The Klein-Gordon Equation, exp(i Action) [Propagator] and Fluctuations

In a previous note, we argued that exp(i Action) for a nonrelativistic particle may be varied independently in T and X, where velocity = X/T. This variation indicates that energy is not kept constant. The solution exp(i Action)= exp(i .5m XX/TT) (for this case) leads to the nonrelativistic Schrodinger equation. 

     In this note, we consider the free particle relativistic Lagragian (1): mo sqrt(1-vv) where v=X/T. We argue that the equation describing independent fluctuations of X and T is the Klein-Gordon equation.