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.

在PT量子力学中，物理系统动力学的生成者不一定是自伴哈密顿量。现在很明显，这种选择不会阻止获得统一的时间演化和哈密顿量的真实频谱，即使在大多数情况下，人们被迫处理双正交集而不是基于特征向量的正交向量也是如此。在本文中，我们考虑了海森堡代数动力学的一些扩展版本，并将此分析与吉布斯状态的某些广义版本及其与之类似的KMS条件相关联。我们还将在上下文中讨论富田-竹崎理论的一些初步方面。