Holonomy difference

The extra holonomy difference due to the rotating platform is the same as for electromagnetic waves ... 

It can be shown straightforwardly, as follows, that there is no holonomy difference if the phase factor (154) is applied to the problem of the Sagnac effect with U(l) covariant derivatives. In other words, the Dirac phase factor [4] of U(l) electrodynamics does not describe the Sagnac effect. For C and A loops, consider the boundary... 

The Sagnac effect caused by the rotating platform is therefore due to a rotation in the internal gauge space ((1),(2),(3)), which results in the frequency shift in Eq. (171). The frequency shift is experimentally the same to an observer on and off the platform and is independent of the shape of the area Ar. The holonomy difference (172) derived theoretically depends only on the magnitudes and ff, and these scalars are frame-invariant, as observed experimentally. There is no shape specified for the area Ar in the theory, and only its scalar magnitude enters into Eq. (172), again in agreement with experiment. 

Unfortunately, it is not possible to automatically generalize the Abelian Stokes theorem [e.g., Eq. (4)] to the non-Abelian one. In the non-Abelian case one faces a qualitatively different situation because the integrand on the l.h.s. assumes values in a Lie algebra g rather than in the field of real or complex numbers. The picture simplifies significantly if one switches from the local language to a global one [see Eq. (5)]. Therefore we should consider the holonomy (7) around a closed curve C ... 

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