by Jose G. Vargas, 138 Promontory Rd., Columbia, SC 29209, USA. firstname.lastname@example.org
Inspired by a similar, more general treatment by Kaehler, 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 concepts such as flows of vector fields for its definition. Enter Quantum Mechanics based on the Kaehler 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. Kaehler 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 -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.
Source: Email from J. Vargas, josegvargas_at_earthlink.net