Geometric Electro-Magnetic Field Equations for Particles with Charge and Spin

The complete and self-consistent "Maxwell" equations are displayed algebraically, describing (time-delayed) force and torque interactions between moving particles with charge and spin. The algebra is the 4-grade Grassmann algebra describing 3D Euclidean space. Here, the {X,Y,Z} coordinate vectors are derived from Points, representing the relative particle positions, geometric structures, and geometric interactions. In essence, the 4 basis elements and 2^4=16 graded elements are requied to represent forces, torques and energy transfer from particle motion, spin, and helicity. In contrast, both the standard "vector algebra" and "Cifford geometric algebra" are inadequate.

With the full Grassmann division algebra, two forces can add to become a torque; two flows can add to become a circulation; and helicity is given geometric meaning. With similar clarity, the algebra describes the (v/c)^1 "coordinate transformations" between vector electric E and bivector magnetic B, through simple motion of an origin point. Consequently, the algebra explicitly distinguishes between conduction currents leading to charge-separation energy, and the orthogonal induction currents considered in MHD.

Most interestingly, the fulll algebra interprets the non-linear "photon" transferring energy between two particles as the non-linear product of two oppositely-traveling linear fields. For this symmetry, both fundamental "length times energy" constants, e^2 and ħc, must be included in the graded algebra.

Full poster at NNP.ucsd.edu/GeoLocCau2 .