Superoperator identity

In this form, it is necessary to perform the inversion of a superoperator which is very cumbersome, so an inner p ojection technique (79-81) or fte resolution of the superoperator identity (49),... 

Such a treatment can, with advantage, be expressed in terms of the superoperators introduced in Eq. (4.19) and in terms of a basis of field operators. The basis of fermion-like operators Xj = a, aj[aja, ,a aja, a ap, - is chosen, such that the electron field operators correspond to the SCF spin orbitals. The field operator space supports a scalar product (XjlXj) = ([A , X,]+) = Tr /9[Xl,Xj]+, where p is the density operator defined in Eq. (4.33). The superoperator identity and the superoperator hamiltonian operate on this space of fermion-like field operators and, in particular, Xi HXj) = [x/, [H,Xj - J. ) = Tt p[xI[H,X ] U. ... 

The state employed in the definition of the superoperator binary product is often called the reference state and need not be the ground state of the system. The transformations working on the vectors in this vector space of operators, i.e. the ( erators, are called superoperators and are here denoted with a wide hat as, e.g. in O. Commonly, only the superoperator Hamiltonian and the superoperator identity operator I are used, which are defined as... 

With this complete set of excitation and de-excitation operators ft-n, arranged either as column vector h or as row vector hP, we can, like in other vector spaces, also find an expression for the superoperator identity operator /, which is called a resolution of the superoperator identity. 

The desired matrix representation of the superoperator resolvent is then obtained in two steps by the inner projection technique (Pickup and Goscinski, 1973), where the superoperator resolvent is projected in the space of the complete set of excitation and de-excitation operators hn - First, we insert the resolution of the superoperator identity twice in Eq. (3.149) leading to... 

This may be further transformed by an inner projection onto a complete set of excitation and de-excitation operators, h this is equivalent to inserting a resolution of the identity in the operator space (remember that superoperators work on operators). 

Through introduction of superoperators and a corresponding metric [13], the propagator may be represented more compactly [2, 6]. Superoperators act on field operator products, X, where the number of annihilators exceeds the number of creators by one. The identity superoperator, I, and the Hamiltonian superoperator, H, are defined by... 

The general expression for the transform F of an operator F, as a function of F and S, is given by an identity relation we shall demonstrate. Let us consider the superoperators s acting on the operators of the Hilbert space, and defined as follows ... 

In summary, the idea of a complete set of operators has been extended to the superoperator binary product so as to introduce the powerful concept of a completeness relation. This completeness relation can now be exploited to derive an equation that permits i A B E expressed in a computationally more useful form (Simons, 1976). We begin by writing the identity... 

We now focus on the comparison between the CODDE [Eq. (2.21)] and the POP-CS-QDT [Eqs. (2.17), (B.4) and (B.6)]. These two formulations share the identical long time and thermal equilibrium behaviors characterized by their common field-free dissipation superoperator TZsj but differ at their correlated driving-dissipation dynamics. With the parameterization expressions for the bath spectral functions (Sec. 2.3) the correlated driving-dissipation dynamics effects may be numerically studied in terms of equations of motion via a set of auxiliary operators, which are Ko, i i, in the CODDE [Eq. (2.21)], and o, in the POP-... 

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