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Extra resources for Advances in Chemical Physics, Vol.119, Part 3. Modern Nonlinear Optics (Wiley 2001)
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.
Jeffers Dirac attempted to remedy this flaw on the U(1) level by defining the electromagnetic phase factor by  þ g ¼ exp ig Am ðxm Þdxm ð297Þ On the O(3) level, vacuum gauge theory is defined by a Clifford algebra ð1Þ ð2Þ ð3Þ þ Að1Þ þ Að3Þ Am ¼ Að2Þ m e m e m e ð2Þ ð1Þ A¼A e ð1Þ ð2Þ þA e ð3Þ ð3Þ þA e ð298Þ ð299Þ where Am is a vector in the internal gauge space of O(3) symmetry and a 4-vector in Minkowski spacetime. In the internal gauge space, the Maxwell vector potential is defined as A ¼ AX i þ AY j þ AZ k ¼ Að2Þ eð1Þ þ Að1Þ eð2Þ þ Að3Þ eð3Þ ð300Þ indicating by ansatz the existence of a nonzero A(3) in the vacuum.
As shown in Eqs. (263), the matter and vacuum fields have the same structure. The coefficient g in the vacuum field is k=Að0Þ and is e="h in the matter field. The process hk ! eAð0Þ " ð281Þ is therefore a transfer of photon linear momentum to an electron, as in the Compton effect. As soon as " h is introduced, Planck quantization is also introduced. Since e is a property of neither the electromagnetic field nor the Dirac electron, the equation hk ¼ eAð0Þ " ð282Þ can be regarded [47–61] as a Planck quantization of the factor g in the vacuum: g¼ k e ¼ " Að0Þ h ð283Þ The Lehnert equations are a great improvement over the Maxwell–Heaviside equations [45,49] but are unable to describe phenomena such as the Sagnac effect and interferometry , for which an O(3) internal gauge space symmetry is needed.