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Rigged Hilbert spaces associated with Misra–Prigogine–Courbage theory of irreversibility

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  • Ordóñez, Adolfo R

Abstract

It is proved that, in the Misra–Prigogine–Courbage Theory of Irreversibility using the Internal Time superoperator, fixing its associated non-unitary transformation Λ, amounts to rigging the corresponding Hilbert–Liouville space. More precisely, it is demonstrated that any Λ determines three canonical riggings of the Liouville space L: first one with a Hilbert space with a norm greater than the relative one from L; a second one with a σ-Hilbertian space, which is a Köthe space if Λ is compact and is a nuclear space if Λ has certain nuclear properties; and finally a third one with a smaller σ-Hilbertian space with a still stronger topology which is nuclear if Λn is Hilbert–Schmidt, for some positive integer n. In contrast any rigging of this type, originated in a dynamical system having an Internal Time superoperator, defines a Λ in a canonical way.

Suggested Citation

  • Ordóñez, Adolfo R, 1998. "Rigged Hilbert spaces associated with Misra–Prigogine–Courbage theory of irreversibility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 252(3), pages 362-376.
    • Handle: RePEc:eee:phsmap:v:252:y:1998:i:3:p:362-376
    DOI: 10.1016/S0378-4371(97)00643-2
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    1. Antoniou, I. & Suchanecki, Z. & Laura, R. & Tasaki, S., 1997. "Intrinsic irreversibility of quantum systems with diagonal singularity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 241(3), pages 737-772.
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