A Novel Energy-Frequency Relation with Relativistic Saturation from Spin-Field Coupling

Abstract

We derive and analyze a novel energy-frequency relation emerging from the coupling of relativistic massive particles with oscillatory spin-dependent fields. The resulting expression:

E = sqrt((m c^2)^2 + (kappa chi S omega)^2) * tanh(m c^2 / (kappa chi S omega))

Exhibits unique saturation behavior where energy approaches the rest mass energy m c^2 in both low-frequency and high-frequency limits, with a characteristic peak at intermediate frequencies. This formula interpolates between non-relativistic perturbative corrections and fully relativistic saturation regimes, suggesting applications in quantum field theory, condensed matter systems with spin-orbit coupling, and high-intensity laser-matter interactions.