Operator particle-hole

It should be observed, however, that one cannot in general expect that an arbitrarily chosen basis B in the operator space should satisfy these three conditions, and this applies particularly to the particle-hole operator bases commonly used in the special propagator methods or the EOM approach. In such a case, the new operator basis B(1) defined by the linearly independent elements in the four matrices (2.31) may serve as a new starting basis. Due to the construction, the basis B(,) is automatically closed under adjunction (t). It is also clear that the basis B(l) is closed under multiplication due to the fact that one has the multiplication rule... 

If

truncated basis of order n in the carrier space, then the corresponding basis P = Pk in the operator space is of order m = n2. It may further be shown4 that, if n —> °° and the basis tp = < becomes complete in the carrier space, then the basis P = Pki becomes complete in the HS operator space. It should perhaps be observed that this completeness theorem is somewhat different in nature from the completeness theorems for products of particle-hole operators which are proven to be... 

It should be observed that it does not matter whether one describes the operator space in terms of particle-hole operators in the language of second quantization or in terms of ket-bra operators. Choosing the ket-bra basis P = Pu constructed from the orthonormal set

carrier space as our basis B, one obtains for r = (k, /) and s = (m, n) ... 

If a Hamiltonian has particle-hole (or charge-conjugation) S3unmetry then it is invariant under the transformation of a particle into a hole under the action of the particle-hole operator, J ... 

The particle-hole excitations, defined in Section 3.5, are eigenstates of the noninteracting Hamiltonian, but they are not eigenstates of the particle-hole operator, J, introduced in Section 2.9.2. To see this, consider the operation of J on the singlet excitation, A ) ... 

The electric dipole operator is antisymmetric with respect to the particle-hole operator and thus it connects states of opposite particle-hole symmetry. The proof is identical to that for the inversion operator. 

In terms of the particle-hole operators a and d (defined as a for the creation and destruction... 

Substitution of the particle-hole operators also in equations (8.7) and (8.8) yields, after elementary calculation, ... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

The reconstruction functionals, derived in the previous section through the particle-hole duality, may also be produced through the theory of cumulants [21,22,24,26,39,55-57]. We begin by constructing a functional whose derivatives with respect to probe variables generate the reduced density matrices in second quantization. Because we require that additional derivatives increase the number of second quantization operators, we are led to the following exponential form ... 

In fact, it turns out that the excitation operators in normal order in the particle—hole sense can be written as hnear combinations of operators in (the original) normal order with respect to the genuine vacuum. We put a tilde on operators in the new normal ordering. We get then... 

This result represents the most important advantage of the particle-hole formalism. Many-body perturbation theory (MBPT) consists mainly in the evaluation of expectation values (with respect to the physical vacuum) of products of excitation operators. This is easily done by means of Wick s theorem in the particle-hole formalism. 

The results of the last section, which are essentially a reformulation of the traditional particle-hole formalism for excitation operators, were first presented in 1984 [8]. At that time it was not realized that only two very small steps are necessary to generalize this formalism to arbitrary reference states. Only after Mukherjee approached the formulation of a generalized normal ordering on a rather different route [2], did it come to our attention how easy this generalization actually is, when one starts from the results of the last section. 

Note that a dot ( ) always means a matrix element of the antisymmetrized electron interaction g, a cross (x) a matrix element of the one-particle operator /, while an open square ( ) collects the free labels in any of these contractions. If the reference function is a single Slater determinant, all cumulants X vanish one is then left with particle and hole contractions, like in traditional MBPT in the particle-hole picture. 

Here t may represent Is or j, whereas z-projection of the quasispin operator is determined by the difference between the particle number operator and the hole number operator in the pairing state (a, / )... 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

Antilinear Particle-Hole Conjugation Operators in Jahn-Teller Theory... 

Four decades ago, Bell [3] introduced a particle-hole conjugation operator CB into nuclear shell theory. Its operator algebra is essentially isomorphic to that of Cq (for example, CB is unitary), the filled Dirac sea now corresponding to systems with half-filled shells. This was later extended to other areas of physics. For example,... 

Our procedure for securing basis independence follows a group-tensor algebraic approach to shell theory, and examines the algebraic interplay of particle-hole conjugation operators with quasispinors and quasispin tensors. The problem with Cs may be remedied while retaining an antilinear transformation only by replacing Z with another antilinear operation which is physical. Apart from an unimportant phase this can be identified as time reversal T, so that C = CT. Hence, the operators to be examined for physical interest are just two in number C and CT. In a later work we will explore the consequences of the work of Ceulemans [7,8,10] from this perspective. 

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