Internal gauge space

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. 

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... 

Information is also lost if we replace the ((1),(2),(3)) basis by the (X, Y, Z) basis for the internal gauge space. The reason is that the former basis is essentially dynamical and the latter is essentially static. This is again a self-consistent result, because electrodynamics, by definition, requires the movement of charge. The misnamed subject of magnetostatics also requires the movement of charge, and so is not static. 

This is an expression for the observed phase difference with the platform at rest in the Sagnac experiment [47] it is a rotation in the internal gauge space. In U(l) electrodynamics, there is no phase difference when the platform is at rest, as discussed already. 

When the platform is rotated in the Sagnac effect, there is an additional rotation in the internal gauge space described by... 

The Sagnac effect caused by the rotating platform is therefore due to a rotation in the internal gauge space ((1),(2),(3)), which results in the frequency shift in Eq. (171). The frequency shift is experimentally the same to an observer on and off the platform and is independent of the shape of the area Ar. The holonomy difference (172) derived theoretically depends only on the magnitudes and ff, and these scalars are frame-invariant, as observed experimentally. There is no shape specified for the area Ar in the theory, and only its scalar magnitude enters into Eq. (172), again in agreement with experiment. 

The principle of interferometry in 0(3) electrodynamics follows from the fact that it is caused by a rotation in the internal gauge space... 

The explanation of the IFE in the Maxwell-Heaviside theory relies on phenomenology that is self-inconsistent. The reason is that A x A 2 is introduced phenomenologically [56] but the same quantity (Section III) is discarded in U(l) gauge field theory, which is asserted in the received view to be the Maxwell-Heaviside theory. In 0(3) electrodynamics, the IFE and third Stokes parameter are both manifestations of the 3 held proportional to the conjugate product that emerges from first principles [11-20] of gauge held theory, provided the internal gauge space is described in the basis ((1),(2),(3)). 

The ansatz, upon which these results are based, is that the configuration of the vacuum is described by the doubly connected group 0(3), which supports the Aharonov-Bohm effect in Minkowski spacetime [46]. More generally, the vacuum configuration could be described by an internal gauge space more general than 0(3), such as the Lorentz, Poincare, or Einstein groups. The 0(3)... 

This equation in vector notation for the internal gauge space can be developed as three equations in reduced units... 

Gauge theory of any symmetry must have two mathematical spaces Minkowski spacetime and the internal gauge space. If electromagnetic theory in the vacuum is a U(l) symmetry gauge field symmetry, there is a scalar internal space of U(l) symmetry in the vacuum. This internal space is the space of the scalar A and A used in the foregoing arguments. In geometric form... 

This process cannot be described classically, because positrons are the result of the Dirac equation, but it illustrates the fact that a vacuum current (of photons) is made up of the interaction of two Dirac currents, one for the electron, one for the positron, and these are both matter currents. Therefore, there is a transmutation of matter current to vacuum current. On the classical level, this can be described in the scalar internal gauge space as... 

The Lehnert equations are a great improvement over the Maxwell-Heaviside equations [45,49] but are unable to describe phenomena such as the Sagnac effect and interferometry [42], for which an 0(3) internal gauge space symmetry is needed. 

Gauge theory can be developed systematically for the vacuum on the basis of material presented in Section II. Before doing so, recall that, on the U(l) level, AM exists in Minkowski spacetime and there is a scalar internal gauge space that can be denoted... 

The internal gauge space has local symmetry, and is a physical space. In complex circular notation, the vector in the internal gauge space can be written as... 

There is an interrelation between the A and A vectors of the scalar internal gauge space and components of A and AM 2 in the vacuum... 

So it becomes clear that the description of the vacuum in gauge theory can be developed systematically by recognizing that, in general, A is an -dimensional vector. On the U(l) level, it is one-dimensional on the 0(3) level, it is three-dimensional and so on. The internal gauge space in this development is a physical space that can be subjected to a local gauge transform to produce physical vacuum charge current densities. 

In the 0(3) gauge group, Ma are rotation generators, and Aa are angles in three-dimensional space, which coincides with the internal gauge space. Rotation about the Z axis leaves the B(3) field unaffected. In matrix notation, this can be demonstrated by... 

A closely similar complex circular basis has been described by Silver [112] for three-dimensional space. This space forms the internal gauge space in 0(3) electrodynamics, as argued already. In the complex circular basis, the unit vector dot product is... 

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