Unification of the Fundamental Forces in Higher-Order Riemannian Geometry

Abstract

In Part I of the present series of papers, we adumbrate our idea of Riemannian geometry to higher order in the infinitesimals and derive expressions for the appropriate generalizations of parallel transport and the Riemannian curvature tensor. In Part II, the implications of higher-order geometry for the general theory of relativity beyond Einstein are developed. In the present Part III, we expand on the framework of Part II so as to take up the problem of field-theoretical unification. Employing the form of the Einstein-Hilbert action to higher order as proposed in Part II, we show how in nearly flat space the higher-order terms give rise to a gauge theory of Yang-Mills type. At the 2-jet level, the electroweak force emerges after imposition of gauge fixing. In fact, the proposed form of the Einstein-Hilbert action permits us to say more: we argue that the equivalence principle results in a Proca term that brings about the spontaneous symmetry breaking of the standard model. Two empirical predictions support our reasoning: first, we obtain a theoretical value for the Weinberg angle and second, we find that -- without any adjustable parameters -- the implied value of Coulomb's constant agrees well with experiment. The final section examines the 3-jet level. The same mechanism that produces Glashow-Weinberg-Salam electroweak theory at the 2-jet level eventuates in a chromodynamical force having $SU(3)$ symmetry at the 3-jet level.