Gauges covariant

Considering the loop-after-loop diagram Fig. 1(a), one should keep in mind that the corresponding contribution is not completely gauge invariant. It is invariant in any covariant gauge but not, e.g., in the Coulomb gauge. This means that, strictly speaking, the loop-after-loop contribution has no well-defined phys-... 

For the photon fields we have chosen to work in the covariant gauge [125], which relies on the Gupta-Bleuler indefinite metric quantization and leads to the gauge fixing term... 

The freedom to choose any of the covariant gauges for the photon field results from the gauge invariance of the Lagrangian (1) A gauge transformation of the photon field,... 

It seems worthwhile to emphasize that all covariant gauges lead to the same renormalization scheme. 

Co-representation matrices explicit forms, 733 multiplication of, 731 of the nonunitary group, 732 Corliss, L. M., 757 Corson, E. M., 498 Coulomb field Dirac equation in, 637 Coulomb gauge, 643,657,664 Counting functions, 165 Covariance matrix, 160 Covariant amplitude of one-particle system, 511 of one, two, etc. particle systems, 511... 

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]. 

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]... 

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... 

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... 

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. 

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... 

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