Stokes theorem electrodynamics

The non-Abelian Stokes theorem gives the homogeneous field equation of 0(3) electrodynamics, a Jacobi identity in the following integral form ... 

These field equations are therefore the result of a non-Abelian Stokes theorem that can also be used to compute the electromagnetic phase in 0(3) electrodynamics. It turns out that all interferometric and physical optical effects are described self-consistently on the 0(3) level, but not on the U(l) level, a result of major importance. This result means that the 0(3) (or SO(3) = SU(2)/Z2) field equations must be accepted as the fundamental equations of electrodynamics. 

Therefore, the distinction between the topological and dynamical phase has vanished, and the realization has been reached that the phase in optics and electrodynamics is a line integral, related to an area integral over Bt3> by a non-Abelian Stokes theorem, Eq. (553), applied with 0(3) symmetry-covariant derivatives. It is essential to understand that a non-Abelian Stokes theorem must be applied, as in Eq. (553), and not the ordinary Stokes theorem. We have also argued, earlier, how the non-Abelian Stokes explains the Aharonov-Bohm effect without difficulty. 

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. 

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