From Poincare Invariance to Gauge Theories: Yang-Mills and General Relativity

TL;DR

This work shows that starting from Poincaré invariance and using Fock-Ivanenko covariant derivatives, one can derive the interacting gauge structure of both Yang-Mills theory (λ = 1) and general relativity (λ = 2) as local gauge theories. Lorentz invariance of massless fields induces gauge transformations for tensor fields, with covariant derivatives yielding field strengths and gauge-invariant Lagrangians. The framework naturally connects matter coupling via the interacting Dirac equation and recasts gravity in Palatini and Ashtekar formalisms, highlighting torsion, curvature, and self-dual connections as central geometric objects. This symmetry-centered perspective offers a unifying route to gauge theories and suggests avenues for extending to larger groups and mass-generation mechanisms beyond the Poincaré group.

Abstract

This article is founded on two fundamental principles: the principle field equations introduced in Refs. \cite{S, S1, S2} and the Fock-Ivanenko covariant derivatives \cite{FI, F}. The former yields the equations of motion for free fields of arbitrary spin and helicity. In the massless case, it also dictates that Lorentz transformations for tensor fields acquire an additional term, which takes the form of a gauge transformation \cite{W, S1}. The latter principle, the Fock-Ivanenko derivative, introduces interactions based on the intrinsic and Poincaré groups. This framework allows us to recover a complete Yang-Mills theory, as well as general relativity in the connection-based formulations of Palatini and Ashtekar, both of which are theories with local gauge symmetries. While the standard approach begins with the symmetries of a matter action, we will instead derive dynamics directly from Poincaré invariance. This perspective reveals that for free fields, Lorentz invariance induces the gauge symmetry of massless tensors. A proper definition of these gauge transformations, in turn, requires the covariant derivatives provided by the Fock-Ivanenko approach. Considering matter fields, we derive the interacting Dirac equation in the presence of Yang-Mills and gravitational fields from its free counterpart.