Orthonormal Right-Handed Frames on the Two-Sphere and Solutions to Maxwell’s Equations via de Broglie Waves

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere S2, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on S2 (represented by the unit sphere of R3), which generate the isometries of the sphere under the rotation group SO(3). These fields, realized as functions Kv : S2 → R3, defined by Kv(q) = v×q for a fixed v ∈ R3 and any q ∈ S2, generate a three-dimensional Lie algebra isomorphic to so(3). We establish an isomorphism K : R3 → K(S2), mapping vectors v = au (with u ∈ S2) to scaled Killing vector fields aKu, and analyze its relationship with SO(3) through the exponential map. Subsequently, at a fixed point e ∈ S2, we construct a smooth orthonormal right-handed tangent frame fe : S2 \ {e,−e} → T(S2)2, defined as fe(u) = ( ˆKe(u),u× ˆKe(u)), where ˆKe is the unit vector field of the Killing field Ke. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on S2 \{e,−e} is either fe or a rotation thereof by a smooth map ρ : S2 \{e,−e} → SO(3), reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via SO(3) symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics.