  Inspired by a similar, more general treatment by Kahler, we obtain the spin
operator by pulling to the Cartesian coordinate system the azimuthal partial
derivative of differential forms. At this point, no unit imaginary enters the
picture, regardless of whether those forms are over the real or the complex
field. Hence, the operator is to be viewed as a real operator. Also a view of
Lie differentiation as a pullback emerges, thus avoiding conceps such as flows
of vector fields for its definition. Enter Quantum Mechanics based on the
Kahler calculus. Independently of the unit imaginary in the phase factor, the
proper values of the spin part of angular momentum emerge as imaginary because
of the idempotent defining the ideal associated with cylindrical symmetry. Thus
the unit imaginary has to be introduced by hand as a factor in the angular
momentum operator |and as a result also in its orbital part| for it to have
real proper values. This is a concept of real operator opposite to that of the
previous paragraph. Kahler stops short of stating the antithesis in this pair
of concepts, both of them implicit in his work. A solution to this antithesis
lies in viewing units imaginary in those idempotents as being the real
quantities of square minus 1 in rotation operators of real tangent Clifford
algebra. In so doing, one expands the calculus, and launches in principle a
geometrization of quantum mechanics, whether by design or not.
