Fundamental forces and their dynamics

TL;DR

This paper proposes that the dynamics of fundamental forces should be derived solely from spacetime geometry, similar to general relativity, without relying on an action principle, emphasizing geometric properties like the Bianchi identity.

Contribution

It introduces a geometric principle for deriving force equations directly from spacetime geometry, extending the geometric approach of general relativity to gauge fields.

Findings

Force dynamics derived from geometric properties of spacetime.

Unified geometric perspective on Abelian and non-Abelian gauge forces.

Reflection on duality and unification of fundamental forces.

Abstract

In this essay, we wish to propose a general principle: \it{the equation of motion or dynamics of a fundamental force should not be prescribed but instead be entirely driven by geometry of the appropriate spacetime manifold, and the equation is then obtained by employing only the geometric property without appeal to an action.} The motivation for this pronouncement comes from the fact that the equation of motion of general relativity follows from the geometry of Riemannian spacetime manifold without appeal to anything else from outside. The driving differential geometric property is the Bianchi identity satisfied by the Riemann curvature tensor. Similarly it is geometry of the principal tangent bundle of fibre spacetime manifold that may account for dynamics of the gauge vector fields. It is the classical electric force for the Abelian gauge symmetry group while the non-Abelian symmetry leads to the non-Abelian forces, the weak and the strong. We shall also reflect on a unified picture of the basic forces, and the duality correspondences it may inspire.