Anticommutation relation for fermions

With the anticommutation relation for fermions in Eq. (15) and the second-quantized dehnitions, it has been shown that the connected portions of the two-particle and two-hole RDMs are equal [14, 20, 38] ... 

Further use of the anticommutation relation of fermions for orthonormal orbitals gives... 

Many-body problems in quantum mechanics are usually described by the number of particles N in the system and the probabilities of finding those particles at different locations in space. If the rank of the one-particle basis is a finite number r, an equally valid description of the system may be given by specifying the number of holes r N in the system and the probabilities of finding these holes at different locations in space. This possibility for an equivalent representation of the system by particles or holes is known as the particle-hole duality. By using the fermion anticommutation relation... 

Because the hole and particle perspectives offer equivalent physical descriptions, the p-RDMs and p-HRDMs are related by a linear mapping [52, 53]. Thus if one of them is known, the other one is easily determined. The same linear mapping relates the p-particle and p-hole reduced Hamiltonian matrices ( K and K). An explicit form for the mapping may readily be determined by using the fermion anticommutation relation to convert the p-HRDM in Eq. (18) to the corresponding p-RDM. Eor p = 1 the result is simply... 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

Compare with the case of anticommutation relations valid for fermions see equations (8.4) and (8.6) in Section 8.1.] ... 

It is essential now to use this form of the second quantized Hamiltonian which is expressed over the true fermion operators obeying the anticommutation relations of Eqs. (13.13). The use of Eqs. (13.28) or (13.33) for the Hamiltonian would complicate the following treatment since the appearance of the overlap matrix in the commutation rules would destroy the purely algebraic character of the creation/annihilation operators. The proper anticommutation rules permit us to consider and as abstract operators creating and annihilating electrons, respectively. Accordingly, taking the variation of Eq. (14.8) we get ... 

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