Group of Paths, Holonomy Groups and Nonintegrable Phase Factors in $\overline{\mathrm{Poincar \'{e}}}$ Gauge Theory of Gravity

Abstract

In \overlinePoincar é gauge theory of gravity, which is based on the principal fiber bundle P over the spacetime having the covering group P0 of the proper orthochronous Poincaré group as the structure group, we examine the group of paths, holonomy groups and nonintegrable phase factors: (i) Considering the semi-direct product EP0def =[Path]⊗ P0 , where [Path] stands for the group of paths in a four-dimensional Minkowski space, we construct a linear representation of EP0 by defining the action of this group on the fiber bundle P . The parallel transports in P , the holonomy group and the nonintegrable phase factor are all described by this representation. (ii) Corresponding discussions are given for the affine frame bundle over the spacetime. Remarkable is the fact that the motions of a point and of a vector are both described by the nonintegrable phase factor. Discussions can be easily extended to the case in which \overlinePoincar é gauge and Yang-Mills gauge fields coexist. Nonintegrable phase factors are equally important in \overlinePoincar é gauge theory of gravity as well as in the Yang-Mills gauge theory.