Operator anticommutator

The annihilation operators are written to the right of the creation operators to ensure that g operating on an occupation number vector with less than two electrons vanishes. Using that the annihilation operators anticommute and that the creation operators anticommute it is easy to show that the parameters g can be chosen in a symmetric fashion... 

Hereby, the exact sequence of the creation and annihilation operators can be chosen rather arbitrary within each of these blocks (up to some phase factor), since all the operators anticommute for each block independently,... 

In words, the creation operators anticommute. Equation (2.11) is nothing but the second quantized representation of the Pauli principle for electrons. The... 

If the creation and annihilation operators anticommute properly, they change the occupation numbers in an abstract Hilbert space of particle number representation which can be considered the same even if the physical orbitals do move (change). For this reason the true fermion operators need not be varied either. Their algebraic properties are determined by the relevant commutation rules, which are also independent of the physical properties of the system or of the nature of the basis orbitals. These anticommutation properties of the operators are the same after and before the variation. 

Thus, all pairs of creation and/or annihilation operators anticommute except for the conjugate pairs of operators such as ap and ap. From these relationships, all other properties of the creation and annihilation operators - often referred to as the elementary operators of second quantization - follow. We note that equation (103) holds only for orthonormal sets of spin orbitals. For nonorthonormal spin orbitals, the Kronecker delta in equation (103) must be replaced by the overlap integral between the two spin orbitals. 

Antisymmetry of electrons implies that annihilation operators anticommute [Xa,Xb = X Xb + XbXa... 

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... 

It is required, in accordance with the Fermi character of particles and antiparticles, to be separately antisymmetric in the particle and antiparticle variables, which in turn requires that the operator b and d satisfy the following anticommutation rules ... 

If we restrict ourselves to the case of a hermitian U(ia), the vanishing of this commutator implies that the /S-matrix element between any two states characterized by two different eigenvalues of the (hermitian) operator U(ia) must vanish. Thus, for example, positronium in a triplet 8 state cannot decay into two photons. (Note that since U(it) anticommutes with P, the total momentum of the states under consideration must vanish.) Equation (11-294) when written in the form... 

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

The only operation used for obtaining this partitioning is the anticommutation rule of the fermion operators. Note, that by adding the F and G terms one falls into the unitarily invariant Absar and Coleman partitioning [32,33] which was obtained by using a Group theoretical approach. 

From the fundamental rule of anticommutation of an annihilator with a creator operator it follows, in our orbital representation, that ... 

The result of commuting/anticommuting (for N even/odd) N annihilator operators with N creator operators is ... 

Relations (44,45) describe the general form of the N-order condition However, some terms must be eliminated from relation (45) because they do not occur when the anticommutation/commutation operations are carried out explicitly. We call these terms spin — forbidden because in all of them the spin correspondence which should exist between the creator and the annihilators forming the p-RO (which generates the p-RDM) is not maintained. These spin-forbidden terms are those having a transposition of at least two indices in their p-RDM. For instance ... 

Equations (45) and (49) stress the direct connexion existing between the elements and classes of the Symmetric Group of Permutations and the terms derived by com-muting/anticommuting groups of fermion operators after summing with respect to the spin variables. 

The set of operators Ox, Oy, anticommute, a property which we demonstrate for the pair Ox, Oy as follows... 

Recalling that the anticommutation rules for the creation and annihilation operators are (App. C)... 

The components of the a and 3 operators are independent of space and time coordinates and satisfy the anticommutator relationship... 

Flere, S represents an even operator, that is one that has no matrix elements between positive and negative energy components while is an odd operator having only matrix elements between positive and negative energy components. The conditions for an operator to be even or odd can be expressed more formally an even operator must commute with d [ ,(d] = 0 while an odd operator must anticommute with [S, S]+ =0. 

The expectation values of the anticommutator/commutator of -electron operators lead to expressions of the type [5]... 

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. 

This equation expresses an antisymmetrized product of two Kronecker deltas in terms of RDMS and HRDMs. By combining it with the expression of the simple Kronecker delta previously used (Eq. (14)), one can replace the antisymmetrized products of three/four Kronecker deltas, which appear when taking the expectation values of the anticommutator/commutator of three/four annrhrlators with three/four creator operators. With the help of the symbolic system Mathematica [55], and by separating as in the VCP approach the particles from the holes part, one obtains... 

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. 

Recalling the basic relation obtained by taking the expectation value of the anticommutator of two fermion operators ... 

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... 

The rewriting of commutators and anticommutators is guided by the simple rule that the particle rank of the operator should be reduced. The particle rank of an operator consisting of a string of p creation and q annihilation operators is 2 (p+q). A reduction in the particle rank by one can... 

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