The description of 3-space as a spacelike 3-surface of the spaceH=M 4 ×CP 2 (product of Minkowski space and two-dimensional complex projective space CP2) and the idea that particles correspond to 3-surfaces of finite size inH are the basic ingredients of topological geometrodynamics, an attempt to a geometry-based unification of the fundamental interactions. The observations that the Schrödinger equation can be derived from a variational principle and that the existence of a unitaryS 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 spaceSH, consisting of the spacelike 3-surfaces ofH. In this paper a formal realization of this idea is proposed. First, the spaceSH is endowed with the necessary geometric structures (metric, vielbein, and spinor structures) induced from the corresponding structures of the spaceH. Second, the concepts of the scalar super field inSH (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 uniqueCP-breaking super d'Alambertian and thus to a unique theory “predicting everything.” Finally, a formal expression for theS matrix of the theory is derived.
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Pitkänen, M. Topological geometrodynamics. III. Quantum theory. Int J Theor Phys 25, 7–54 (1986). https://doi.org/10.1007/BF00669712
- Quantum Theory
- Variational Principle
- Projective Space
- theS Matrix
- Unique Theory