broglie-Bohm trajectory equations.