Sagnac effect 0 electrodynamics

The group space of 0(3) is doubly connected (i.e., non-simply connected) and can therefore support an Aharonov-Bohm effect (Section V), which is described by a physical inhomogeneous term produced by a rotation in the internal gauge space of 0(3) [24]. The existence of the Aharonov-Bohm effect is therefore clear evidence for an extended electrodynamics such as 0(3) electrodynamics, as argued already. A great deal more evidence is reviewed in this article in favor of 0(3) over U(l). For example, it is shown that the Sagnac effect [25] can be described accurately with 0(3), while U(l) fails completely to describe it. 

The Sagnac effect cannot be described by U(l) electrodynamics [4,43] because of the invariance of the U(l) phase factor under motion reversal symmetry (T) ... 

The explanation of interferometric effects in U(l) electrodynamics is in general self-inconsistent, and sometimes, as in the Sagnac effect, nonexistent. In this... 

In U(l) electrodynamics in free space, there are only transverse components of the vector potential, so the integral (158) vanishes. It follows that the area integral in Eq. (157) also vanishes, and so the U(l) phase factor cannot be used to describe interferometry. For example, it cannot be used to describe the Sagnac effect. The latter result is consistent with the fact that the Maxwell-Heaviside and d Alembert equations are invariant under T, which generates the clockwise... 

C) Sagnac loop from the counterclockwise (A) loop [17]. It follows that the phase difference observed with platform at rest in the Sagnac effect [47] cannot be described by U(l) electrodynamics. This result is also consistent with the fact that the traditional phase of U(l) electrodynamics is invariant under T as discussed already in Section (III). The same result applies for the Michelson-Gale experiment [48], which is a Sagnac effect. 

This result is true for all matter waves and also in the Michelson-Gale experiment, where it has been measured to a precision of one part in 1023 [49]. Hasselbach et al. [51] have demonstrated it in electron waves. We have therefore shown that the electrodynamic and kinematic explanation of the Sagnac effect gives the same result in a structured vacuum described by 0(3) gauge group symmetry. 

The Maxwell-Heaviside theory of electrodynamics has no explanation for the Sagnac effect [4] because its phase is invariant under 7 as argued already, and because the equations are invariant to rotation in the vacuum. The d Alembert wave equation of U(l) electrodynamics is also 7 -invariant. One of the most telling pieces of evidence against the validity of the U(l) electrodynamics was given experimentally by Pegram [54] who discovered a little known [4] cross-relation between magnetic and electric fields in the vacuum that is denied by Lorentz transformation. 

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... 

There is no Sagnac effect in U(l) electrodynamics, as just argued, a result that is obviously contrary to observation [44]. In 0(3) electrodynamics, the Sagnac effect with platform at rest is given by the phase factor [44]... 

If we attempt the same exercise in U(l) electrodynamics, the closed loop gives the Maxwell-Heaviside equations in the vacuum, which are invariant under T and that therefore cannot describe the Sagnac effect [44] because one loop of the Sagnac interferometer is obtained from the other loop by T symmetry. The U(l) phase factor is oof kZ + a, where a is arbitrary [44], and this phase factor is also "/ -invariant. The Maxwell-Heaviside equations in the vacuum are... 

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. 

Therefore, it has been shown convincingly that electrodynamics is an 0(3) invariant theory, and so the 0(3) gauge invariance must also be found in experiments with matter waves, such as matter waves from electrons, in which there is no electromagnetic potential. One such experiment is the Sagnac effect with electrons, which was reviewed in Ref. 44, and another is Young interferometry with electron waves. For both experiments, Eq. (584) becomes... 

The received view, in which the phase factor of optics and electrodynamics is given by Eq. (554), can describe neither the Sagnac nor the Tomita-Chiao effects, which, as we have argued, are the same effects, differing only by geometry. Both are non-Abelian, and both depend on a round trip in Minkowski spacetime using 0(3) covariant derivatives. 

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