A central issue in the development of a kinetic model for open systems is the stability of the resulting time-dependent solutions, which depends upon the eigenvalue spectrum of the Liouville superoperator. Zwanzig (1964) has presented an excellent discussion of the properties of superoperators (or tetradics). However, the present analysis requires a somewhat different group of expressions, so the subject will be developed here. The density operators which represent the state of a statistically mixed system themselves form a linear vector space analogous to the space of pure quantum states represented by wavefunctions. A linear combination of density operators might be used to describe the results of superposing two partially polarized beams of particles, for example (using the present normalization of ). Anything which generates linear transformations on a density operator [such as the right-hand side of the Liouville equation (2.3)] is a superoperator. In a finite, discrete system with N states, a wavefunction will be a vector (a singly-indexed object) with N elements, the density operator will be a matrix (a doubly-indexed object) with elements, and a superoperator will be a tetradic (a quadruply-indexed object) with elements. The linear algebra of superoperators is isomorphic to that of ordinary operators, but to define concepts such as Hermiticity or unitarity of superoperators, we must have a definition for the inner product of two ordinary operators. The simplest definition is
where A and B are operators and the notation is introduced to indicate expressions in the linear space of operators. It is easily shown that this satisfies the axioms (Apostol, 1969) defining an inner product on a complex vector space. Then a Hermitian superoperator satisfies
and a unitary superoperator satisfies
Superoperators are usually derived from ordinary quantum observable operators by forming the commutator or anticommutator with the operator being acted upon. For an operator C let us denote these superoperators
If C is Hermitian () the Hermiticity of and follow immediately:
and similarly for . The Hermiticity (or lack thereof) of the Liouville superoperator is the critical issue in formulating a kinetic model of open systems.
Of particular importance are the superoperators generated by the position operator x and the momentum operator :
These superoperators obey the following commutation relations:
Thus, is in some sense conjugate to , and bears a similar relationship to . Of course commutes with for any operator C.