B-field

This correction is added to the energy shift due to the B(3) field to give... 

If A] is phase-free, as discussed in Section III, and in Ref. 15, there are no longitudinal electric field components. This also occurs if A,-3"1 is zero [17]. The B(3) field is then a Fourier sum over modes with operators a qaq and is perpendicular to the plane defined by A and /1<2>. The four-dimensional dual to this term is defined on a time-like surface, following Crowell [17], which can be interpreted as E under dyad vector duality in three dimensions. The ( field vanishes because of the nonexistence of the raising and lowering operators l3 , . The BM is nonzero because of the occurrence of raising and lowering... 

The effect of a local gauge transformation (Sction II) on the classical B(3 field is described as... 

The amplitude contribution from the B(3> field occurs in a second-order process using the sum over all possible fluctuations of B(3> in the virtual photon that causes electron-electron interaction. The amplitude due to B(i) has an ultraviolet divergence [17] described by Crowell. This may be removed by regularization techniques. 

This type of process is missing from U(l) quantum field theory [6] the B(3> field produces quantum vortices [17] that interact with electrons and other charged particles. The vortices are quantized states and exist as fluctuations in the QED vacuum, fluctuations that are associated, not with an E(3) field, but with the = E(1> fields ... 

The absence of an E(3) field does not affect Lorentz symmetry, because in free space, the field equations of both 0(3) electrodynamics are Lorentz-invariant, so their solutions are also Lorentz-invariant. This conclusion follows from the Jacobi identity (30), which is an identity for all group symmetries. The right-hand side is zero, and so the left-hand side is zero and invariant under the general Lorentz transformation [6], consisting of boosts, rotations, and space-time translations. It follows that the B<3) field in free space Lorentz-invariant, and also that the definition (38) is invariant. The E(3) field is zero and is also invariant thus, B(3) is the same for all observers and E(3) is zero for all observers. 

The latter is therefore related to the concept of the B(3) field through the Lehnert equations, which in the vacuum are... 

Therefore, the B(3) field [2] is proved from a particular choice of metric using the irreducible representations of the Einstein group [ 1]. It can be seen from Eq. (21) that the B(3> field is the vector-valued metric field q(3> within a factor QR. This result proves that Bi3) vanishes in flat spacetime, because R 0 in flat spacetime. If we write... 

From general relativity, it can therefore be concluded that the B<3) field must exist and that it is a physical magnetic flux density defined to the precision of the Lamb shift. It propagates through the vacuum with other components of the field tensor. 

The B(3) field [3] of 0(3) electrodynamics is defined in terms of the cross-product of plane wave potentials A(1) = A12 ... 

The preceding section, and a review in Part 1 of this compilation, supply copious empirical evidences of the fact that the B(3) field is part of the topological phase that describes interferometry through a non-Abelian Stokes theorem. Therefore, the early critical papers are erroneous because they argue on a U(l) level. 

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... 

The objects M 3 Yvac) and A/ 2)(vac) depend on the phaseless vacuum magnetic field B(3> and so do not exist as concepts in U(l) electrodynamics. The B<3) field itself is defined through... 

Therefore, on the 0(3) level, the magnetic part of the complete free field is defined as a sum of a curl of a vector potential and a vacuum magnetization inherent in the structure of the B cyclic theorem. On the U(l) level, there is no B(3) field by hypothesis. 

As an example of these methods, consider the B cyclic theorem for multipole radiation, which can be developed for the multipole expansion of plane-wave radiation to show that the B<3) field is irrotational, divergentless, and fundamental for each multipole component. The magnetic components of the plane wave are defined, using Silver s notation [112] as... 

In general, all the off-diagonal elements of the quaternion-valued commutator term [the fifth term in Sachs Eq. (4.19)] exist, and in this appendix, it is shown, by a choice of metric, that one of these components is the Ba> field discussed in the text. The B<3) field is the fundamental signature of 0(3) electrodynamics discussed in Vol. 114, part 2. In this appendix, we also give the most general form of the vector potential in curved spacetime, a form that also has longitudinal and transverse components under all conditions, including the vacuum. In the Maxwell-Heaviside theory, on the other hand, the vector... 

The definition of the B(3 ) field in this manner illustrates that the internal index associated with the extended gauge group is identified with coordinates that are orthogonal to the direction of propagation of the electromagnetic field. This has various implications, which, if interpreted classically, mean that the Stokes parameters of an electromagnetic field determine this field. 

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