Dynamical symmetry breaking in Yang–Mills geometrodynamics

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, 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. Using perturbative analysis from a previous manuscript dealing with the Abelian case, we will demonstrate how to develop an algorithm for constructing local conserved currents inside both local orthogonal planes of stress-energy diagonalization. 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.