Superoperator binary product

Using successively larger and larger operator manifolds we may define a hierarchy of approximate propagator methods based on Eq. (58). Also the choice of reference state influences Eq. (58) through the definition of the superoperator binary products in Eqs (52) and (60). It has been our experience that it is important to maintain a balance between the level of sophistication of the operator manifold and of the reference state. Better results are generally obtained this way, as we will see examples of in the subsequent sections. 

It is conventional to combine the two terms present on the right-hand side of Eq. (6.14) into a single factor by introducing the so-called superoperator binary product. This product, between two operators C and O, is defined as... 

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... 

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... 

Inserting this in the superoperator binary products between the complete set of operators arranged as a matrix, h hP),... 

Even if these formulas in the beginning have a more symbolic meaning, they may be given a strict mathematical interpretation by properly defining the domains of the operators and superoperators involved. Usually the wave function space is considered as a realization of the abstract Hilbert space based on a binary product ( 1 2) of the L2-type ... 

A. General Binary Products and the Concept of Adjoint Pairs of Superoperators... 

It is easily checked that the quantity [A B] satisfies the first four axioms of a binary product in the theory of the abstract Hilbert space, and, for this reason, the quantity [A 5] will be referred to as a general binary product, and the bounded superoperator G as a metric superoperator. Conversely, one may also show that any binary product [A B], which satisfies the first four axioms, may be written in the form (3.2). 

In the definition (2.3), the concept of a pair of adjoint superoperators—Mand M —is related to the HS binary product. Similarly, one may now introduce a pair of adjoint superoperators—M and N—through the general binary product... 

It follows that, if the superoperator M is self-adjoint with respect to any general binary product, it has always real eigenvalues, and its eigen-operators are orthogonal with respect to this binary product. In the definition (3.2), we have defined the general binary product in terms of the HS binary product, but it is evident that we could have used any binary product satisfying the first four axioms as a starting point or reference binary product. ... 

The HS binary product has the property that the Liouvillian superoperator L is self-adjoint according to (2.4) in this representation, and, for this reason, the HS binary product is the natural reference product in the Liouvillian formalism. 

The commutator binary product defined through (1.54) or (3.17) and the GNS binary product (3.48) have played an important role in revealing the mathematical structure of the special propagator methods and their connection with the EOM approach. In the opinion of the author, however, it is of great advantage to work with a binary product with respect to which the Liouvillian superoperator L from the very beginning is self-adjoint, and the HS binary product (2.2) would then be the most natural starting point. 

It should be observed that, even if the Hilbert-Schmidt binary product is a natural device for the study of superoperators and is frequently treated in the mathematical literature, its use in propagator theory has been limited to a few papers, e.g., the development of a Liouvillian perturbation theory by Dalgaard and Simons.33 This is hence a rather unexplored field, and only further studies may reveal how powerful this approach may be. 

Here the super reflects that the "-operators work on operators rather than functions. The binary product corresponding to a bracket is in superoperator space defined as in eq. (10.128). 

This series can be expressed in a more compact form by using the so-called superoperator formalism (Goscinski and Lukman, 1970). We introduce this formalism here, as we had introduced the interaction picture in Section 3.8, in order to facilitate our derivations. The final equations will, however, be written without any superoperators. The superoperator formalism is one level of abstraction higher than the Hilbert vector space of quantum mechanics. In the infinite-dimensional Hilbert space the vectors of the vector space are given as quantum mechanical wavefunctions and the transformations performed on the vectors in the vector space are given by the quantum mechanical operators. The binary product defined in Hilbert space is the overlap integral /) between two wavefunctions, and 4 . In the superoperator formalism we now have an infinite-dimensional vector space, where the quantmn... 

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