Anticommutator

The matrix y5 = y°y1y2y3[(y5)2 = +1], anticommutes with all the y s. It therefore has the property that ysyu(y5) 1 — — / . Hence, a possible choice for D is y6. Further properties of the matrices A,B,D can be obtained as follows Consider for example the matrix A. Upon taking the hermitian adjoint of Eq. (9-259) and substituting therein Eq. (9-259) again, 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 Ue anticommutes with the total charge, clearly the only interesting eigenstates of Uc are those with zero total charge. Furthermore, if A> is such an eigenstate, i.e., if... 

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... 

Since Oy and oy anticommute with Oz as represented in (7.24), we must have... 

These conditions define a matrix algebra which requires at least four anticommuting, traceless (hence even-dimensioned) matrices. The smallest even dimension, n = 2, can only accommodate three anti-commuting matrices, the Pauli matrices ... 

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. 

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)... 

Also, the fermion anticommutation rules interrelate the ROMs with the HRDMs they render these matrices antisymmetric with respect to odd permutations of the row or column indices and, finally, they interrelate them with two other families of matrices the G-matrices and the comelation matrices. 

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... 

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... 

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