Anticommutation relation

We have seen that the creation operators anticommute among themselves (1.2.11) and that the same is true for the annihilation operators (1.2.12). We shall now establish the commutation relations between creation and annihilation operators. Combining (1.2.5) and (1.2.16), we obtain 

The phase factors cancel since they appear twice. Adding these equations together, we arrive at the following expression  

Since k) is an arbitrary ON vector, we have the operator identity 

The case P Q s, obtained by taking the conjugate of this equation and renaming the dummy indices. Combination of (1.2.21) and (1.2.25) shows that, for all P and Q, 

Consider a fermion collection, where each state is either empty or occupied, i.e., n G 0,1. If the states are ordered, the total wave function can be expressed in terms of the state occupancy (Davydov 1991), namely, 

Let us now define a creation operator, c , such that, if nk = 0, its operation will yield a wave function with nk = 1 and, if nk = 1 already, its operation will give zero, since a fermion can not be created in a state that is occupied. Hence, we have 

Similarly, we define the hermitean conjugate operator, ck, as an annihilation operator, which produces a wave function with a fermion missing from the fcth state, if it was occupied, or zero, if not. Thus, in this case, we have 

Let us first treat the operation on a single state, for which the relevant four equations are 

we investigate the two states i and j, where i j. Here, we have 

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Since the field operators satisfy the anticommutation relations (21,129)... 

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

The coefficients Cpq (Cpg) in eqs. (25, 26) are determined so that Up (a ) satisfy fermion anticommutation relation. The coefficients d pg d pg) in eqs. (27,28) are determined so that briber) satisfy bosqn commutation relation. Finally we ask fermions dp dp) to commute with bosons br br ) This means that we can write similarly as in (5) the total wave function P(r, R) as a product of fermion wave function p r, R) and boson wave function as 0( r, R)... 

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... 

Again this term has the same functional form for particles and holes. Note that for odd p the corrections must have opposite signs to cancel in the anticommutation relation (26). As with 4, the proportionality factor k is equal to the number of distinct ways of distributing the five particles between a group of three particles and a group of two particles thus ks = 10. 

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] ... 

Following Ziesche [35, 55], in order to develop the theory of cumulants for noncommuting creation and annihilation operators (as opposed to classical variables), we introduce held operators /(x) and / (x) satisfying the anticommutation relations for a Grassmann held. 

Valdemoro [28] achieved a close approximation to the 2-RDM by using the anticommutating relation of fermion operators, or what is equivalent, the 7/-representability conditions. This work indicated that the development of 1-RDM functional theories should be couched in terms of explicitly antisymmetric reconstructions of the 2-RDM. 

It is interesting to note that this approach works equally well for commutation relations as well as for anticommutation relations. 

The creation operators aj are the hermitian adjoint of the operators a . The properties of a can be inferred from the above equations. From Eq. (1.12) the hermitian conjugated operators are seen to satisfy the anticommutation relation... 

Using Eqs. (4.19) and (4.20a) it is easily verified that the anticommutation relations hold also for the transformed creation- and annihilation-operators. In Eq. (4.19) we have determined a unitary matrix that describes the... 

The operators in the orthonormal basis a satisfies the usual anticommutation relations. We therefore have... 

No signs of nonorthogonality showed up in these formulas. The anticommutation relation between a creation operator and an annihilation operator becomes... 

Show explicitly that the annihilation and creation operators fulfill the anticommutator relations (3 5). 

Second-quantization formalism was introduced into the theory of many-electron atoms by Judd [12]. This formalism enables one to give a simple and elegant description of both the rotation symmetry of a system and its permutational symmetry the tensorial properties of wave functions are translated to electron creation and annihilation operators, and the Pauli exclusion principle stems automatically from the anticommutation relations between these operators. 

Since these two equations hold for any one-determinant wave function, and the functions on the right side of these equations only differ in sign, we arrive at the following anticommutation relation for the creation operators ... 

Subjecting this to Hermitian conjugation, we find that the same anticommutation relation is also obeyed by the annihilation operators... 

From these anticommutation relations, specifically, we obtain the relation... 

The above anticommutation relations for second-quantization operators... 

The anticommutation relations between the electron creation and annihilation operators, accounting for (14.14), become... 

Using second-quantization, it is often necessary to transform complicated tensorial products of creation and annihilation operators. If, to this end, conventional anticommutation relations (14.19) are used, then one proceeds as follows write the irreducible tensorial products in explicit form in terms of the sum over the projection parameters of conventional products of creation and annihilation operators, then place these operators in the required order, and finally sum the resultant expression again over the projection parameters. On the other hand, the use of (14.21) enables the irreducible tensorial products of second-quantization operators to be transformed directly. 

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