Bose operators effect

Substituting the expansion (3.197) into (3.19) and (3.29) we obtained the desired expansion of the Hamiltonian in powers with respect to Bose operators, when not only the dynamic, but also the kinetic interaction is taken into account. The new anharmonicity terms do not contain kinematic corrections. The role of this anharmonicity in the theory of third order nonlinear optical effects has been discussed in the article by Ovander (92). 

Nonlinear optical effects in crystals can be investigated also microscopically without using the phenomenological Maxwell equations. In the framework of this approach one has to keep, in the Hamiltonian of the crystal (formed, for example, by multilevel molecules), not only quadratic but also terms of third, fourth, etc. order with respect to the Bose amplitudes of excitons and photons. The part of the Hamiltonian which is quadratic with respect to the Bose amplitudes (see Ch. 4), can be diagonalized by making use of new Bose operators s(k) and j(k) (see eqn 4.16) so that... 

Energy excitations in 1-d Fermi systems are effectively Bose excitations with zero mass. A suitable representation of Fermion field operators in terms of Bosons has been given by... 

In spite of considering two-photon processes, we still find the energy levels (8.17) to be equidistant with respect to the photon occupation numbers n, -I- i. This suggests that it is permissible to neglect the effect of the electron-photon interaction on statistics and to occupy the electron states and the photon states according to Fermi statistics and to Bose statistics. We shall check this question in Section 8.4 by studying the creation and annihilation operators of the resulting quasi-electrons and quasi-photons. 

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