In the paper4 the notion of local KMS condition, introduced in Ref. 3 and extended in Ref. 2, was shown to open new possibilities in the study of the problem of Bose–Einstein Condensation (BEC). In this paper we analyze the general structure of states on the CCR algebra over a pre-Hilbert space that satisfies the local KMS condition with respect to a free Hamiltonian [Formula: see text] and to a given inverse temperature function [Formula: see text]. The replacement of Hilbert space by a pre-Hilbert space allows one to deal with test functions more singular than those usually considered in the theory of distributions (thus allowing e.g., fractal critical surfaces) and is equivalent (in the language of Weyl algebras) to consider a degenerate symplectic form. It is precisely this degeneracy that allows one to introduce in an intrinsic way the notions of [Formula: see text]-critical subspace (resp. [Formula: see text]-critical surface) and of states exhibiting BEC, independently of infinite volume limits and of boundary conditions. We prove that the covariance of any local KMS state is uniquely determined by the pair [Formula: see text], through a nonlinear extension of the Planck factor. For a large class of such states (including all known examples) the covariance splits into a sum of two mutually singular terms: one corresponding to a regular state, the other one with support on a critical surface (or more generally a critical subspace) uniquely determined by [Formula: see text] and [Formula: see text]. In particular we prove that, if such a state is gauge invariant Gaussian (quasi-free), then the [Formula: see text]-equilibrium condition uniquely determines the regular part of the state, while the singular part is arbitrary.
We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable
H∈Md(C) with real spectrum, the biorthogonal Gibbs functional
ωbi(A)=Zbi−1∑ne−βEn⟨φn∣A∣ψn⟩ satisfies
ωbi(A†A)≥0 for all
A if and only if
H is quasi-Hermitian. The proof reconstructs the metric
η directly from the eigenprojectors of
ωbi via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the
η-Gibbs state
ωη(A)=Zη−1Tr[ηe−βHA] satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever
[η,h]=0, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the
η-Gibbs state in finite dimensions, while the construction of a compatible
C∗-norm and the proof of
σ-weak continuity remain open.
We consider the dynamics of systems of lattice bosons with infinitely many degrees of freedom. We show that their dynamics defines a group of automorphisms on a
C∗--algebra introduced by Buchholz, which extends the resolvent algebra of local field operators. For states that admit uniform bounds on moments of the local particle number, we derive propagation bounds of Lieb--Robinson type. Using these bounds, we show that the dynamics of local observables gives rise to a strongly continuous unitary group in the GNS representation. Moreover, accumulation points of finite-volume Gibbs states satisfy the KMS condition with respect to this group. This, in particular, proves the existence of KMS states.
The limit states of canonical ensembles of a large number of interacting bosons at a given temperature, which are confined by harmonic forces, are studied in the framework of the resolvent algebra. It is shown that the limits satisfy the KMS condition or are ground states, regardless of the type of interaction. In case of attractive forces, where the ensembles collapse, observables that become meaningless in the limit disappear from the limit representations. For repulsive forces, this can also happen if condensates with an infinite number of particles in the same state (proper condensates) appear in the limit. The resulting structures and their interpretation are illustrated by a simple model. The study of vanishing harmonic forces (thermodynamic limit) involves changes of the dynamics. It is conveniently based on derivations acting on the algebra. They are given by the commutator of the Hamiltonians with the elements of the algebra. To ensure that the images remain in the algebra, the interaction must be regularized. This is accomplished in a manner that has only a minor impact on the dynamics and may be of broader interest. With this input a relation between the strength of the confining harmonic forces and the number of particles in the ensembles is derived from the condition that the limit states are to be stationary (invariant) under the adjoint action of the unconfined, spatially homogeneous limit dynamics. This relation encompasses the conditions that are frequently used in studies of Bose-Einstein condensates.
The Kubo–Martin–Schwinger (KMS) condition is a widely studied fundamental property in quantum statistical mechanics which characterizes the thermal equilibrium states of quantum systems. In the seventies, Gallavotti and Verboven, proposed an analogue to the KMS condition for infinite classical mechanical systems and highlighted its relationship with the Kirkwood–Salzburg equations and with the Gibbs equilibrium measures. In this paper, we prove that in a certain limiting regime of high temperature the classical KMS condition can be derived from the quantum condition in the simple case of the Bose–Hubbard dynamical system on a finite graph. The main ingredients of the proof are Golden–Thompson inequality, Bogoliubov inequality and semiclassical analysis.
Abstract We consider infinite volume limit Gibbs states of a nonrelativistic quantum Bose gas consisting of one species of spinless particles with positive interaction potentials. The finite volume reduced density matrices are dominated by the corresponding matrices for the noninteracting gas, and as a consequence all infinite volume limit states are regular, locally normal, and analytic on the appropriate CCR algebra. For sufficiently short range repulsive two-body interactions, the cyclic vector associated with the limit state is separating for the σ-weak closure of the algebra in the associated representation.