Affine connection representation of gauge fields

Abstract

There are two ways to unify gravitational field and gauge field. One is to represent gravitational field as principal bundle connection, and the other is to represent gauge field as affine connection. Poincaré gauge theory and metric affine gauge theory adopt the first approach. This paper adopts the second.An affine connection is used to establish a unified coordinate description of gauge field and gravitational field. This theory has the following advantages.(i) Gauge field and gravitational field can both be represented by affine connection; they can be described by a unified spatial frame.(ii) Time can be regarded as the total metric with respect to all dimensions of internal coordinate space and external coordinate space. On-shell can be regarded as gradient direction. Quantum theory can be regarded as a geometric theory of distribution of gradient directions. Hence, gravitational theory and quantum theory obtain the same view of time and space and a unified description of evolution in affine connection representation of gauge fields.(iii) Chiral asymmetry, coupling constants, MNS mixing and CKM mixing can appear spontaneously as geometric properties in affine connection representation, whereas in U(1) x SU(2) x SU(3) principal bundle connection representation they can just only be artificially set up. Some postulates of the Standard Model can be turned into theorems in affine connection representation, so they are not necessary to be regarded as postulates anymore.(iv) The unification theory of gauge fields that are represented by affine connection can avoid the problem that a proton decays into a lepton.(v) Since the concept of point particle is thoroughly abandoned, this theory is not required to be renormalized.(iv) There exists a possible geometric interpretation to the color confinement of quarks.The Standard Model is not possessed of the above advantages. In the affine connection representation, we can get better interpretations of these physical properties. This is probably a necessary step towards the ultimate theory ofphysics.