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By Harnwell G. P.

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There is again a relation between the internal vector A and components in the vacuum of the four vector Am. For example Að1Þ ¼ iAX eð1Þ ; Að2Þ ¼ ÀiAY eð2Þ ; Að3Þ ¼ AZ k ð302Þ So it becomes clear that the description of the vacuum in gauge theory can be developed systematically by recognizing that, in general, A is an n-dimensional vector. On the U(1) level, it is one-dimensional; on the O(3) level, it is threedimensional; and so on. The internal gauge space in this development is a physical space that can be subjected to a local gauge transform to produce physical vacuum charge current densities.

Exp ðiÃðxm ÞÞAÃ ð137Þ and the gauge-invariant Lagrangian (126) becomes 1 L ¼ ðqm A þ ig Am AÞðqm A À ig Am AÃ Þ À F mn Fmn 4 ð138Þ Finally, the inhomogeneous field equation in the vacuum becomes qn F mn ¼ ÀigcðAÃ Dm A À ADm AÃ Þ ð139Þ 28 m. w. evans and s. jeffers in SI units. This form has the advantage of eliminating any geometric variable such as Ar from the vacuum charge current density. 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 .

For this to be the case, the vacuum polarization must be such that the displacement D(1) is not the complex conjugate of the displacement D(2). It can be seen as follows that for this to be the case, polarization must develop asymmetrically as follows: Dð1Þ ¼ e0 Eð1Þ þ aPð1Þ Dð2Þ ¼ e0 Eð2Þ þ bPð2Þ ð86Þ If there is no vacuum polarization, then the photon mass resides entirely in the vacuum current. In the preceding analysis, commutators of covariant derivatives always act on an eigenfunction, so, for example:  à ½Dm ; Dn Šc ¼ qm À ig Am ; qn À ig An c ¼ ðqm qn À qn qm Þc À ig Am qn c þ igqn ðAm cÞ À igqm ðAn cÞ þ ig An qm c À g2 ½Am ; An Šc ¼ Àig Am qn c þ igqn Am c þ ig Am qn c À igðqm An Þc ð87Þ À ig An qm c þ ig An qm c À g2 ½Am ; An Šc ¼ Àigðqm An À qn Am À ig½Am ; An ŠÞc giving the field tensor for all gauge groups: Gmn ¼ qm An À qn Am À ig ½Am ; An Š ð88Þ In the literature, the operation ½Dm ; Dn Šc is often written simply as ½Dm ; Dn Š but this shorthand notation always implies that the operators act on the unwritten c.

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