Speculation on Quantum Mechanics and Special Relativity

In this note, we consider the classical Action= Integral L dt which is equal to Et - px with E=.5mo vv and p=mo v. We also consider the factor exp(ipx-iEt) for a photon which follows from Maxwell’s equations. We argue that a probabilistic view of a free quantum particle could lead to the idea of exp(iEt-px) (nonrelativistic). This expression already links p=mov with x i.e. p is a wavelength. Given that p=mov i.e. rest mass flux, we argue that E=.5movv might actually be E=b1 mo + b2.5movv with b2 rendering the second term small. (This places a limit then on the maximum v.). Next, we argue that Et - px should be invariant when viewed from different frames on the basis of probability. Finally, we note that Et-px is related to Et(1-vv/cc). For a particle at rest, E=mob1, but if E and t are on the same footing from quantum probabilistic ideas, then mob1 and to are energy and time in the rest frame and mob1/sqrt(1-vv) and to/sqrt(1-vv) are energy and time seen from the lab frame for invariance to hold. Such a picture yields the nonrelativistic kinetic energy .5movv upon expanding E for small v. Thus, we speculate that one may obtain special relativistic results from quantum probabilistic arguments. [As a note, in (1) Lorentz boosts are obtained by analyzing transformations of quarks together with the Pauli principle, but the idea is to also argue that special relativity follows from quantum mechanics.