Reflection/Refraction and Quantum Dynamic Probability Included in Special Relativity?

In a previous note (1) we argued that quantum square root probabilities follow from the consideration of light reflecting and refracting at a surface along the y direction at x=0. No knowledge of Maxwell’s periodic photon electric field is needed. The photon number and energy are conserved at a single photon level, but flux/intensity may be included through (2)  AA/c1 = BB/c1 +CC/c2  where AA is the intensity of the incident beam (AA>0), BB the reflected and CC the refracted. There are two unknowns B,C if A is known and one equation which may be broken into two  A+B=C and A/c1 -B/c1 = C/c2. For n1<n2 (n=index of refraction B<0). Thus we argue that there is an underlying square root probability.

     In this note we first heuristically suggest the form for the square root probability. A key point is that “particle” features such as E and p (for a photon) govern the dynamical wavelike probability. This also applies to a particle with rest mass. 

We also ask why this square root probability has not appeared in dynamical theories and argue that it appears in special relativity (i.e. a Lorentz boost) for a particle with rest mass mo at x=0 at t=to. As seen from a frame moving at constant speed -v, this particle moves with speed v. Thus x’=vt’, yet a 2x2 Lorentz transformation is applied to (x=0,t=to). Only one relation is needed because x’=vt’ and p’=E’/cc v suffice to yield the variables x’ and p’. Thus we argue that a Lorentz transformation describes more than is necessary for a simple particle. The 2x2 nature of the problem we suggest describes more than a simple particle. It leads to the Lorentz invariant -Et+px ,but x=vt so why should x and t appear as if independent? We argue that -Et+px=A is compatible with dA/dx |constant t = p and dA/dt |constant x = -E. P and E are “particle/photon” properties, but exp(ipx) or exp(-iEt) represent eigensolutions of -id/dx and id/dt. Thus if x and t are independent exp(ipx) may be interpreted as existing at all x points (as there is no time). This is the picture of an unusual probability which is two dimensional, with components which may be positive or negative as in the reflection/refraction problem. Thus we argue that the notion of a wave probability or square root probability not only exists in the reflection/refraction problem, but in the case of special relativity with what looks like an unnecessary 2x2 matrix from the point of view a particle (mo, x=0, t=to) describing an associated square root wave like probability completely dependent on the particle properties E and p.