Notes on Gauge Field Theories (II): Local Electromagnetic Dual Transformation The-ory, Generalized Kaluza-Klein Emergent Yang-Mills Gauge Field Theory and Higher-Dimensional Gravitational Gauge Field Theory

Abstract

This paper continues to explore and review the two topics of “synthetic” gauge field and “emergent” gauge field, which have already been considered in the last paper entitled “Notes on Gauge Field Theories (I)”. The present paper on “Notes on Gauge Field Theories (II)” includes three topics: i) A theory of local electromagnetic dual transformation symmetry and dual gauge field is suggested based on the previous theories of magnetic charge and dual transformation, and it is pointed out that the effect of dual gauge potential makes an electromagnetic wave in vacuum seem to propagate in an anisotropic medium whose permittivity and permeability are both tensors; ii) A non-Abelian version or generalized Kaluza-Klein theory is given in detail. The purpose is to introduce a theory of fundamental interaction that unifies Einstein’s general-relativity gravity and Yang-Mills gauge interaction. In this model, the Yang-Mills gauge potential is a higher-dimensional gravitational me-tric-field off-diagonal component emerging in the ordinary four-dimensional spacetime, or in other words, the Yang-Mills gauge field is essentially a non-Abelian Kaluza-Klein higher-dimensional gravitational field; iii) A theory of higher-dimensional spin-connection gravitational gauge field theory, of which the gravitational Lagrangian density is quadratic in the Riemannian curvature, is reviewed. The higher-dimensional spin-affine connection (the Lorentz connection) can serve as a Yang-Mills gauge potential and the spin currents of vectorial and spinorial matter fields play a role of Yang-Mills gauge charge currents in the four-dimensional spacetime, and so the gravitational interaction and the Yang-Mills gauge interaction can be unified into the present higher-dimensional spin-connection gravitational gauge theory, which was suggested by us. There have been many theories of gravitation in the literature. Although the merits and weaknesses of only the non-Abelian version of Kaluza-Klein theory and the gauge theory of gravitation are reviewed in this paper, we expect that the analysis of these two theories would still help readers to draw parallels among the relevant gravity theories and to understand the stylistic characteristics, advantages and disadvantages of various gravitation theories in the literature.