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Talking with Texas Writers: Twelve Interviews by Patrick Bennett
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.
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