Rigged Hilbert Spaces associated with Misra-Prigogine-Courbage Theory of Irreversibility

TL;DR

This paper demonstrates how fixing the non-unitary transformation in the Misra-Prigogine-Courbage theory of irreversibility induces specific riggings of the Hilbert-Liouville space, linking mathematical structures to physical irreversibility.

Contribution

It establishes a canonical correspondence between the non-unitary transformation and riggings of the Hilbert-Liouville space in the context of irreversibility theory.

Findings

Three canonical riggings of the Liouville space are identified.

Rigging properties depend on nuclear and Hilbert-Schmidt properties of the transformation.

A reverse construction links riggings to the dynamical system's internal time superoperator.

Abstract

It is proved that, in the Misra-Prigogine-Courbage Theory of Irreversibility using the Internal Time superoperator, fixing its associated non-unitary transformation $\Lambda$, amounts to rigging the corresponding Hilbert-Liouville space. More precisely, it is demonstrated that any $\Lambda $ determinates three canonical riggings of the Liouville space $\QTR{cal}{L}$: a first one with a Hilbert space with a norm greater than the relative one from $\QTR{cal}{L}$; a second one with a $\sigma $-Hilbertian space, which is a K\"{o}the space if $\Lambda$ is compact and is a nuclear space if $\Lambda$ has certain nuclear properties; and finally a third one with a smaller $\sigma $-Hilbertian space with a still stronger topology which is nuclear if $\Lambda ^{n}$ is Hilbert-Schmidt, for some positive integer n. Viceversa: any rigging of this type, originated in a dynamical system having an Internal Time superoperator, defines a $\Lambda$ in a canonical way.