Are Maxwell's Electromagnetic Equations Probabilistic?
Francesco R. Ruggeri · Zenodo (CERN European Organization for Nuclear Research) · 2023
We have argued in previous notes (1) that the “wave” nature of quantum mechanics follows from the Lorentz invariant for a constant p: A= -Et+px which holds for both a particle with rest mass and a photon. In A, t and x are separated indicating independence which violates the Newtonian view x(t). “A” leads to dA/dx partial = p and dA/dt= -E. Creating eigenvalue equations yields: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt). Thus, x and t may be separated. Given that exp(ipx) exists in all of space (because there is no time), it represents a somewhat spatially invariant probability. In the case of a photon, Maxwell showed one may write wave equations for the electric El and magnetic B fields using his electromagnetic equations with no charge or currents.. He showed the wave travels at the speed of light in a vacuum and thus characterized light as an electromagnetic wave. This is not the same statement as exp(ipx) which is a probability wave. One may note that Maxwell’s equations are Lorentz invariant, so the ideas of the first paragraph should hold. In this note, however, we consider the Poynting equation: d/dt partial (.5( eoElEl+1/uoBB)) + 1/uo grad dot El x B = function(charge and current). 1/uo ElxB is considered to be a momentum density. We note Maxwell’s equations contain d/dt partial and d/dx partial, so time and space are considered independent (unlike x(t)), and act on El and B. The presence of charge and current densities breaks spatial invariance i.e. creates local regions. We argue in this note that d/dx and d/dt are generators of translations in space and time. We further argue that Maxwell’s theoretical creation of a photon follows from introducing spatial invariance into the problem, i.e. removing all charge and current densities. Spatial invariance in Newtonian mechanics, however, is associated with a constant i.e. a constant speed and momentum and equal probability to be at any x. Given that d/dx and d/dt appear, one has a function of space and time which seems to be completely at odds with spatial invariance and the idea of a constant momentum. We have argued in (2), however, that constant motion, even though represented by v=constant, is associated with the force that created the momentum, and this manifests itself in a spatial density which is probabilistic. Thus the presence of d/dt and d/dx in Maxwell’s electromagnetic equations, even when charge and current densities are removed, implies the same thing. Thus we argue that Maxwell’s electromagnetic equations are probabilistic. We investigate these ideas in this note. We further argue that it is spatial invariance, together with the presence of d/dx partial which yields the periodic form exp(ipx). Thus it is not a coincidence that wave equations appear for El and B.