Covariant derivative electrodynamics

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. 

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... 

In 0(3) electrodynamics, the covariant derivative on the classical level is defined by... 

Therefore, it is always possible to write the covariant derivative of the Sachs theory as an 0(3) covariant derivative of 0(3) electrodynamics. Both types of covariant derivative are considered on the classical level. 

The curvature tensor is defined in terms of covariant derivatives of the spin-affine connections fip, and according to Section ( ), has its equivalent in 0(3) electrodynamics. 

In this section, we suggest a resolution of this > 70-year-old paradox using 0(3) electrodynamics [44]. The new method is based on the use of covariant derivatives combined with the first Casimir invariant of the Poincare group. The latter is usually written in operator notation [42,46] as the invariant P P 1, where P1 is the generator of spacetime translation ... 

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. 

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. 

Electrodynamics is locally gauge invariant because all derivatives occur in special combinations D, called covariant derivatives which do have the property that... 

By analogy with electrodynamics we seek a covariant derivative such that... 

In Eq. (5), the product q q is quaternion-valued and non-commutative, but not antisymmetric in the indices p and v. The B<3> held and structure of 0(3) electrodynamics must be found from a special case of Eq. (5) showing that 0(3) electrodynamics is a Yang-Mills theory and also a theory of general relativity [1]. The important conclusion reached is that Yang-Mills theories can be derived from the irreducible representations of the Einstein group. This result is consistent with the fact that all theories of physics must be theories of general relativity in principle. From Eq. (1), it is possible to write four-valued, generally covariant, components such as... 

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