Gauge field covariant derivative

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]. 

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]... 

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... 

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... 

It is also possible to consider the holonomy of the generic A in the vacuum. This is a round trip or closed loop in Minkowski spacetime. The general vector A is transported from point A, where it is denoted Aa 0 around a closed loop with covariant derivatives back to the point Aa () in the vacuum. The result [46] is the field tensor for any gauge group... 

Here,, 4 ( is the vector 4-potential introduced in the vacuum as part of the covariant derivative, and therefore introduced by spacetime curvature. The electromagnetic field and the topological charge g are the results of the invariance of the Lagrangian (868) under local U(l) gauge transformation, in other words, the results of spacetime curvature. 

The 5(7(2) gauge field W), has three components, corresponding to the isospin vector of matrices r, with no relationship to the coordinate space ct, x. By implication, the fermion field i// is a set of spinors, one for each value of the isospin index. The covariant derivative... 

For interacting fields, the Maxwell field energy is not separately conserved. A gauge-covariant derivation follows from the inhomogeneous field equations (Maxwell equations in vacuo),... 

As a model for classical gauge fields, the energy-momentum conservation law can be derived directly in covariant notation. The 4-divergence... 

The local conservation law for the interacting gauge field can be derived from the covariant field equations, as was done above for the Maxwell field. Using the SU(2) field equations and expanding (3VW0>-) WV/l as... 

Eq. (3.198) just represents the inhomogeneous Maxwell equations in covariant form, cf. Eq. (3.172). We have thus derived the inhomogeneous Maxwell equations as the natural equations of motion for the gauge potential A. The sources, as described by the charge-current density are considered as external variables which do not represent dynamical degrees of freedom, i.e., only the action (or effect) of the sources on the gauge fields is taken into account. 

All gauge theory depends on the rotation of an -component vector whose 4-derivative does not transform covariantly as shown in Eq. (18). The reason is that i(x) and i(x + dx) are measured in different coordinate systems the field t has different values at different points, but /(x) and /(x) + d f are measured with respect to different coordinate axes. The quantity d i carries information about the nature of the field / itself, but also about the rotation of the axes in the internal gauge space on moving from x + dx. This leads to the concept of parallel transport in the internal gauge space and the resulting vector [6] is denoted i(x) + d i. The notion of parallel transport is at the root of all gauge theory and implies the introduction of g, defined by... 

The effect of the local gauge transform is to introduce an extra term 8M A in the transformation of the derivatives of fields. Therefore, 8 A does not transform covariantly, that is, does not transform in the same way as A itself. These extra terms destroy the invariance of the action under the local gauge transformation, because the change in the Lagrangian is... 

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