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The description of 3-space as a spacelike 3-surfaceX of the spaceH = M
4 ×CP
2 (Product of Minkowski space and two-dimensional complex projective spaceCP
2) and the idea that particles correspond to 3-surfaces of finite size inH are the basic ingredients of topological geometrodynamics (TGD), an attempt at 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 for some kind of superfield (describing both fermions and bosons) in the configuration space consisting of the spacelike 3-surfaces ofH. This idea as such has not led to a calculable theory. The reason is the wrong realization of the general coordinate invariance. The crucial observation is that the space Map(X, H), the space of maps from an abstract 3-manifoldX toH, inherits a coset space structure fromH and can be given a Kahler geometry invariant under the local M4×SU(3) and under the group Diff ofX diffeomorphisms. The space Map(X, H) is taken as a basic geometric object and general coordinate invariance is realized by requiring that superfields defined in Map(X, H) are diffeo-invariant, so that they can be regarded as fields in Map(X, H)/Diff, the space of surfaces with given manifold topology. Superd'Alembert equations are found to reduce to a simple algebraic condition due to the constant curvature and Kähler properties of Map(X, H). The construction of physical states leads by localM
4× SU(3) invariance to a formalism closely resembling the quantization of strings. The pointlike limit of the theory is discussed. Finally, a formal expression for theS-matrix of the theory is derived and general properties of theS-matrix are discussed.
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 Schrdinger 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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