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

 In a previous note (Part 1), we argued that for both a relativistic and nonrelativistic free particle Lagrangian, id/dT exp(i Action) = E exp(i Action) and -id/dX exp(i Action) = p exp(i Action) where velocity= X/T. We further argued that these results allow one to convert an energy conservation equation (or even a Dirac type equation) in E and p into a differential equation in terms of d/dT and d/dX. Thus, d/dT and d/dX seem to represent fluctuations because velocityt=X/=constant is not imposed.For the case of a free particle with no velocity related potential such as vA(x), we argued Action= L(v) T as v is constant. Furthermore, L(v) so d/dt and d/dx both make use of dL/dv. In this note, we consider the presence of a magnetic vector potential term vA(x), but no electric field so the particle does not accelerate.