Finite yet Exact Expansion of the Electromagnetic Hamiltonian for Electrons and Positrons in Static Fields
This poster is the third complement to the talk by the present author "A unifying calculus for physics: a demonstration in relativistic quantum mechanics".
If m is the dominant energy, the wave function can be written with a phase factor with "-imt" in the exponent, factor that multiplies a function slowly varying with time. But one certainly retains the option of having "-iEt" as exponent (where E is the proper value of the energy) multiplying a function of the spatial coordinates only, if the energy is a constant. It follows from this that the time derivatives of large (small) components in the dominant energy approach are proportional to the large (respectively small) components themselves. This considerably simplifies the mother system. Exact (meaning closed form) separated equations for large and small components are then obtained, in the case of both, electrons and of positrons.
In the case of static fields, some important features of the exact results obtained for the large and small components of both are:
a) The Hamiltonian for the small components contains the same type of terms as for the large components, plus an additional term which is the negative of twice the rest energy, thus, again, as if the rest energy were -2mc2 and everything else remained the same. This is achieved without resorting to time and charge reversal.
b) (To be viewed with the perspective that, in the "Kaehler version of the Pauli approximation", the Hamiltonian contains the electric potential energy term plus a second term that contains first components of mass/kinetic energy and Pauli's magnetic moment contributions). In the exact equation for large components in the case of static fields, one obtains generalizations of those two energy terms (thus further expansion of kinetic energy, as well as corrections to magnetic moment) plus an additional term which is a general form of the Darwin and Lamb shift terms (The "corrections" implicit in those generalized expressions are, however, much smaller than those that pertain to the actual radiative corrections for magnetic moment and Lamb shift, which the Kaehler-Dirac equation is not meant to address).