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Additional resources for Advances in Chemical Physics, Vol.119, Part 3. Modern Nonlinear Optics (Wiley 2001)
It is convenient for our purposes to identify f with a scalar component B of the electromagnetic field in the vacuum. Therefore, there are two independent magnetic flux density components: 1 B ¼ pﬃﬃﬃ ðB1 þ iB2 Þ 2 1 BÃ ¼ pﬃﬃﬃ ðB1 À iB2 Þ 2 ð118Þ ð119Þ The Lagrangian governing these scalar components is L ¼ ðqm BÞðqm BÃ Þ ð120Þ and is invariant under global gauge transformation, also known as ‘‘gauge transformation of the first kind’’ B ! eÀiÃ B; BÃ ! eiÃ BÃ where Ã is any real number. The Euler–Lagrange equation qL qL ¼ qn qB qðqn BÞ ð121Þ ð122Þ 26 m.
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.
In complex circular notation, the vector in the internal gauge space can be written as A ¼ Að2Þ eð1Þ þ Að1Þ eð2Þ ð285Þ indicating two states of circular polarization. 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.