COMPLEX NUMBERS IN QUANTUM MECHANICS
http://modelingnts.la.asu.edu/pdf/Spin&uncert.pdf

VII. COMPLEX NUMBERS IN QUANTUM MECHANICS
The prominent role of complex numbers in quantum mechanics is usually taken for
granted and justiØed by its implications. I would like to call attention to an important
fact which shows that a deeper analysis is possible with potentially important physical
implications. That fact is that the unit imaginary in the Schroedinger wave function is
the generator of rotations about the axis of spin. (See the Appendix for a mathematical
formulation.)
Let us be clear about the ontological status of this purported fact. It is not an interpretation
arbitrarily imposed on the Schrƒodinger wave function by external considerations as
is the case with wave-particle duality. Rather it comes from requiring that quantum mechanics
be internally consistent, speciØcally, that the interpretation as well as the equations
of Schrƒodinger theory be derived from the Dirac theory.28
The consistency argument establishes the relation of spin to the unit imaginary in the
Schrƒodinger wave function for electrons but not for pions, which are also commonly described
by complex wave functions. Therefore, some new facts and ideas will be required to
extend what has been established in electron theory to a coherent physical interpretation
of all complex wave functions in quantum theory. By way of suggestion, let us speculate
brie°y on some possibilities.
Two facts of general signiØcance are available to guide our speculations: First, as already
mentioned, the unit imaginary in the electron wave function has a deØnite geometrical
interpretation by virtue of its relation to spin. Second, distinct wave functions in quantum
mechanics are related by the group structure of elementary particle theory. By combining
these facts in a unique model, we may be able to establish a geometrical interpretation in
spacetime of all wave functions and dynamical transformation groups relating them. This
is obviously a tall order, so the best that can be done here is to give some hint as to how
one might try to fulØll it.