Operators of electric and magnetic fields

In this subsection we derive formulas for the operators of the electric and the magnetic field in crystals, expressing them in terms of the Bose amplitudes and p. 

Using the Coulomb gauge for the vector potential we can decompose the electric field operator into two parts  

The crystal Hamiltonian, which appears in (4.40), in both cases of the retarded and Coulomb interaction, is expressed in terms of and p by the formula (4.19). As concerning the operator A(r), making use of (4.5) and (4.16) we find 

Inserting (4.19) into (4.41) and (4.40) and making use of the commutation rules for the Bose amplitudes p and we obtain the following expression for the operator E  

To find the operator EH(r), we consider only the long-wavelength limit of the longitudinal field, where some well-known relations from phenomenological theory can be applied. In particular, in virtue of the solenoidal character of the induction vector (divD = 0), we have kD(w, k) = 0 for plane waves. Simultaneously D = E + 4-7tP (see also Section 4.4) so that the longitudinal parts of the vectors E and P are related by 

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