Born's Rule W*(x,t)W(x,t) and Determinism

 In (1) it is argued that a quantum particle’s probability P(x,t) satisfies a Nelson stochastic equation: d/dt partial P(x,t) + d/dx partial (b(x,t)P(x,t)) = D d/dx d/dx partial P. ((1))

where b(x,t) = hbar/m d/dx partial  S(x,t)  + 2D d/dx partial ln(R(x,t)) ((2)) with W(x,t)=wavefunction= R(x,t) exp(i S(x,t). Thus both a quantum wavefunction (satisfying the Schrodinger equation) and a separate P(x,t) are present. If P(x,0)= W*(x,0)W(x,0) ((3)), i.e. Born’s rule, then this rule holds for later times. It is, however, suggested in (1) that if P(x,0) not= W*(x,0)W(x,0) then the system may relax over time into a situation for which ((3)) holds. 

       In this note, we argue that for a free particle moving with constant p in one direction, P(x,t)=constant, but there is an underlying two dimensional wavefunction exp(ipx). We ascribe this to two x-probability distributions associated with motion. In particular: v= (xfinal - xintial) / delta t. If xfinal and xinitial are both distributions such that on average v = constant, one needs to give a probability for both i.e. have two dimensions or an ordered pair. The two probability distributions, however, are not independent as they must describe a constant velocity i.e all points of space being the same. This leads to exp(ipx) such that exp(-ipx) exp(ipx) =1 . In other words, one may constrain two probability distributions to yield a deterministic type result for a single particle which appears as a probabilistic result for a steady stream.Born’s Rule automatically applies to a free particle and is not a “postulate” as argued in (1). The form exp(ipx) follows from the Lorentz invariant -Et+px we argue. 

   We further argue that if one creates a linear superposition of exp(ipx)s i.e. W(x)=Sum over p a(p)exp(ipx) then  Integral dx W*(x)W(x) is always linked to a steady stream probability picture ie. Sum over p a*(p)a(p) = 1. This is why given a p, one does not know its x point- it is associated with a steady stream. As a result, we argue that there is always a meaning for Born’s rule W*(x,t)W(x,t). As a result P(x,t=0) not= W*(x,t=0)W(x,t=0) must describe a probability which differs from the steady stream scenario linked with exp(ikpx).