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Additional info for Advances in Chemical Physics, Vol.119, Part 3. Modern Nonlinear Optics (Wiley 2001)
The complete vacuum inhomogeneous equation is qn Gmn ¼ ÀgAn Â Gmn À gðDm AÞ Â A ð200Þ If m ¼ 2 and n ¼ 1, the left-hand side vanishes because G21 contains only B3, which is phaseless. The right-hand side gives the equation B3 ¼ g A1 A 2 ð201Þ which reduces in the notation that we have been using to Bð3Þ ¼ ÀigAð1Þ Â Að2Þ ð202Þ In the usual complex circular basis used for O(3) electrodynamics , this is the definition of the field B(3). Therefore, a check for self-consistency has been carried out for indices m ¼ 2 and n ¼ 1.
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 !
However, in special relativity, the number Ã is a function of the spacetime coordinate xm . This property defines the local gauge transformation m B ! eÀiÃðx Þ B; m BÃ ! eiÃðx Þ BÃ 1 L ¼ ðqm BÞðqm BÃ Þ À igðBÃ qm B À Bqm BÃ ÞAm þ g2 Am Am BÃ B À F mn Fmn 4 1 mn m Ã m Ã ¼ ðqm B þ ig Am BÞðq B À ig A B Þ À F Fmn 4 ð125Þ ð126Þ or gauge transformation of the second kind. The Lagrangian (120) is invariant under the local gauge transformation (125) if it becomes : The 4-potential becomes 1 Am ! Am þ q m Ã g ð127Þ where Ã is any number and the derivative qm becomes the covariant derivatives: Dm B ¼ ðqm þ ig Am ÞB Dm BÃ ¼ ðqm À ig Am ÞBÃ ð128Þ ð129Þ acting respectively on B and B*.
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