Quantum topological geometrodynamics

Abstract

The description of 3-space as a spacelike 3-surface X of the space H = M/sup 4/ x CP/sub 2/ (Product of Minkowski space and two-dimensional complex projective space CP/sub 2/) and the idea that particles correspond to 3-surfaces of finite size in H are the basic ingredients of topological geometrodynamics (TGD), an attempt at a geometry-based unification of the fundamental interactions. The observations that the Schrodinger equation can be derived from a variational principle and that 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 for some kind of superfield (describing both fermions and bosons) in the configuration space consisting of the spacelike 3-surfaces of H. 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-manifold X to H, inherits a coset space structure from H and can be given a Kahler geometry invariant under the local M/sup 4/ x SU(3) an under the group Diff of X diffeomorphisms. The space Map(X,H)more » 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 Kahler properties of Map(X,H). The construction of physical states leads by local M/sup 4/ x 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 the S-matrix of the theory is derived and general properties of the S-matrix are discussed.« less