Vibrational Coordinates for the Description of Physical Systems
Keywords: simple harmonic oscillator, complex coordinates, Compton frequency, Hermitian metric symmetry, Hermitian?]Riemannian geometry, Lorentz force, wave function, anti?]Hermitian curvature symmetry, Maxwell
The introduction of special complex coordinates, motivated by vibrational systems, proves to be very useful in a comprehensive unification of gravitation with electromagnetism when applied to Einstein\'s Hermitian metric form. The geometry that follows is Hermitian?]Riemannian, with all symmetries modified accordingly. Charge appears in the field equations as the product of mass and frequency, preserving the equivalence principle, while maintaining a double?]valued Lorentz force term. Finally, quantization rules can be derived from various complex coordinate expansions involving only classical mechanical principles.