Commutation relations elementary

The E operators are the elementary operators, the generators, in unitary group theory. Below we give some commutator relations between E operators, e operators, creation and annihilation operators that are important for developing a good strategy for evaluating many of the quantities that show up in the later derivation. 

As the matrix product is an elementary example of a noncommutative product, it is very natural to look for n x n matrices in order to satisy the anticommutation relations (5). Here we show that matrices a and 0 obeying the anti-commutation relations have to be at least four-dimensional. 

It is inevitable that one be selective in choosing topics for a book such as this. This book emphasizes ground state MO theory of molecules more than do most introductory texts, with rather less emphasis on spectroscopy than is usual. Angular momentum is treated at a fairly elementary level at various appropriate places in the text, but it is never given a ftiU-blown formal development using operator commutation relations. Time-dependent phenomena are not included. Thus, scattering theory is absent,... 

Before discussing size-extensivity, we consider the commutation relations between two operators Op and Ob associated with fragments A and B, respectively. We first assume that both operators represent a single string of elementary operators and recall that the elementary operators of Oa anticommute with those of Ob. The operators Op and Ob therefore anticommute if an odd number of transpositions of elementary operators is required to reorder Op and Ob and otherwise commute. The number of transpositions is equal to the number of elementary operators in Op times the number of elementary operators in Ob. Therefore, Op and Ob anticommute if both operators contain an odd number of elementary operators and otherwise commute ... 

Consider the excitation operator k (k corresponding to the elementary transition relating k (root state) to a target state k) F,f acting from the right to commutator [H, k)(k ] yields ... 

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... 

---