Anticommutation operator

For scalar /> and vector a, however, serious contradictions arise (Problem 3.6.1), which were "fixed" by defining ax, ay, az, and j8 as anticommuting operators, representable by the following traceless 4x4 matrices ... 

The commutator and anticommutator operations in Hilbert space can thus be implemented with a single multiplication by a and + superoperator, respectively. We further introduce the Liouville space-time ordering operator T. This is a key ingredient for extending NEGFT to superoperators when applied to a product of superoperators it reorders them so that time increases from right to left. We define (A(t)) = Tr A(f)Peq where peq = p(t = 0) represents the equilibrium density matrix of the electron-phonon system. It is straightforward to see that for any two operators A and B we have... 

The same holds for the string of creation operators (X g t ), too. In general, this result is always true if the string consists of an odd number of anticommuting operators. If we have an even number of anticommution operators, a cyclic permutation will change its sign. 

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 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... 

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... 

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... 

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