Topological geometrodynamics, Part I: General theory

The recent status of topological geometrodynamics (TGD) is reviewed. One can end up with TGD either by starting from the energy problem of general relativity or from the need to generalize hadronic or superstring models. The basic principle of the theory is `Do not quantize!ʹ meaning that quantum physics is reduced to Kähler geometry and spinor structure of the infinite-dimensional space of 3-surfaces in 8-dimensional space H=M4+×CP2 with physical states represented by classical spinor fields. General coordinate invariance implies that classical theory becomes an exact part of the quantum theory and configuration space geometry and that space-time surfaces are generalized Bohr orbits. The uniqueness of the infinite-dimensional Kähler geometric existence fixes imbedding space and the dimension of the space-time highly uniquely and implies that superconformal and supercanonical symmetries acting on the lightcone boundary δM4+×CP2 are cosmologies symmetries. The work with the p-adic aspects of TGD, the realization of the possible role of quaternions and octonions in the formulation of quantum TGD, the discovery of infinite primes, and TGD inspired theory of consciousness encouraged the vision about TGD as a generalized number theory. The vision leads to a considerable generalization of TGD and to an extension of the symmetries of the theory to include superconformal and Super-Kac-Moody symmetries associated with the group P×SU(3)×U(2)ew (P denotes the Poincaré group) acting as the local symmetries of the theory. Quantum criticality, which can be seen as a prediction of the theory, fixes the value spectrum for the coupling constants of the theory. The proper mathematical and physical interpretation of the p-adic numbers has remained a long-lasting challenge. Both TGD inspired theory of consciousness and the vision about physics as a generalized number theory suggest that p-adic space-time regions obeying p-adic counterparts of the field equations are geometric correlates of mind in the sense that they provide cognitive representations for the physics in the real space-time regions representing matter. Evolution identified as a gradual increase of the infinite p-adic prime characterizing the entire Universe is basic prediction of the theory. S-matrix elements can be identified as Glebsch–Gordan coefficients between interacting and free Super-Kac-Moody algebra representations and it is now possible to give Feynmann rules for the S-matrix in the approximation that elementary particles correspond to the so-called CP2 type extremals.