Towards a unified theory of the fundamental physical interactions based on the underlying geometric structure of the tangent bundle

Abstract

AbstractThis paper pursues the hypothesis that the tangent bundle (TB) with the central extended little groups of the SO(3,1) group as gauge group is the underlying geometric structure for a unified theory of the fundamental physical interactions. Based on this hypothesis as a first step, I recently presented a generalized theory of electroweak interaction (including hypothetical dark matter particles) (Herrmann in Eur Phys J C 79:779, 2019). The vertical Laplacian of the tangent bundle possesses the same form as the Hamiltonian of a 2D semiconductor quantum Hall system. This explains fractional charge quantization of quarks and the existence of lepton and quark families. As will be shown, the SU(3) color symmetry for strong interactions arises in the TB as an emergent symmetry similar to Chern–Simon gauge symmetries in quantum Hall systems. This predicts a signature of quark confinement as a universal large-scale property of the Chern–Simon fields and induces a new understanding of the vacuum as the ground state occupied by a condensate of quark–antiquark pairs. The gap for quark–antiquark pairing is calculated in the mean-field approximation, which allows a numerical estimation of the characteristic parameters of the vacuum such as its chemical potential, the quark condensation parameter and the vacuum energy. Note that a gauge theoretical understanding of gravity was previously achieved by considering the translation group T(3,1) in the TB as gauge group. Therefore, the theory presented here can be considered as a new type of unified theory for all known fundamental interactions linked with the geometrization program of physics.