KMS equilibrium and local Born quantum spin model

Adopt, for this bounded finite-dimensional quantum spin model, the following physical interpretation and equilibrium rule. A prepared state is a positive normalized functional $\omega$ on the quasi-local observable algebra of Infinite algebraic state and finite-region restrictions. The model assigns the Born probability $\omega (P)$ to a local projective-measurement outcome represented by a projection $P\in {\mathcal{A}}_{\mathrm{\Lambda }}$; the probabilities of the mutually orthogonal spectral projections of a finite-region self-adjoint observable sum to one. For a physical equilibrium preparation at temperature $T>0$, adopt as the model's equilibrium criterion that $\omega$ is a $\beta$-KMS state for the thermodynamic dynamics ${\alpha }_t$, with $\beta =(k_{B}T{)}^{-1}$ and the physical-time strip of width $\beta \mathrm{\hslash }$ seconds as in Physical-time KMS strip equilibrium condition. This is an adopted model postulate, not a deduction from the algebraic state definition or from the finite-volume Gibbs identity.

The postulate does not claim that an actual material has the specified interaction parameters, that a system dynamically relaxes to a KMS state, that equilibrium is unique, or that an apparatus realizes a specified projective measurement. It reports no experiment or measured value. Finite-volume approximations, when used, remain distinct from the infinite-volume state and its local restrictions.

The entropy-density construction in the physical-quantities definition is a separate conditional quantity under its stated Countable Choice and full Axiom of Choice hypotheses. This postulate uses only the model state, local measurement and KMS/unit conventions; it does not assume an entropy limit or assert that a material realizes the model.