Unification of the Fundamental Forces in Higher-Order Riemannian Geometry

TL;DR

This paper develops a higher-order Riemannian geometric framework to unify fundamental forces, deriving gauge theories and predicting physical constants without adjustable parameters.

Contribution

It introduces a higher-order geometric approach to unify gravity with gauge theories, leading to predictions of the Weinberg angle and Coulomb's constant.

Findings

Electroweak force emerges at the 2-jet level.

Spontaneous symmetry breaking of the standard model is explained.

Predicted values of physical constants match experimental data.

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.