Creation operators unitary transformations

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

The new creation and annihilation operators are related to the original operators by a unitary transformation = U ... 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

Corresponding to a transformation of the orbitals we have a set of transformed creation operators. The one to one mapping between the orbitals and the creation operators implies that the unitary matrix for the orbitals can be used also to transform the creation operators. 

Rotation of any irreducible tensor in quasispin space transforms it according to the D-matrix of relevant dimensionality. For example, unitary transformations of a(qls yield the creation operators... 

According to the general relationship (5.9), rotations in isospin space transform the electron creation operators by the D-matrix of rank 1/2. If we go over from these operators to the one-electron wave functions they produce, then we shall have the unitary transformation of radial orbitals... 

Secondly, the commutator is the Lie product33 of the operators X Xs and Xu this choice of multiplication is particularly appropriate when one realizes that the X XS are the generators of the semisimple compact Lie group U , which is associated with the infinitesimal unitary transformations of the Euclidean vector space R (e.g., the space of the creation operators).34 With the preceding comments, the action of the transformation operator on the creation operators can formally be written in the usual form of the transformation law for covariant vectors,33... 

It may be readily verified that the matrix exp( - iA) is in fact unitary, provided the matrix A is Hermitian. The fact that the matrix exp( — iA) is unitary also means that the operator exp( — iA) is unitary and its matrix representation in the full ket expansion space, with matrix elements , is a unitary matrix. An analogous relation holds for transformations of the electron annihilation operators a, but it is the creation operator expansion that is most important for the MCSCF method. Substitution of the operator transformation into the expression of an arbitrary determinant gives the relation... 

The creation operators and the basis transform in the same manner, so when we apply a general unitary transformation to the basis, the creation operators suffer the same transformation. We can write in matrix form... 

Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141 134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented. 

The Fock space as introduced in Chapter I is defined in terms of a set of orthonormal spin orbitals. In many situations - for example, during the optimization of an electronic state or in the calculation of the response of an electronic state to an external perturbation - it becomes necessary to carry out transformations between different sets of orthonormal spin orbitals. In this chapter, we consider the unitary transformations of creation and annihilation operators and of Fock-space states that are generated by such transformations of the underlying spin-orbital basis. In particular, we shall see how, in second quantization, the unitary transformations can be conveniently carried out by the exponential of an anti-Hermitian operator, written as a linear combination of excitation operators. 

Let Op and ap be the elementary creation and annihilation operators associated with the untransformed spin orbitals and let 0) be any state in Fock space expressed in terms of the untransformed elementary operators. We shall in this section demonstrate that the elementary operators Op and ap and state 0) generated by the unitary transformation (3.2.1) and (3.2.2) can be expressed in terms of the untransformed operators and states as... 

In this exercise, we consider the spin-flip operators - that is, unitary operators that transform alpha operators into beta operators and visa versa. Although we shall investigate only the effect of the spin-flip operators on the creation operators, we note that the corresponding results for the annihilation operating are readily obtained by taking the adjoints of the relations for the creation operators. 

To demonstrate (3E.4.3)-(3E.4.6), we shall evaluate the BCH expansions of the unitary transformations. For this purpose, we must first evaluate the commutators of the creation operators with Sx and Sy. Expressing Sx and Sy in terms of the spin-shift operators... 

Using Eqs. (4.19) and (4.20a) it is easily verified that the anticommutation relations hold also for the transformed creation- and annihilation-operators. In Eq. (4.19) we have determined a unitary matrix that describes the... 

Another special case of the general quasiparticle transformation is when the coefficients Aj = 0. This corresponds to interchanging the creation and annihilation operators. The canonical conditions of Eq. (16.3) then require matrix B to be unitary. With B being the unit matrix, this is the particle-hole transformation we have considered in Sect. 10.2. 

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