Dynamical symmetry breaking in Yang-Mills geometrodynamics
Alcides Garat
TL;DR
This work presents a classical, geometrical framework for dynamical symmetry breaking in Yang–Mills geometrodynamics by treating local gauge transformations as Lorentz tetrad rotations on two orthogonal planes (blades) that locally diagonalize the stress-energy tensor. It develops Abelian and non-Abelian tetrads, a gauge-invariant block-diagonalization procedure, and a first-order perturbation scheme to study how external perturbations tilt the symmetry planes and alter local currents, leading to a continuous evolution of symmetries and generation of curvature rather than mass. A key result is a theorem describing the dynamic evolution of the local planes of symmetry and their associated currents (LB1/LB2 frames) under perturbations, implying that old symmetries are broken while new ones arise. The formalism provides a geometric lens on anomalies in quantum field theories and links symmetry evolution to curvature changes induced by external interactions. The approach hinges on extremal and block-diagonalized field constructs, including $f_{eta\u001beta}$, $\epsilon_{\mu\nu}$, and isospace projections, and is substantiated by detailed appendices connecting gauge and Lorentz structures.
Abstract
We will analyze through a first order perturbative formulation the local loss of symmetry when a source of non-Abelian Yang-Mills and gravitational fields interacts with an external agent that perturbes the original geometry associated to the source. Then, as the symmetry in Abelian and non-Abelian field structures in four-dimensional Lorentzian spacetimes is displayed through the existence of local planes of symmetry that we previously called blades one and two. These orthogonal local planes diagonalize the stress-energy tensor and every vector in these planes is an eigenvector of the stress-energy tensor. The loss of symmetry will be manifested by the tilting of these planes under the influence of the external agent. It was also found already that there is an algorithm to block diagonalize the Yang-Mills field strength in a local gauge invariant way. The loss of symmetry will also be manifested by the tilting of these planes that block diagonalize the Yang-Mills field strength under the influence of the external agent. As the interaction proceeds, the planes will tilt perturbatively, and in this strict sense the original local symmetry will be lost. But we will prove that the new orthogonal planes or blades at the same point will correspond after the tilting generated by perturbation to a new symmetry, with associated new local currents, both on each new local planes of symmetry. Old symmetries will be broken, new symmetries will arise. There will be a local symmetry evolution in the non-Abelian case as well. This result will produce a new theorem on dynamic symmetry evolution. This new classical model will be useful in order to better understand anomalies in quantum field theories.