Stochasticity at a Single Particle Level?

 In previous notes (1), we argued that stochasticity (in x and t for a fixed E energy and p momentum) may occur at a single particle level based on the Lorentz invariant A = -Et+px ((1)). This invariant is also the relativistic/nonrelativistic classical action for v=x/t. The argument made in (1) was that a fluctuation in t of constant/E could cancel one in x of constant/p. We further argued one could create eigenvalue equations using id/dt and -id/dx and obtain the two dimensional probabilities exp(-iEt) and exp(ipx), i.e obtain a description of free particle quantum mechanics. 

   Experimentally, both exp(-iEt) and exp(ipx) have been verified through interference patterns. Theoretically, however, ((1)) was known about 1905 and no single particle stochasticity was associated with it. In this note, we try to argue why there should be an associated stochasticity associated with ((1)) based on motion through a constant potential for a particle with rest mass and motion through a medium with an index of refraction for a photon. We in fact argue that one cannot solve certain photon problems such as one dimensional reflection/refraction without explicitly considering the stochasticity which is usually left out when describing p2=pn2 and c2=c/n2 where n2 is the index of refraction in the second medium and p and c are values in the first medium with n1=1.

   We further consider exp(ipx) and exp(-iEt) as arising from the maximization of Shannon’s entropy subject to certain constraints. The product of these two two-dimensional periodic probabilities is exp(-iEt+ipx) which we argue must hold as seen from observers moving with constant velocity. Thus we argue that the Lorentz invariant -Et+px follows from statistical considerations alone. Given this expression (a subset of the whole picture), one may try to create a deterministic viewpoint by setting x/t=v=constant. This allows one to consider transformations which define E and p in terms of mo and v and yields the invariant EE = pp+momo  (c=1). Thus we argue that the conservation of energy equation ultimately follows from a deterministic view of the argument of a statistical probability. In other words, it is only a subset of the full picture i.e the quantum mechanical statistical one.