Fock-space unitary transformation

In the method based on the unitary transformation, we start by writing the exact wavefunction th in terms of the reference function and a unitary transformation operator in Fock space ... 

There are also some unexpected problems, related to the fact that the stationarity conditions do not discriminate between ground and excited states, between pure states and ensemble states, and not even between fermions and bosons. The IBQ give only information about the nondiagonal elements of y and the Xk, whereas for the diagonal elements other sources of information must be used. These elements are essentially determined by the requirement of w-representability. This can be imposed exactly to the leading order of perturbation theory. Some information on the diagonal elements is obtained from the lCSE,t, though in a very expensive and hence not recommended way. The best way to take care of -representability is probably via a unitary Fock-space transformation of the reference function, because this transformation preserves the -representability. 

In the case of quantum field theory the section determines the Hilbert space of states under a certain gauge. This choice of gauge then determines the unitary representation of the Hilbert space. We may then replace the section with the fermion field /, which acts on the Fock space of states. It is then apparent that a gauge transformation A t > A t + 84 is associated with a unitary transform of the fermion field v / > v / I 8 /. The unitary transformation of the fermion... 

Kutzelnigg and Koch/77/ also introduced a unitary Fock space cluster operator O = exp(o) with an antihermitian Fock space cr, as in eq. (7.2.3). In this case, the transformed Fock space hamiltonian L should be brought into a form such that... 

Now the door is open for the representation of unitary transformations in Hilbert space required for quantum computing by transformations in Fock space. Consider Hermitian operator H in one-dimensional Hilbert space Tilt has the form Ho + Hi where... 

Some properties of the Fock space transformations W and effective Hamiltonians h and, thus, of the resulting h, appear to differ from those obtciined by Hilbert space transformations. For example, their canonical unitary W is not separable and yields an h and, thus, an h with disconnected diagrams on each degenerate subspace. However, the analogous U of Eq. (5.13) may be shown to be separable 71), and the resulting He on each complete subspace flo is fully linked , as proven by Brandow [8]. These differences are not explained. 

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... 

EXPONENTIAL UNITARY TRANSFORMATIONS OF STATES IN FOCK SPACE... 

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