Klein-Gordon Approximation


The direction in the 4D Space-Time continuum is lost in the scalar product.


This equation describes only Bosons – for example: photons

(It is the wave equation obtained from Maxwell’s equations of electrodynamics.)



Schrődinger Approximation



The expansion separates space from time in the 4D Space-Time continuum.


This allows a coordinate separation among radial, angular, and time coordinates


                         Y(r,q,f,t) = R(r)Y(q,f)T(t)


This permits the spin to be treated as an angular coordinate, and combined with the spatial rotations


                                   Y(q,f) ®  Y(q,f)c(spin)


And here the orbital and spin angular momentum of many electrons can be combined.



Dirac approximation


Here the 4D Space-Time continuum causes spin to be an intrinsic dynamic quantity, associated with its 4D nature (and NOT angular variable).


It is bound individually to the electron (up/down) and positron (up/down) solutions.


Thus, the result is relativistically correct for a single electron charge-coupled to a nucleus.


Virtual photon exchange with all other electrons in the universe are neglected.


It includes only the connection of the spin-own-orbit interaction, and cannot include spin-other-orbit or orbit-other-orbit interactions.