Gibbs states, algebraic dynamics and generalized Riesz systems

TL;DR

This paper explores generalized Gibbs states and algebraic dynamics in PT-quantum mechanics, focusing on non-self-adjoint Hamiltonians, biorthogonal systems, and related KMS conditions, with preliminary insights into Tomita-Takesaki theory.

Contribution

It introduces extended algebraic dynamics and connects them to generalized Gibbs states and KMS-like conditions in PT-quantum mechanics.

Findings

Relation between algebraic dynamics and generalized Gibbs states.

Extension of KMS conditions to non-self-adjoint Hamiltonians.

Initial discussion of Tomita-Takesaki theory in this context.

Abstract

In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita-Takesaki theory in our context.