B cyclic theorem

If this is used in the third equation of Eq. (83), the B cyclic theorem [47-61] is recovered self-consistently as follows. Without considering vacuum polarization and magnetization, the third equation of Eqs. (83) reduces to... 

Independent confirmation of the invariance of the B cyclic theorem was next produced by Dvoeglazov [93], but he did not argue on the 0(3) level as required. His argument is therefore only partially valid, but produces the correct result. 

Therefore, the definition of the field tensor in 0(3) electrodynamics gives the first two components of the B cyclic theorem [47-62]... 

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. 

The B cyclic theorem is a Lorentz invariant construct in the vacuum and is a relation between angular momentum generators [42], As such, it can be used as the starting point for a new type of quantization of electromagnetic radiation, based on quantization of angular momentum operators. This method shares none of the drawbacks of canonical quantization [46], and gives photon creation and annihilation operators self-consistently. It is seen from the B cyclic theorem ... 

The operator f3 is therefore also an intrinsic spin, and can be identified in this novel quantization method based on the B cyclic theorem with the intrinsic spin of a photon with mass, with eigenvalues —1,0, and +1. 

All three of ea>, e(2 e(3] can be expressed in terms of vector spherical harmonics. Thus, in addition to the nonlinear B cyclic theorem, the following linear relations occur... 

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... 

Using Eqs. (768a) and (769), it is seen that the product is unity if we sum over all multipole components with 1 —> oo in Eq. (768). In all other cases, the B cyclic theorem is... 

As argued, infinitesimal field generators appear as a by-product of this novel quantization scheme, so that B° is rigorously nonzero from the symmetry of the Poincare group and the B cyclic theorem is an invariant of the classical field. The basics of infinitesimal field generators on the classical level are to be found in the theory of relativistic spin angular momentum [42,46] and relies on the Pauli-Lubanski pseudo-4-vector ... 

On the 0(3) level, particular solutions of the E2> Lie algebra (796) give a total of six commutator relations. Three of these form the B cyclic theorem (B(0) = 1 units) ... 

From this analysis, it is concluded that the B<3> component is identically nonzero, otherwise all the field components vanish in the B cyclic theorem (812) and Lie algebra (809). If we assume Eq. (803) and at the same time assume that B(3) is zero, then the Pauli-Lubanski pseudo-4-vector vanishes for all A0. Similarly, in the particle interpretation, if we assume the equivalent of Eq. (803) and assume that J(3) is zero, the Pauli-Lubanski vector vanishes. This is contrary to the definition of the helicity of the photon. Therefore, for finite field helicity, we need a finite B(3). 

The precise correspondence between field and photon interpretation developed here indicates that E(2) symmetry does not imply that Ii(3) is zero, any more than it implies that J<3> = 0. The assertion B(3> = 0 is counterindicated by a range of data reviewed here and in Ref. 44, and the B cyclic theorem is Lorentz-covariant, as it is part of a Lorentz-covariant Lie algebra. If we assume the particular solutions (809) and (810) and use in it the particular solution (803), we obtain the cyclics (809) from the three cyclics Eq. (810) thus we obtain... 

This is also the relation obtained in the hypothetical rest frame. Therefore, the B cyclic theorem is Lorentz-invariant in the sense that it is the same in the rest frame and in the light-like condition. This result can be checked by applying the Lorentz transformation rules for magnetic fields term by term [44], The equivalent of the B cyclic theorem in the particle interpretation is a Lorentz-invariant construct for spin angular momentum ... 

It is concluded that the B(3) component in the field interpretation is nonzero in the light-like condition and in the rest frame. The B cyclic theorem is a Lorentz-invariant, and the product B x B<2> is an experimental observable [44], In this representation, B(3> is a phaseless and fundamental field spin, an intrinsic property of the field in the same way that J(3) is an intrinsic property of the photon. It is incorrect to infer from the Lie algebra (796) that Ii(3) must be zero for plane waves. For the latter, we have the particular choice (803) and the algebra (796) reduces to... 

To complete the derivation, we multiply both sides of Eq. (828) by the phase factor el s>e lis> to obtain the B cyclic theorem. The latter is therefore equivalent to a commutator relation of the Poincare group between infinitesimal magnetic held generators. Similarly... 

These equations reduce in turn [42,44,47-62] to the B cyclic theorem ... 

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