Stokes theorem invariance

In the non-Abelian Stokes theorem (482), on the other hand, the boundary conditions are defined because the phase factor is path-dependent, that is, depends on the covariant derivative [50]. On the U(l) level [50], the original Stokes theorem is a mathematical relation between a vector field and its curl. In 0(3) or SU(2) invariant electromagnetism, the non-Abelian Stokes theorem gives the phase change due to a rotation in the internal space. This phase change appears as the integrals... 

Equation (482) is a simple form of the non-Abelian Stokes theorem, a form that is derived by a round trip in Minkowski spacetime [46]. It has been adapted directly for the 0(3) invariant phase factor as in Eq. (547), which gives a simple and accurate description of the Sagnac effect [44], A U(l) invariant electrodynamics has failed to describe the Sagnac effect for nearly 90 years, and kinematic explanations are also unsatisfactory [50], In an 0(3) or SU(2) invariant electrodynamics, the Sagnac effect is simply a round trip in Minkowski space-time and an effect of special relativity and gauge theory, the most successful theory of the late twentieth century. There are open questions in special relativity [108], but no theory has yet evolved to replace it. 

The basic problem in a U(l) invariant description of the Aharonov-Bohm effect is that the field B is zero outside the solenoid, so outside the solenoid, V x A is zero, whereas A is not zero [46]. At the same time, the U(l) Stokes theorem states that... 

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