Gauge symmetry covariant derivative

The development just given illustrates the fact that the topology of the vacuum determines the nature of the gauge transformation, field tensor, and field equations, as inferred in Section (I). The covariant derivative plays a central role in each case for example, the homogeneous field equation of 0(3) electrodynamics is a Jacobi identity made up of covariant derivatives in an internal 0(3) symmetry gauge group. The equivalent of the Jacobi identity in general relativity is the Bianchi identity. 

The P on the left-hand side of Eq. (162) denotes path ordering and the P denotes area ordering [4]. Equation (162) is the result of a round trip or closed loop in Minkowski spacetime with 0(3) covariant derivatives. Equation (161) is a direct result of our basic assumption that the configuration of the vacuum can be described by gauge theory with an internal 0(3) symmetry (Section I). Henceforth, we shall omit the P and P from the left- and right-hand sides, respectively, and give a few illustrative examples of the use of Eq. (162) in interferometry and physical optics. 

The non-Abelian Stokes theorem is a relation between covariant derivatives for any gauge group symmetry ... 

The covariant derivative in the vacuum for any internal gauge group symmetry is therefore dehned by... 

In order to form a self-consistent description [44] of interferometry and the Aharonov-Bohm effect, the non-Abelian Stokes theorem is required. It is necessary, therefore, to provide a brief description of the non-Abelian Stokes theorem because it generalizes the ordinary Stokes theorem, and is based on the following relation between covariant derivatives for any internal gauge group symmetry ... 

Essentially, this replacement means that the spacetime changes from one that is conformally flat to one that is conformally curved in other words, the axes vary from point to point whenever a covariant derivative is used for any gauge group symmetry. It is this variation of the axes that introduces energy into a pure gauge vacuum. The covariant derivative in the latter is... 

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