[Paper Review] Klein-Gordon equation from Maxwell-Lorentz dynamics

Ricardo J. Alonso-Blanco|arXiv (Cornell University)|Jan 27, 2012

Geometric Analysis and Curvature Flows4 references3 citations

TL;DR

This paper demonstrates that the Klein-Gordon equation arises naturally from classical Maxwell-Lorentz dynamics when solutions are constrained to Lagrangian submanifolds of a symplectically modified phase space. By imposing geometric consistency conditions on particle trajectories under Lorentz forces, the authors derive the Klein-Gordon equation without any quantum postulates or semiclassical limits, offering a classical-geometric origin for a fundamental quantum equation, thus addressing foundational inquiries by Dirac.

ABSTRACT

We consider Maxwell-Lorentz dynamics: that is to say, Newton's law under the action of a Lorentz's force which obeys the Maxwell equations. A natural class of solutions are those given by the Lagrangian submanifolds of the phase space when it is endowed with the symplectic structure modified by the electromagnetic field. We have found that the existence of this type of solution leads us directly to the Klein-Gordon equation as a compatibility condition. Therefore, surprisingly, quite natural assumptions on the classical theory involve a quantum condition without any process of limit. This result could be a partial response to the inquiries of Dirac.

Motivation & Objective

• To explore whether the Klein-Gordon equation can emerge from classical field theory without invoking quantum mechanics or limits.
• To investigate the geometric constraints imposed by Lagrangian submanifolds in the symplectic phase space of Newtonian dynamics with electromagnetic forces.
• To establish a direct link between classical field solutions and the relativistic wave equation, particularly in the context of Dirac’s foundational inquiries.
• To show that the Klein-Gordon equation arises as a compatibility condition for consistent field solutions under Lorentz force dynamics.

Proposed method

• Formulates Newton’s second law using a symplectic structure on the tangent bundle TM, modified by the electromagnetic 2-form F.
• Introduces a modified symplectic form ω_F = ω + F, which allows the Lorentz force to be expressed as a Hamiltonian vector field for the kinetic energy T.
• Defines field-solutions as velocity fields u that are Lagrangian submanifolds in (TM, ω_F), corresponding to u^* = df for some smooth function f.
• Applies differential geometric tools such as the Hodge star operator, codifferential δ, and the Laplace-de Rham operator Δ to analyze the field equations.
• Imposes the Lorentz gauge condition δA = 0 and assumes the current J satisfies δF = J^*, leading to Δf = 0 for the phase function f.
• Derives the Klein-Gordon-type equation by computing Δ(e^{if/ħ}) and identifying the effective mass term from the metric contraction T_2(df, df).

Experimental results

Research questions

• RQ1 Can the Klein-Gordon equation be derived from classical Maxwell-Lorentz dynamics without introducing quantum postulates?
• RQ2 What geometric constraints on particle trajectories lead to relativistic wave equations in classical field theory?
• RQ3 How does the existence of Lagrangian submanifold solutions in a symplectically modified phase space lead to the Klein-Gordon equation?
• RQ4 Is there a classical-geometric interpretation of the mass term in the Klein-Gordon equation within the framework of Newtonian dynamics with electromagnetic forces?

Key findings

• The Klein-Gordon equation arises as a necessary compatibility condition for the existence of Lagrangian submanifold solutions in the symplectic phase space of Maxwell-Lorentz dynamics.
• For conservative geodesic fields (δu = 0) with u^* = df, the function ψ = e^{if/ħ} satisfies the standard Klein-Gordon equation (Δ + m²/ħ²)ψ = 0, where m² = T₂(u,u).
• In the presence of a Maxwell field F = dA with δA = 0 and J = δF, the solution ψ = e^{if/ħ} satisfies the modified Klein-Gordon equation (Δ - 2iħ⁻¹A^* + ħ⁻²(m² - ||A||²))ψ = 0.
• The mass parameter m² is identified as the constant norm squared of the current vector field J, i.e., m² = T₂(J,J), and is preserved under the geometric constraints.
• The derivation shows that quantum-like behavior (the Klein-Gordon equation) emerges purely from classical geometric and symplectic structures, without any limiting process or ad hoc quantization.