Stochasticity in Newton's Second Law, Special Relativity and Quantum Mechanics

Newton’s law F= dp/dt is considered to be associated with deterministic kinematics. The problem, we argue, is that dt is not really zero (only very small on Newtonian scales). If, for the sake of argument, one considers a finite dt, one must then consider the workings of force in a very tiny time and there is no reason to believe this is a deterministic situation. It is more likely that there is some stochasticity at the level of dt and dx and so t and x within dt and dx are stochastic and associated with probability. On Newtonian scales, this would not be noticed, but on other scales (quantum ones), it could be. A key point seems to be that even if one has x=vt ( a deterministic equation), one does not really know the x and t values exactly, there is a stochastic uncertainty of dt, dx (because the particle was put in motion by an interaction). This means that even a free particle should have this inherent stochasticity in x and t, even if p and E are considered exact. Now expressions for p and E formally follow from special relativity and so if there is some dx,dt stochasticity and it has a certain range (i.e. a maximum value for dx and dt in a sense), this range should appear in special relativity as one must know dx,dt and dx’, dt’. We have suggested in previous notes that possible dx,dt ranges in fact appear in the Lorentz invariant A = -Et+px because for x,t (center of mass description) yielding a certain. Values, off center-of-mass values, x1=x+hbar/p and t1=t+hbar/E also yield the same A. We suggest that these are the “maximum” dx and dt values with uncertainty because the momentum or energy interaction can occur with probability within dx and dt and need not occur at the center-of-mass. It is dx and dt which differ in different frames, e.g. hbar/p’ and hbar/E’, but there is uncertainty in these ranges, i.e. a probability distribution. This uncertainty is associated with a free particle and should manifest itself in an interaction (e.g. 2-slit interference), but it also means that two body elastic scattering cannot ever be considered fully deterministic. There is still the stochastic, uncertain dx, dt. If one wishes to consider conservation of energy and momentum, however, one uses the notion of a free particle x=vt. Even though actual interaction hits may occur at different places, if one has two particles P(p1,x1) P(p2,x2) in an AND situation, one must have x1=x2 in order to show that momentum is conserved at the same x. This leads to the notion of exp(ipx) as a probability, or more fully exp(-iEt+ipx) which is consistent with the special relativistic condition on the range of dx and dt. If one were to consider special relativity, then two particles together have E1+E2 and p1+p2 and one uses the same x and t and so dx= hbar/(p1+p2) and dt= hbar/(E1+E2). This matches the P(p1,x)P(p2,x) scheme for ranges of dx and dt. Alternatively, one may consider that one has a uniform distribution for any outcomes of E1,E2, p1,p2 elastic scattering which conserve energy and momentum. This leads to exp(iC1E) and exp(iC2p), but these must display the dt, dx ranges and so one has exp(-iEt+ipx) in order to have a Lorentz invariant probability. The above arguments have been based on Newton’s second law dp/dt=F and special relativity which means that the two must exist together, which in fact they do. In particular, special relativity is not simply about viewing particles from different frames, but must also be linked with interactions which create the equivalent motion. As shown in (1), one may start with number and vector concepts E,p without knowing their explicit forms and then use special relativity to calculate them. In (1), however, this involved the notion of dp = Fdt and dke= Fdx even for the form of p unknown. Thus, dp/dt =F and special relativity are directly intertwined and the notion of a finite stochastic dt and hence dx in dp/dt=F should have effects on special relativity. One, however, has to directly impose this stochasticity in dt (and hence dx) on dp/dt=F by arguing that dt has a finite value with probabilistic t value within and so one must also look for the stochastic dx and dt range in special relativity, i.e. by considering A= -Et+px as described above. The stochasticity is an assumption, but the finite dt and t+hbar/E, x+hbar/p are not.