Physical Constraints Simplify 5 Dimensions in Relativistic Waves

The classical mechanics of Newton—where momentum energy is less than rest mass energy, pc < m_oc² = E_o—transitions, with increasing momentum, to the relativistic mechanics of Einstein and Klein-Gordon—where pc > m_oc². The transition replaces the classical rest mass energy constant E_o by Einstein’s varying relativistic energy $E=m^{\prime }{c}^2=\sqrt{p^2{c}^2+m_o^2{c}^4}$ . We proceed to apply this 5-dimensional relativistic equation to dispersion dynamics in electron microscope probes¹. The two scalers, E and m_o are initially presented as concentric, spherically-symmetric, spheres that are constrained by the Pythagorean triangle that is implicit in the relativistic expression $E^2=p^2{c}^2+m_o^2{c}^4$ , where the speed of light c is constant. Then p, E and m_o are co-planar, and all act like vectors in 5 independent dimensions. The fact is significant because E is conjugate to time t in the free particle wave equation. Band structures and band gaps are derived. Remarkable physical effects are explained with mathematics hardly more complex than Pythagoras’s theorem in a realistic framework of conventional and verified hypotheses. Consistent with Occham’s razor, a proper economy in dimensionality is retained. On this firm footing, other dimensions extend to 2-dimensional displays of physical properties