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**Read or Download Advances in Chemical Physics, Vol.119, Part 3. Modern Nonlinear Optics (Wiley 2001) PDF**

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**Extra info for Advances in Chemical Physics, Vol.119, Part 3. Modern Nonlinear Optics (Wiley 2001)**

**Sample text**

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*. The Lagrangian (126) is gauge invariant under a U(1) gauge transformation that introduces the electromagnetic field tensor F mn. Using the Euler–Lagrange equation (100) gives the vacuum field equation: qn F mn ¼ ÀigðBÃ qm B À Bqm BÃ Þ þ 2g2 Am jBj2 ¼ igðBÃ Dm B À BDm BÃ Þ ð130Þ ÀgJ m ðvacÞ where J m ðvacÞ ¼ iðBÃ Dm B À BDm BÃ Þ ð131Þ the present status of the quantum theory of light 27 Therefore J m ðvacÞ is a covariant conserved charge current density in the vacuum.

Meanwhile, the Jacobi identity (40) implies, in vector notation, the identities Að2Þ Bð3Þ À Bð2Þ Að3Þ 0 Að3Þ Bð1Þ À Bð3Þ Að1Þ 0 ð62Þ Að1Þ Bð2Þ À Bð1Þ Að2Þ 0 and ð3Þ ð2Þ ð1Þ ð3Þ ð2Þ ð1Þ cA0 Bð2Þ À cA0 Bð3Þ þ Að2Þ Â Eð3Þ À Að3Þ Â Eð2Þ 0 cA0 Bð3Þ À cA0 Bð1Þ þ Að3Þ Â Eð1Þ À Að1Þ Â Eð3Þ 0 ð63Þ cA0 Bð1Þ À cA0 Bð2Þ þ Að1Þ Â Eð2Þ À Að2Þ Â Eð1Þ 0 It has been shown elsewhere [42] that the identities (63) correspond with the B cyclic theorem [42,47–61] of O(3) electrodynamics: Bð1Þ Â Bð2Þ ¼ iBð0Þ Bð3ÞÃ ...

These must be complex in order to define the globally conserved charge: ð Q ¼ J 0 dV ð277Þ from the globally invariant current: J m ¼ iðAÃ qm A À Aqm AÃ Þ ð278Þ in the internal U(1) space of the gauge theory. The existence of a vacuum charge current density in the vacuum was first introduced phenomenologically by Lehnert [45,49], and it has been shown that the Lehnert equations can describe phenomena that the Maxwell–Heaviside equations are unable to describe. The reason for this is now clear. The vacuum Maxwell–Heaviside equations do not conserve action under a local gauge transformation in the internal scalar space of a U(1) gauge field theory.