Quantum Theory in Liouville Space

S. Twareque Ali, Eduard Prugovečki

Masashi Ban

Gonzalo E. Ordonez

Bryan J. Dalton

The theory of sequential quantum processes has been extended to Liouville space via the use of non-Hermitian projection operators in order to treat the evolution of the quantum density operator and to enable physically important matrix elements of the density operator to be calculated. The formal relationship of master equation methods to the theory of sequential quantum processes is established, and a new set of coupled master equations is derived. Special choices of projection operators lead to further simplification of the results. The Markoff approximation is also examined.

Ioannis E. Antoniou, Z. Suchanecki

Shousuke Ohmori, Junichi Takahashi

We discuss a super bra-ket formalism for quasi-Hermitian Liouvillian operators within the framework of rigged Hilbert spaces (RHS). An RHS in terms of the Liouville space, referred to as a rigged Liouville space (RLS), is reconstructed by exploiting the mathematical fact that the space of Hilbert-Schmidt operators is unitarily equivalent to the tensor product of Hilbert spaces. The obtained RLS endows a rigorous foundation of the construction for the super bra-ket and for the spectral decompositions of both Hermitian and quasi-Hermitian Liouville operators, which are characterized by the generalized eigenvectors in the dual spaces. Furthermore, within this framework, the non-Hermitian Liouvillian operator and its adjoint can be constructed symmetrically, with their symmetric structure preserved. As an application of our RLS methodology, we examine the Liouville operators corresponding to Hermitian and non-Hermitian harmonic oscillators and elucidate the differences between their spectral decomposition forms.