Quantum Free Particle exp(-iEt+i p dot r) Based on Relativistic Path Probability

  In a recent paper, (1) introduces a path related probability scheme in order to derive quantum mechanics probability. This probability is a real positive number between 0,1 and so one is classical, but the fact that it is linked to paths and not simply a position, time x,t suggests that this approach differs from strictly classical probability. In particular, the assumptions used by (1) are path related.

  (1)’s path based probability is P(nm) ( f(x,t)), where nm is a path and f(x,t), a Lorentz invariant. Bayesian composition is allowed and P(nm) = P(n)(path1) P(m)(path2). As a result, one may break a probability into a product of probabilities based on paths. For example, path1 might be from a starting point to a point P and path2 from P to a detector. For a linear path along x, this suggests the form exp(-iEt+ipx)=W(x,t). To obtain a real probability, one needs (1)’s assumption of time reversal invariant, i.e. P(-Et+px)=P(Et-px), so P = W*(x,t)W(x,t). We argue that (1)’s quantum form follows from the last two (of three assumptions).  Formally, one suggests that quantum mechanics arises from (A) “Kolmogorov additivity restricted to one or two path contributions”, (B) time reversal symmetry and (C )  Bayesian composition. 

   We go on to argue that the path related approach of (1) is equivalent to a momentum-energy conserving probability approach of (2).