Superoperator Algebra

To demonstrate how one goes about finding an equation that permits /4 B to be directly computed, let us return to Eq. (6.1) and rewrite the 

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 

The tools needed for evaluating the above matrix elements of the super-operator resolvent are based upon the idea of operators (of the same type as A and B ) forming complete sets (Manne, 1977 Dalgaard, 1979). For example, if A and are number-conserving operators (e.g., r s), then the set of operators (oL P y - p q r —) 

The above results having to do with completeness of operator manifolds permit us to write a resolution of the identity as 

To better appreciate the meaning of Eq. (6.22), we write in detail some elements of the overlap matrix (7V r/) for the one-electron addition operator case (recall the definition of the occupied and unoccupied orbitals, 0, 

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We shall make use of Eqs. (9) and (10) in discussing the retarded and advanced Green functions in Appendix 14D. Superoperator algebra was surveyed in Ref. [49]. 

The linear operators on a vector space forms themselves a vector space, called operator space. In this context, the original vector space is called the carrier space for the operators. The operator space is sometimes normed, but usually not. Since operator products are defined, we have here a vector space where a product of vectors to give a vector is defined. Such a vector space is also called a linear algebra. Operations and functions can be defined in the operator space thus we can define superoperators for which the operator space is the carrier space. The hierarchy is not usually driven any further. Functions are usually named in analogy to their analytical counterparts. To be specific, assume that A has a spectral resolution... 

These challenges can be dealt with the powerful mathematical tools of quantum chemistry, as advocated by Per-Olov Lowdin.[l, 2, 3, 4] In our studies, linear algebras with matrices,[4] partitioning techniques,[3] operators and superoperators in Liouville space, and the Liouville-von Neumann... 

The powerful mathematical tools of linear algebra and superoperators in Li-ouville space can be used to proceed from the identification of molecular phenomena, to modelling and calculation of physical properties to interpret or predict experimental results. The present overview of our work shows a possible approach to the dissipative dynamics of a many-atom system undergoing localized electronic transitions. The density operator and its Liouville-von Neumann equation play a central role in its mathematical treatments. 

The superoperator formalism that has been used in previous publications is outlined here [2, 9, 29], The alternative diagrammatic and algebraic-diagrammatic representations can be found in other works [6],... 

An alternative way to derive the perturbational expansion for the electron propagator is to use an algebraic approach based on superoperators [3-6,18]. 

In principle, such a propagator theory would be particularly simple if the metric superoperator G satisfies the extra conditions (3.3). In practice, one has found it algebraically and computationally convenient to choose [Pg.324]

So the FC integral is added to the very few physical systems [18] which are realizations of this particular algebra. Using the Taylor theorem for shift operators due to Sack [19], and the Cauchy relation mentioned above, we can apply this very general idea to the specific case of the harmonic oscillator to obtain the closed formula (5). Recurrence relations can also be obtained by noticing that O is in reality a superoperator which maps normal ladder operators by the canonical transformation ... 

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