Non-Abelian electrodynamics

There are well known similarities between the Riemann curvature tensor of general relativity and the field tensor in non-Abelian electrodynamics. The Riemann tensor is... 

F. A Possible SO( 10) Grand Unification that Includes Non-Abelian Electrodynamics... 

Non-Abelian electrodynamics is an interesting proposal that electrodynamics has a more general gauge structure. The basis advanced by Barrett, Harmuth, and Evans proposes that electrodynamics has a more complex structure than one... 

There have been reports of the inverse Faraday effect that are predicted by non-Abelian electrodynamics. However, these reports are comparatively dated, and no updated results appear to have been reported. In 1998 the Varian... 

Corporation attempted to measure the B3 field. However, the results were null, and an inconclusive direct measurements of the B3 field still remains elusive. On the theoretical front non-Abelian electrodynamics remains controversial and not widely upheld. Some objections are not entirely reasonable. On the other hand, Waldyr Rodriques objected to certain assumptions, proposed by M. W. Evans, that relates coefficients in Whittaker s 1904 paper on electrodynamics to the putative existence of longitudinal modes in non-Abelian electromagnetic waves in vacuum. Rodrigues objections appear reasonable. However, this response was quite forceful and direct, and resulted in his refusal to consider anything involving non-Abelian electrodynamics. 

There is rub to this construction. This Proca equation is really only applicable on a scale that approaches high-energy physics where the A 3 boson has appreciable influence. This will be only at a range of 10 17 cm. On the scale of atomic physics 10 3 cm, where quantum optics is applicable, this influence will be insignificant. In effect on a scale where the Al 3 does not exist, as it has decayed into pion pairs, the duality is established and there is no Lagrangian for the B 3 field. This puts us back to square one, where we must consider non-Abelian electrodynamics as effectively U(l) electrodynamics plus additional nonLagrangian and nonHamiltonian symmetries. 

This suggests that electromagnetism may in fact have a deeper non-Abelian structure. In what follows it is assumed that the B(3 field exists. It is likely that the B 3 field exists only as a manifestation of nonlinear optics. This is an aspect of non-Abelian electrodynamics that has been quite under studied. Later, a discussion of squeezed state operators in connection to non-Abelian electrodynamics is mentioned. However, its role in nonlinear optics is an open topic for work. 

The difference this derivation has in comparison to the previous derivation of the nonlinear Schrodinger equation is that the nonlinearity is more fundamentally due to the non-Abelian wavefunction rather than from material coefficients. In effect these material coefficients and phenomenology behave as they do because the variable index of refraction is associated with non-Abelian electrodynamics. Ultimately these two views will merge, for the mechanisms on how photons interact with atoms and molecules will give a more complete picture on how non-Abelian electrodynamics participates in these processes. However, at this stage we can see that we obtain nonlinear terms from a non-Abelian electrodynamics that is fundamentally nonlinear. This is in contrast to the phenomenological approach that imposes these nonlinearities onto a fundamentally linear theory of electrodynamics. 

This is the nature of non-Abelian electrodynamics in a nonrelativistic regime. It leads to various predictions that appear to obtain for electromagnetic fields in media. As yet there have not been the appearance of these types of effects for fields in a vacuum. Just why it is that nonlinear optics appears to be associated... 

Non-Abelian electrodynamics has been presented in considerable detail in a nonrelativistic setting. However, all gauge fields exist in spacetime and thus exhibits Poincare transformation. In flat spacetime these transformations are global symmetries that act to transform the electric and magnetic components of a gauge field into each other. The same is the case for non-Abelian electrodynamics. Further, the electromagnetic vector potential is written according to absorption and emission operators that act on element of a Fock space of states. It is then reasonable to require that the theory be treated in a manifestly Lorentz covariant manner. 

This example, within 1/(1) electrodynamics can then be seen in the light of non-Abelian electrodynamics. This may simply be seen by the replacement where f is a structure constant that obeys [f, tb] = 2eabcf. The time ordered product is written as... 

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