A brief geometric derivation of some fundamental physics

Abstract

Several fundamental results in physics are derived from the simple starting point of two commuting orthogonal unit vectors. The combination of these unit vectors leads to spherical harmonics and Schwinger's expression of the second-quantized angular momentum states in terms of bosonic operators. Commuting unit vectors can be turned into anticommuting ones by the restriction to a single unit vector. This leads to Pauli spin matrices. By including hyperbolic rotations, vectors can be given a finite norm and results from special relativity and Dirac's equation are found. The assumption that the change in four-momentum is due to the change in four-potential leads to the electromagnetic field tensor and the Lorentz force. Mawell's equations are obtained by viewing the four-potential as an harmonic oscillator driven by the four-current. The Schr\"odinger equation is obtained from the nonrelativistic limit of the four-momentum.