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Extra info for Advances in Chemical Physics, Vol.119, Part 3. Modern Nonlinear Optics (Wiley 2001)
These are used as an illustration only, and in general the vacuum charge current densities of O(3) electrodynamics are richly structured, far more so than in U(1) electrodynamics, where vacuum charge current densities also exist from the first principles of gauge theory as discussed already. 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).
The kinetic energy T appearing in Eq. (250) has no role in defining the ground state of the field theory, because the ground state is defined by the minimum of V with respect to a given variable, as just argued. In the present status of the quantum theory of light 43 these equations, the physical A and A* are excitations above the ground state or vacuum, and the vacuum gives no contribution to the global Lagrangian (250). The potential energy V is part of the locally gauge-invariant Lagrangian that gives the field equation (253), a relation between vacuum properties.
W. evans and s. jeffers under some group of transformations on xm and A. It follows  that there exist conserved quantities that are combinations of fields and derivatives, which are invariant under these transformations: energy, momentum, angular momentum, and charge. For example, it can be shown that the energy momentum tensor due to A is  1 ymn ¼ qm Aqn A À dmn qs Aqs A ð270Þ 2 For translation of the origin of space and time , Noether’s theorem gives 1 Jnm ¼ Àymn ¼ Àqm Aqn A þ dmn qs Aqs A 2 The conserved quantity in this case is the energy momentum ð d y0 d 3 x ¼ 0 dt n ð271Þ ð272Þ in the internal gauge space.