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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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Additional resources for Advances in Chemical Physics, Vol.119, Part 3. Modern Nonlinear Optics (Wiley 2001)
Sample text
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 Bà 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 ! eÀià B; Bà ! eià Bà where à 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 [46] 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 [46] 1 ymn ¼ qm Aqn A À dmn qs Aqs A ð270Þ 2 For translation of the origin of space and time [46], 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.
