Pure gauges

Thus, the existence of a (matrix-type) phase g represents the pure-gauge case and the nonexistence of g represents the nonpure YM field case, which cannot be tiansformed away by a gauge. 

Suppose that we want this to be transformed away by a pure gauge factor having the form... 

Now, we recall the remarkable result of [72] that if the adiabatic electronic set in Eq. (90) is complete (N = oo), then the curl condition is satisfied and the YM field is zero, except at points of singularity of the vector potential. (An algebraic proof can be found in Appendix 1 in [72]. An alternative derivation, as well as an extension, is given below.) Suppose now that we have a (pure) gauge g(R), that satisfies the following two conditions ... 

However, this procedure depends on the existence of the matrix G(R) (or of any pure gauge) that predicates the expansion in Eq. (90) for a full electronic set. Operationally, this means the preselection of a full electionic set in Eq. (129). When the preselection is only to a partial, truncated electronic set, then the relaxation to the truncated nuclear set in Eq. (128) will not be complete. Instead, the now tmncated set in Eq. (128) will be subject to a YM force F. It is not our concern to fully describe the dynamics of the truncated set under a YM field, except to say (as we have already done above) that it is the expression of the residual interaction of the electronic system on the nuclear motion. 

In Chapter IV, Englman and Yahalom summarize studies of the last 15 years related to the Yang-Mills (YM) field that represents the interaction between a set of nuclear states in a molecular system as have been discussed in a series of articles and reviews by theoretical chemists and particle physicists. They then take as their starting point the theorem that when the electronic set is complete so that the Yang-Mills field intensity tensor vanishes and the field is a pure gauge, and extend it to obtain some new results. These studies throw light on the nature of the Yang-Mills fields in the molecular and other contexts, and on the interplay between diabatic and adiabatic representations. 

Putting this to a test, we consider the representative case of the pure gauge plasma,... 

M. W. Evans, P. K. Anastasovski, T. E. Bearden, et al., Energy inherent in the pure gauge vacuum, Physica Scripta (in press). 

Therefore, a check for self-consistency has been carried out for indices p 2 and v = 1. It has been shown, therefore, that in pure gauge theory applied to electrodynamics without a Higgs mechanism, a richly structured vacuum charge current density emerges that serves as the source of energy latent in the vacuum through the following equation ... 

The above is a pure gauge field theory. The Higgs mechanism on the U(l) level provides further sources of vacuum energy as discussed already. On the 0(3) level, the Higgs mechanism can also be applied, resulting in yet more sources of energy. 

The simplest example of the generation of energy from a pure gauge vacuum is to consider the case of an electromagnetic potential plane wave defined by... 

In a U(l) invariant theory, the pure gauge vacuum is defined by a scalar internal gauge space in which there exist the independent complex scalar fields ... 

It can be shown as follows that the transition from a pure vacuum to a pure gauge vacuum is described by the spacetime translation generator of the Poincare group. The pure vacuum on the U(l) invariant level is described by the held equations ... 

Essentially, this replacement means that the spacetime changes from one that is conformally flat to one that is conformally curved in other words, the axes vary from point to point whenever a covariant derivative is used for any gauge group symmetry. It is this variation of the axes that introduces energy into a pure gauge vacuum. The covariant derivative in the latter is... 

The action is therefore not invariant under local gauge transformation. To restore invariance the four potential, A must be introduced into the pure gauge vacuum to give the Lagrangian... 

The total Lagrangian if I X I if2 is now invariant under the local gauge transformation because of the introduction of the 4-potential A, which couples to the current of the complex A of the pure gauge vacuum. The field A also contributes to the Lagrangian, and since if + ifj + if2 is invariant, an extra term if3 appears, which must also be gauge-invariant. This can be so only if the electromagnetic field is introduced... 

If we are working with local gauge transformations where A is flat, we can work with the pure gauge term (dg)d 1 = idX as the gauge connection. 

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