Dual Symmetry in Gauge Theories

TL;DR

This paper explores continuous dual symmetry across gauge theories and gravitation, proposing dual-invariant Lagrangians, analyzing instanton solutions, and deriving new gravitational equations that generalize Einstein's theory.

Contribution

It introduces a dual-invariant Lagrangian for nonlinear electrodynamics, extends dual symmetry concepts to Yang-Mills theory and gravitation, and derives new gravitational equations from duality considerations.

Findings

Dual-invariant nonlinear electrodynamics with dual angle parameter.

Yang-Mills equations admit instanton solutions similar to conventional theory.

New gravitational equations generalize Einstein's equations, reducing to Nordstrom theory.

Abstract

Continuous dual symmetry in electrodynamics, Yang-Mills theory and gravitation is investigated. Dual invariant which leads to badly nonlinear motion equations is chosen as a Lagrangian of the pure classical dual nonlinear electrodynamics. In a natural manner some dual angle which is determined by the electromagnetic strengths at the point of the time-space appears in the model. Motion equations may well be interpreted as the equations of the standard Maxwell theory with source. Alternative interpretation is the quasi-Maxwell linear theory with magnetic charge. Analogous approach is possible in the Yang-Mills theory. In this case the dual-invariant non-Abelian theory motion equations possess the same instanton solutions as the conventional Yang--Mills equations have. An Abelian two-parameter dual group is found to exist in Gravitation. Irreducible representations have been obtained: the curvature tensor was expanded into the sum of twice anti-self-dual and self-dual parts. Gravitational instantons are defined as (real) solutions to the usual duality equations. Central-symmetry solutions to these equations are obtained. The twice anti-self-dual part of the curvature tensor may be used for introduction of new gravitational equations generalizing Einstein's equations. However, the theory obtained reduces to the conformal-flat Nordstrom theory.