Gibbs States, Algebraic Dynamics and Generalized Riesz Systems
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Complex Analysis and Operator Theory
Gibbs States, Algebraic Dynamics and Generalized Riesz Systems F. Bagarello1,2 · H. Inoue3 · C. Trapani4 Received: 28 April 2020 / Accepted: 10 September 2020 © The Author(s) 2020
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. Keywords Gibbs states · Non-Hermitian Hamiltonians · Biorthogonal sets of vectors · Tomita–Takesaki theory
Communicated by Sergey Naboko. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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C. Trapani [email protected] F. Bagarello [email protected] https://www1.unipa.it/fabio.bagarello H. Inoue [email protected]
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Dipartimento di Ingegneria, Università di Palermo, 90128 Palermo, Italy
2
Sezione di Napoli, INFN, L’Aquila, Italy
3
Center for Advancing Pharmaceutical Education, Daiichi University of Pharmacy, 22-1 Tamagawa-cho, Minami-ku, Fukuoka 815-8511, Japan
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Dipartimento di Matematica e Informatica, Università di Palermo, 90123 Palermo, Italy 0123456789().: V,-vol
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1 Introduction In the past 25 years or so it has become clearer and clearer that the role of selfadjointness of the observables of some given microscopic system can be, sometimes, relaxed, without modifying the essential benefits of dealing with, for instance, a selfadjoint Hamiltonian. In fact, we can still find real eigenvalues, a unitary time evolution and a preserved probability even if the requirement of the Hamiltonian being selfadjoint is replaced by some milder assumption, like in PT- or in pseudo-hermitian quantum mechanics. We refer to [1–5] for some references on these approaches, both from a more physical point of view and from their mathematical consequences. Considering a non-selfadjoint Hamiltonian H = H ∗ may lead to the appearance of new and often unpleasant features; for instance, the set {ϕn } of eigenstates of H , if any, in general is no longer an orthonormal system, but this set {ϕn } and the set {ψn } of the eigenstates of H ∗ turn out to be biorthogonal i.e., (ϕn |ψm ) = δn,m . Also, in concrete examples they are not bases for the Hilbert space H where the model is defined, but they may still be complete in H. This is the reason why the notion of D-quasi bases was proposed in [6]. This concept can be thought as a suitable extension of Riesz biorthogonal bases, and similar biorthogonal sets are found in several concrete phys
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