Topological geometrodynamics. III. Quantum theory

The description of 3-space as a spacelike 3-surface of the space H = M4 x CP2 (product of Minkowski space and two-dimensional complex projective space CP2) and the idea that particles correspond to 3-surfaces of finite size in H are the basic ingredients of topological geometrodynamics, TGD, an attempt to a geometry-based unification of the fundamental interactions. The observations that the Schroedinger equation can be derived from a variational principle and that the existence of a unitary S matrix follows from the phase symmetry of this action lead to the idea that quantum TGD should be derivable from a quadratic phase symmetric variational principle in the space SH consisting of the spacelike 3-surfaces of H. In this paper a formal realization of this idea is proposed. First, the space SH is endowed with the necessary geometric structures (metric, vielbein, and spinor structures) induced from the corresponding structures of the space H. Second, the concepts of the scalar super field in SH (both fermions and bosons should be describable by the same probability amplitude) and of super d'Alambertian are defined. It is shown that the requirement of a maximal symmetry leads to a unique CP-breaking super d'Alambertian and thus to a unique theory ''predicting everything.'' Finally, a formal expression for the S matrix of the theory is derived