Quantum Mechanics and A Breakdown of Deterministic Motion?

  Newtonian mechanics focuses on dx and dt units which tend to zero and so presents, it seems, a smoothed over or averaged type of motion. This motion in principle extends to dx and dt values which are almost zero. There is, however, a priori no reason not to expect that such a smooth averaging is at some length scale based on  another kind of motion. The question is: What should that “other” kind of motion be? If it is deterministic in x and t, one may reapply the idea of this motion breaking down at yet some smaller scale. Thus it seems one should consider decoupling of x and t because one cannot follow a particle from one x point to another in time within some small length scale as this would be deterministic. 

   The opposite approach seems to be a probabilistic one with t and x decoupled. The external ruler and clock which exist in the Newtonian picture, however, do not disappear simply because a length scale is small. Thus having x positions based on probabilities at some length scale still uses the notion of the probability itself being smoothed over because it contains x which may take on any value. This is not necessarily a problem because the probabilistic behaviour must still appear to be Newtonian on average and so x and t based on the external clock/ruler with infinite resolution need to “appear” again. (For example, an electron has a wavelength and exhibits interference from two slits, but while moving towards the slits, distance = velocity* time with distance and time measured by an external clock. It is a mathematical generalization, however, to expect that d=vt actually fully describes motion (i.e. other than on average) within a wavelength length. If it did there would be no interference pattern.

   The idea of a smoothed over (or averaged motion) based on probabilistic motion also appears in the classical world, but for many particles. For example, a Maxwell-Boltzmann gas with no potential has uniform density. If one considers a hypothetical box containing many molecules, it is like a Newtonian stationary object with a balance of pressure. In the case of a potential (1-D) one may consider a dx unit (with many molecules) and balance Force(x) density(x) with the difference of pressure to find the density distribution proportional to exp(- V(x)/T). These ideas are completely Newtonian, but they are smoothed over average notions. At a microscopic level molecules undergo probabilistic collisions linked to T (temperature). Thus the microscopic picture is not static - it is only the equilibrium average or big picture which is static. Even in the case of no potential, focusing on a tiny region of space smaller than the interparticle average separation and arguing that density is constant seems to be an average argument which bypasses much of the reality.

     If x and t may then become probabilistic and -Et+px is a Lorentz invariant, then why should there not be stochasticity linked with E and p? In fact we argue they are == as shown by Maxwell’s solution of a photon from his electromagnetic equations. E and p (energy and momentum) are average values. Their density is given by:   .5C1( Electric field dot Electric field + C2 Magnetic field dot Magnetic field) and the Poynting vector for momentum. Both the Electric field and Magnetic field fluctuate as cos(-Et+px) with E and p being average values. At first this was ascribed to wave motion in a medium, but in 1905 Einstein’s photoelectric arguments against energy being spread out seem to suggest a breakdown of deterministic physics describing a wave in a medium. Instead a photon is probabilistic and has a wavefunction of Electric + i Magnetic field. (2)