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

This paper explores some geometric 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, which generate the isometries of the sphere under the rotation group SO(3). These fields, represented as Kv(q)=v×q for v∈R3, form a 3-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 adjoint action and exponential map, highlighting the geometric and algebraic unity of spherical symmetries. Subsequently, we focus on constructing a smooth orthonormal right-handed tangent frame fe:S2∖{e,−e}→T(S2)2, defined as fe(u)=(K^e(u),u×K^e(u)), where K^e(u)=e×u|e×u| is the unit vector of the Killing field Ke(u)=e×u. We verify its smoothness, orthonormality, and right-handedness, characterized by K^e(u)×(u×K^e(u))=u. 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 contractible 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. We define a frame g=(r,s,u) over the dual Minkowski space M4*∖Π, with u=k|k|, extended from fe, satisfying u×s=−r. The complex field wk(x)=ei〈k,x〉(rk+isk), where ηk(x)=ei(k·x−ωt) is the de Broglie family, is shown to satisfy Maxwell’s equations for light-like 4-wavevectors k, with ∇×wk=|k|wk and i∂wk∂t=c|k|wk. The frame’s orientation allignes with circularly polarized plane wave solutions. The deeper scientific significance lies in reuniting 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 spacetime). The construction offers a geometric perspective on electromagnetic waves, with potential applications in optics, photonics, quantum field theory and gauge theories. This paper blends mathematical analysis, differential geometry, Schwartz distributions with quantum physics and Maxwell’s electromagnetism.