Towards a Topological Formulation of Fundamental Interactions

Abstract

A thought experiment is formulated to unify quantum mechanics and general relativity in a topological manner. An analysis of the interactions in Nature is then presented. The universal ground state of the constructed theory derives from the cyclic properties ($S^1$ homotopy) of the topological manifold $Q=2T^3\oplus 3S^1\times S^2$ which has 23 intrinsic degrees of freedom, discrete $Z_3$ and $Z_2\times Z_3$ internal groups, an SU(5) or SO(10) gauge group, and leads to an anomalous U(1) symmetry on a lattice. These properties can in principle reproduce the standard model with a stable proton. The general equation of motion for the unified theory is derived up to the Planck energy and leads to a Higgs field with possible inflation. The thermodynamic properties of $Q$ are discussed and yield a consistent amplitude for the cosmic microwave background fluctuations. The manifold $Q$ possesses internal energy scales which are independent of the field theory defined on it, but which constrain the predicted mass hierarchy of such theories. In particular the electron and its neutrino are identified as particle ground states and their masses are predicted. The mass of the electron agrees very well with observations. A heuristic argument for the occurrence and magnitude of CP violation is given. Future extensions of the presented framework are discussed.