Commutator 616 INDEX

Again we have dropped the mode index j momentarily. Note that the two exponentials cannot be combined since they do not commute. To evaluate the integral it is convenient to reintroduce the complete set of final states [cf. Eq. (19.22)] ... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

For computational purposes it is sometimes convenient to introduce a di-adic notation where a square n x n matrix with indices (j, k) is transformed into a column matrix with index J running from 1 to n2. [7,10] This transforms p to the n2 x 1 column matrix a and allows us to write the commutators in terms of n2 x n2 matrices A and B, so that the equations are now of the diadic form ... 

As in section 2, we introduce the identity, position and momentum operators, labelling the two subsystems with a = 1,2, Ia,Xj,aJlaDjta (as before, j = 1,..., n is a vector index in Rn), whose commutation relations are... 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

In Ch. 3 we have applied the second quantization for investigation of exciton states. The first step was to express the crystal Hamiltonian in terms of creation and annihilation Pauli operators Pjf and P/ of single-molecule excited states, where the index s indicates the lattice points where the molecule is placed, and / labels the molecular excited states. When taking into account only one fth excited molecular state, the operators P%f and P/ satisfy the following commutation rules (see eqn 3.28)... 

The standard model of physics is based on the direct product gauge group SU 3)c X SU 2)i, x f/(l)Y- SU(3)c is the non-Abelian gauge group of quantum chromodynamics (QCD) that describes the colour interaction between the quarks (hence the index C), while SU(2)l x U(1)y is the gauge group of the electroweak model [13-15]. Since the SU(3)c transformations of QCD commute with the 517(2)l x t/(l)Y transformations, the QCD part is not discussed here any further. 

The first commutator has been neglected in in both Eqs. (131) and (132), whereas the remaining two commutators were neglected only in the T2 equation. The removal of entire commutators assures the eize-extensivity of the CCSD(F12) energy [5, 41]. The Eqs. (130)-(132) are of a general form that is not yet suitable for the implementation. In the present work very often the expressions vector function and residual are used. They always refer to the many-index quantity, defined by the right hand sites of these equations. The working expressions of the coupled-cluster Ti, T2 and T2/ residuals are discussed in next subsections. 

In this theorem, we did not imply any concrete Hamiltonian system, but described the properties of the whole class of fields of the form sgrad h generated by the annihilator of the covector of general position. A particular case of Theorem S.1.1 is, of course, the classical Liouville theorem. Indeed, if the maximal linear subalgebra of functions G is commutative then its index r = ind G is equal to its dimension k and, theorefore, the maximality condition becomes A + A = dim Af = 2n, that is A = n. In this case, all the tori 17 from Theorem 3.1.1 are ordinary n-dimensional Liouville tori. 

Theorem 4.4.4 (BrailOV). For any Lie algebra G and for any commutative associative Frobenius algebra A, of dimension N, with unity the index ind Ga is equal to N ind G. 

Theorem 5.3.1 (Taymanov). If the fundamental group ni(M ) of a closed Riemannian manifold is not almost commutative, that is, does not contain a commutative subgroup of hnite index, then the geodesic Bow on does not admit a geometrically simple set of Brst integrals. 

Proof. We need to check that the map p commutes with the differentials, and it is enough to do this for a generator t], where suppty = V T). Comparing the incidence mmiber from (20.15) for Horn (T, G) with the one for the simplicial complex Honi4-(T, G), we see that the difference is the multiplicative functor (—1) , where t is the index of the simplex in the direct product where the new vertex is added. On the other hand, by the definition of c t]) we know that (—1)°0) = (— ) =( ) ( i) which proves that the incidence numbers coincide once C (Hom+(T, G)) is replaced with X T, G). ... 

Because a tensor product is a scalar quantity in index notation, the commutative law can be used within the sums ... 

As such, by using probabilities conditionally independent on categories with a lower index value and following the same geometric transformation as in Olbrich s work we can obtain a commutative formula for the angle. This is given in the following theorem. The proof follows the same form as that for Theorem 1. It is left to the reader. 

Prove that when the operators of the unitary group U(n) and the permutation group S v act on the symmetry-adapted hmctions (10.3.1) the two types of operation commute and hence that the CFs displayed schematically in (10.3.2) behave as indicated in the text. [Hint The basic tensor products are Qk, with K = kyk2 - ks, and the unitary operator U induces a transformation of the IV-index tensor components in which... 

At this point, one selects a single electron (index j) that occupies a particular one-electron state j j) in the initial state of the solid and considers the rate of photoemission from this state into the continuum state (cf, k). For photon energies in the UV and soft X-ray region, the associated wavelengths are large compared to the atomic distances and the vector potential A(r,-, t) is no longer dependent on rj, that is, it commutes with the momentum operator pj. For weak-enough radiation fields, the linear terms in A(t) dominate over the quadratic term and the interaction operator reduces to... 

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