From Maxwell’s equations to relativistic Schrödinger equations via Schwartz Linear Algebra and Killing frames on the two-sphere

Abstract

In this work, we develop a rigorous framework partially unifying classical electromagnetism and relativistic quantum mechanics through the language of Schwartz Linear Algebra and via continuous symmetries of the 2-sphere. Building upon our recent analysis of Killing vector fields on the two-sphere and the associated orthonormal right-handed frames, we construct explicit correspondences between two fundamental distribution spaces:

V = S'(M₄, C),   W = S'(M₄, C³),

where M₄ is Minkowski space-time. The space V hosts scalar tempered wave distributions (the de Broglie basis of relativistic quantum mechanics), while W hosts vector-valued tempered fields with natural Maxwellian structure. For each unit vector e ∈ S², we introduce a canonical orthonormal tangent frame f_e arising from the Killing field K_e on the sphere (canonically associated with e) and we define the associated families of de Broglie waves η^e and Maxwell-de Broglie fields w^e. By these families we construct two continuous and Schwartz-linear operators:

F_e : V_e → W_e : ψ ↦ ∫M₄* (ψ)η w^e,   Ψ_e : W_e → V_e : F ↦ ∫M₄* (F)w^e η^e,

which are shown to be inverse isomorphisms of each other. The operator F_e transforms a wave distribution ψ into a Maxwellian field F_ψ, while the projection Ψ_e recovers the wave distribution from an electromagnetic-like field. We prove that these isomorphisms are dynamically compatible: a scalar distribution ψ ∈ V_e solves the relativistic Schrödinger equation

ih ∂_t ψ = H_m0 ψ,

if and only if its Maxwellian counterpart F_ψ = F_e(ψ) solves the corresponding generalized Maxwell-Schrödinger equation in W_e:

ih ∂_t F_ψ = H^e_m0 F_ψ.

This duality holds in both the massless (photonic) case --- where H^e₀ reduces to the curl operator multiplied times the Planck's constant and the speed of light --- and in the massive case --- where H^e_m0 is the spectral principal square root of the Schwartz diagonalizable and (strictly) positive operator

c²(h∇×)² + (m₀c²)² I_W^e.

The construction provides a precise mathematical mechanism to pass from Maxwell theory to the relativistic Schrödinger picture, mediated by Killing frames on the two-sphere. Beyond its intrinsic geometric interest, this approach suggests new perspectives in the analysis of wave propagation, polarization, and spectral synthesis, as well as possible applications to inverse problems in electromagnetism and the inclusion of curvature from general relativity.