Finite yet Exact Expansion of the Electromagnetic Hamiltonian for Electrons and Positrons in Static Fields
Abstract:
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).