Partitioning the Hilbert space

Let us partition the Hilbert space spanned by the y), y>a into — orthogonal subspaces. Each subspace is associated with one of the that are doubly occupied in We can choose an orthogonal basis in any of these subspaces and label these basis orbitals as We also define = < r. 

This is a very useful expression for considering it associated with the mono-density operators when the many-fermionic systems are treated, although similar procedure applies for mixed (sample) states as well. There is immediate to see that for Af formally independent partitions the Hilbert space corresponding to the iV-mono-particle densities on pure states, we individually have, see Eqs. (4.176), (4.181), (4.182) and (4.184),... 

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