Operators and matrix elements in second-quantization representation

The mathematical apparatus of the angular momentum theory can be applied to describe the tensorial properties of electron creation and annihilation operators in the space of occupation numbers of a certain definite one-particle state a). It follows from (13.29) and (13.30) that the operators 

The expressions that define the action of second-quantization operators 4 and aa on wave functions a) and 0) can be presented in terms of the conventional relation 

The finite rotations in the space of occupation numbers for the one-electron state define the transition to the quasiparticle wave functions 

the change-over from the particle to the hole treatment can be represented in this case by a special kind of the aforementioned transformations. Indeed, by selecting in the D-matrix the values of parameters defining the rotation through angle n about the Oy-axis, we obtain 

The pertinent transformation of the second-quantization operators now yields creation and annihilation operators for holes (13.34) and (13.35). 

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Operators and matrix elements in second-quantization representation... 

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