Spinors, Minimal Surfaces... and the Hopf Map.

A re-reading of Cartan's book on "Spinors" (and Chadrasekhar's book on "Black Holes") leads to the thought that there is a connection between Spinors, Minimal Surfaces, Chirality, Helicity, Topological Torsion, Spin, Orientability, Point particles, Fractals, Polarization, the Light Cone and the Hopf Map.

It is remarkable that both Cartan and Chandresekhar do not mention the fact that an isotropic (complex null) vector - which is used to define the spinor - is related to the generator of a Minimal Surface. Neither do the two authors mention the fact that the expressions they utilize to define spinors are essentially linear combinations of the Hopf Map. This is surprising to me, as Cartan was a differential geometer and knew about minimal surfaces. The relationship of spinors to minimal surfaces is ignored by many other authors as well as Cartan and Chandrasekhar. My recent (independent) appreciation of the relationship turns out to be predated by a note given by D. Sullivan in 1989. A few references to the concept can be found in the more recent math literature. None of the literature of my experience connects the Hopf map to the minimal surface spinor representation. The complex null vector (Spinor S) consists of two orthogonal Hopf vectors, S = H1 + i H2.

The idea that minimal surfaces come in "conjugate" pairs, and the fact that these pairs represent extremes of "polarization" turns out to be a useful idea that generalizes the theory of coherent optics. The idea that a complex analytic function generates a minimal surface implies that all functional iterates are also minimal surfaces. Using functional iterates of Z^2+C ultimately generates the Mandelbrot set, a process which thereby connects fractals to minimal surfaces. Note that there should exist polarization conjugates for fractals. The idea that the Spinor has a zero quadratic form, implies that complex "point" particles in the form of Spinors can have "interior" structure. These concepts have no explicit dependence on the ideas of Quantum Mechanics. The fact that the conjugate pairs of minimal surfaces are all related to the same light cone implies that the observable change of polarization state could proceed at speeds faster than the speed of light.