Commutators and anticommutators

In the manipulation of operators and matrix elements in second quantization, the commutator 

Before considering the evaluation and simplihcadon of commutators and anticommutators, it is useful to introduce the concepts of operator rank and rank reduction. The (particle) rank of a string of creation and annihilation operators is simply the number of elementary operators divided by 2. For example, Ihe rank of a creation operator is 1/2 and the rank of an ON operator is 1. Rank reduction is said to occur when the rank of a commutator or anticommutator is lower than the combined rank of the operators commuted or anticommuted. Consider the basic anticommutation relation 

Here the combined rank of the creation and annihilation operators is 1 whereas the rank of the anticommutator itself is 0. Anticommutation thus reduces the rank by 1. 

We now return to the evaluation of commutators and anticommutators. One useful strategy for the evaluation of such expressions is based on their linear expansion in simpler commutators or 

Note that the commutators and anticommutators on the right-hand side of these expressions contain fewer operators than does the commutator or the anticommutator on the left-hand side. For proofs, see Exercise 1.3. In deciding what identity to apply in a given case, we follow the principle of rank reduction - that is, we try to expand the expression in commutators or anticommutators that, to the greatest extent possible, exhibit rank reduction. 

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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... 

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... 

Show by performing the following commutators and anticommutators that one achieves a lowering of the particle rank of the operators involved ... 

The evaluation of averages of the commutators and anticommutators arising from (at yaj.a / ) uses the results... 

The preceding examples should suffice to illustrate the usefulness of the expansions (1.8.4)-(1.8.9) for evaluating commutators and anticommutators involving elementary operators. 

QED theory is based on two distinct postulates. The first is the dynamical postulate that the integral of the Lagrangian density over a specified space-time region is stationary with respect to variations of the independent fields Atl and ijr, subject to fixed boundary values. The second postulate attributes algebraic commutation or anticommutation properties, respectively, to these two elementary fields. In the classical model considered here, the dynamical postulate is retained, but the algebraic postulate and its implications will not be developed in detail. 

It seems, therefore, that the strong orthogonality between two states implies, according to Eq.(7), that the operators which create and annihilate those states commute, or anticommute, according to whether (qr) is even or odd respectively. 

The historically first attempt to achieve the block-diagonalisation of the Dirac Hamiltonian Hois due to Foldy and Wouthuysen and dates back to 1950 [42]. The central idea of this time-honoured approach is to partition the Hamiltonian into even and odd terms and to classify these terms according to their order in Xjrru . Even terms are block-diagonal and commute with whereas odd terms are off-diagonal and anticommute with... 

Any operator can be separated into odd and even contributions. The Dirac operator fi commutes with an even operator and anticommutes with an odd operator. The Breit Hamiltonian can be decomposed into even-even (EE), even-odd (EO), odd-even (OE) and odd-odd (OO) components, depending on whether the part of the operator acting on electrons one and two, respectively, is even or odd... 

The fields if/ and ij/ and the sources J and anticommute, while j and commute. The a term in (3.43) takes care, as usual, of the vacuum nonvanishing polarization ... 

The operator (t) generates a normalized 7V-electron spin eigenfunction as specified by the genealogical vector t. Since CSFs may be written as linear combinations of Slater determinants, the tensor r rator in (2.6.3) must be a linear combination of operator strings each containing N creation operators. The tensor operator therefore commutes with creation operators for even N and anticommutes for odd N. 

The second expression follows from the first since the creation operator to the right commutes or anticommutes with the second spin tensor of)erator, and likewise the annihilation operator to the left... 

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 ... 

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 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)... 

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... 

Using the anticommutation properties of electron creation and annihilation operators we can establish any necessary commutation relations for their tensorial products. For example, [103]... 

Further we shall show that such linear combinations have a certain rank in the quasispin space of n lN2n2lN2 configuration. The commutation relations between tensors (17.8) or (17.9) are completely defined by the anticommutation relations for creation and annihilation operators. These relations can be written as... 

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