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Additional resources for Advances in Chemical Physics, Vol.119, Part 3. Modern Nonlinear Optics (Wiley 2001)
Therefore, we have Amð1Þ and Amð2Þ in the vacuum. Circular polarization becomes a prerequisite for the conserved Q of Eq. (277). In the notation of Eq. (285) 1 Að1Þ ¼ pﬃﬃﬃ ðAX À iAY Þ; 2 1 Að2Þ ¼ pﬃﬃﬃ ðAX þ iAY Þ 2 ð286Þ the present status of the quantum theory of light 49 Circular polarization appears in general if AX ¼ Að0Þ exp ðÀiðot À kZÞÞ ð287Þ AY ¼ Að0Þ exp ðÀiðot À kZÞÞ ð288Þ where we have included the electromagnetic phase on the U(1) level. The scalar internal space in the vacuum is therefore described by the following two vectors: 1 A ¼ pﬃﬃﬃ ðAX þ iAY Þ; 2 1 AÃ ¼ pﬃﬃﬃ ðAX À iAY Þ 2 ð289Þ Global gauge transformation on these vectors produces a shift in the electromagnetic phase AX !
218) from Eq. (206) follows from local gauge invariance, and it is always possible to apply a local gauge transform to the vector A, the Maxwell vector potential. The ordinary derivative of the d’Alembert wave equation is replaced by an O(3) covariant derivative. The U(1) equivalent of Eq. (218) in quantum-mechanical (operator) form is Eq. (13), and Eq. (212) is the rigorously correct form of the phenomenological Eq. (25). It can be seen that Eq. (212) is richly structured in the vacuum and must be solved numerically.
The coefficient g is the same on the U(1) and O(3) levels. In SI units, Eq. (130) is qn F mn ¼ ÀigcðBÃ Dm B À BDm BÃ ÞAr ð133Þ and shows that the electromagnetic field in the vacuum has its source in the conserved J m ðvacÞ, which is divergentless. In Eq. (133), Ar is the area of the electromagnetic beam, c the vacuum speed of light and m0 is the vacuum permeability in SI units. The analysis can be repeated by identifying the scalar field f with a scalar component A of the vacuum four potential Am .