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On the U(1) level, for example, the structure of the Lehnert [45] and gaugeinvariant Proca equations is obtained as follows: À Á & þ k 2 An ¼ 0   qm þ ig AÃm Gmn ¼ 0 ð72Þ ð73Þ These are regarded as eigenequations with eigenfunctions An and Gmn in configuration space. In this method, there is no need for the Lorenz condition. The equivalent of Eq. (72) in momentum space is the Einstein equation (2), and this statement is true for all gauge group symmetries. Comparing Eqs. (6) and (7) with Eqs.

258) is that the time rate of change of electromagnetic energy within an arbitrary volume V, combined with the energy flowing out through the boundary surfaces of the volume per unit time, is equal to the negative of the total work done by the field (a vacuum property) on the source, interpreted as vacuum charge current 44 m. w. evans and s. jeffers density. This is a statement of conservation of energy applied within the vacuum and in the absence of matter (electrons). In the received view J m ðvacÞ ¼ 0 ð261Þ and there is no vacuum Poynting theorem, but as argued already, the received view violates gauge invariance, special relativity, and causality.

The covariant derivatives (128) and (129) become Dm A ¼ ðqm þ ig Am ÞA Dm AÃ ¼ ðqm À ig Am ÞAÃ ð140Þ ð141Þ indicating the presence of self-interaction in the terms Am A and Am AÃ . This selfinteraction is observed empirically [47–61] in a number of ways, including the inverse Faraday effect and the third Stokes parameter defining the circular polarization of electromagnetic radiation. So it is also possible to use the form (139) for the vacuum charge current density, a form that eliminates any geometric unit such as Ar that is not fully relativistic.

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Near Perfect Optics

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