Covariant derivatives

Here, Cbc are the structure constants for the Lie group defined by the set of the noncommuting matrices ta appearing in Eq. (94) and which also appear both in the Lagrangean and in the Schrodinger equation. We further define the covariant derivative by... 

It should finally be mentioned that the basic equations (l)-(8) have been derived from gauge theory in the vacuum, using the concept of covariant derivative and Feynman s universal influence [38]. These equations and the Proca field equations are shown to be interrelated to the well-known de Broglie theorem, in which the photon rest mass m can be interpreted as nonzero and be related to a frequency v = moc2/h. A gauge-invariant Proca equation is suggested by this analysis and relations (l)-(8). It is also consistent with the earlier conclusion that gauge invariance does not require the photon rest mass to be zero [20,38]. 

In terms of 0(3) covariant derivatives, Evans et al. [78] have shown that the Proca-type equation represented by the form (22) can be derived without imposing the conventional Lorentz condition L = 0. This result is supported by the following two considerations. 

General gauge held theory emerges when the covariant derivative is applied to v / [6] ... 

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. 

Recall that in general gauge field theory, for any gauge group, the field tensor is defined through the commutator of covariant derivatives. In condensed notation [6]... 

The homogeneous held equation (31) can be expanded in terns of the 0(3) covariant derivative [6,11-20] ... 

In order to understand interferometry at a fundamental level in gauge field theory, the starting point must be the non-Abelian Stokes theorem [4]. The theorem is generated by a round trip or closed loop in Minkowski spacetime using covariant derivatives, and in its most general form is given [17] by... 

To reduce Eq. (153) to the ordinary Stokes theorem, the U(l) covariant derivative is used... 

The P on the left-hand side of Eq. (162) denotes path ordering and the P denotes area ordering [4]. Equation (162) is the result of a round trip or closed loop in Minkowski spacetime with 0(3) covariant derivatives. Equation (161) is a direct result of our basic assumption that the configuration of the vacuum can be described by gauge theory with an internal 0(3) symmetry (Section I). Henceforth, we shall omit the P and P from the left- and right-hand sides, respectively, and give a few illustrative examples of the use of Eq. (162) in interferometry and physical optics. 

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... 

All four terms have been observed empirically. Terms 1-3 are well known, and term 4 has been observed as a magnetization in europium ion doped glasses by van der Ziel et al. [37] as argued already. The RFR term therefore emerges self-consistently with three other well-known and well-observed terms from what is effectively the 0(3) covariant derivative. 

Potential differences are primary in gauge theory, because they define both the covariant derivative and the field tensor. In Whittaker s theory [27,28], potentials can exist without the presence of fields, but the converse is not true. This conclusion can be demonstrated as follows. Equation (479b) is invariant under... 

The physical vacuum is assumed to be defined by the Higgs mechanism, and the SU(2) x SU(2) covariant derivative is... 

The non-Abelian Stokes theorem is a relation between covariant derivatives for any gauge group symmetry ... 

---