Rigged Hilbert Space Formulation of Quantum Thermo Field Dynamics and Mapping to Rigged Liouville Space

TL;DR

This paper develops a rigorous mathematical framework using rigged Hilbert spaces to unify quantum statistical mechanics at finite temperature and connect Thermo Field Dynamics with Liouville space, enabling advanced analysis of quantum systems.

Contribution

It introduces a triplet structure for thermal space in Thermo Field Dynamics using rigged Hilbert spaces and maps it to rigged Liouville space, providing a unified foundation.

Findings

Rigged Hilbert space formalism for thermal states.

Isomorphic mapping between Thermo Field Dynamics and Liouville space.

Framework for analyzing open and non-equilibrium quantum systems.

Abstract

The rigged Hilbert space, a triplet extension of the Hilbert space, provides a mathematically rigorous foundation for quantum mechanics by extending the Hilbert space to accommodate generalized eigenstates. In this paper, we construct a triplet structure for the thermal space arising in Thermo Field Dynamics with the aid of the tensor product formulation of rigged Hilbert spaces, a formalism that reformulates thermal averages as pure-state expectation values in a doubled Hilbert space. We then induce the rigged Liouville space for Liouville space of density operators from the triplet structure for Thermo Field Dynamics; the rigged Liouville space corresponds isomorphically one-to-one with that of Thermo Field Dynamics. This correspondence offers a unified topological foundation for quantum statistical mechanics at finite temperature and establishes a framework for future generalizations to open and non-equilibrium quantum systems.