Bose commutation rules

Using the Bose commutation rules (which means neglecting the deviation from the Bose character of the pair operators), we may derive... 

Equations (15.1)—(15.3) are combinations of commutation rules of Fermi type (when s = s ) and of Bose type (when s s ). The appearance of commutation relations for Fermi operators when s = s means that the number of molecular excitations, i.e. the eigenvalue of the operator Pj-fp/, takes either the value 0 (the molecule in the ground state) or 1 (the molecule is excited). In turn, the Bose commutation rules for s A s are related to the fact that the operators with different indices s are acting on different arguments of the crystal wavefunction. 

By virtue of the commutation rules (9-636) these amplitudes are symmetric under the interchange (k,A( k,A,) thus verifying the Bose... 

This condition applies not only to the vacuum state, but to any arbitrary state which can only mean that aj/zt = 0. To incorporate this condition into the previous set of commutation rules for Bose fields, it is sufficient to change the negative into positive signs, such that... 

For a general a, this wave function is not equal to P for two reasons (i) diagonal terms like xpf xpf, which vanish for single determinants, now survive (ii) ip2 ipi = ipi xp2 without sign change. This latter point affects only the normalization of P, but the former brings in configurations absent in F. Both are consequences of the Bose-like commutation rules [Eq. (18)]. 

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