Actually constructed Gibbs/KMS preparation

Adopt the selected stationary bath of Selected stationary thermal equilibrium on its connected smooth Lorentzian domain. A finite-cell preparation in this item means exactly the setup of Constructed independent cell Gibbs states: finitely many $r\ge 1$ noninteracting cells with finite-dimensional complex Hilbert spaces ${\mathcal{H}}_i$, Hermitian proper Hamiltonians $H_i$ in joules, and dimensionless Hermitian number matrices $Q_i$ with nonnegative integer eigenvalues and $[H_i,Q_i]=0$. Their tensor product is $\mathcal{H}={⨂}_{i=1}^r{\mathcal{H}}_i$; each cell operator acts as the identity on the other factors.

Choose a placement map $p:\{1,\dots ,r\}\to O$ assigning each cell to a spacetime event, and write $p_i:=p(i)$. Let $N_{\xi }(x):=\sqrt{-g_x({\xi }_x,{\xi }_x)}$ be the dimensionless lapse field, and let $T:O\to (0,\mathrm{\infty })$ and $\mu :O\to \mathbb{R}$ be the smooth proper-temperature and chemical-potential fields in the stationarity hypotheses. Evaluate them at the assigned event: $N_i:=N_{\xi }(p_i),T_i:=T(p_i),{\mu }_i:=\mu (p_i)\mathrm{.}$ Assume the same connected stationary geometry, selected future timelike Killing field $\xi$, common charge $q$ in coulombs, global smooth potential $A$, global compensator ${\mathrm{\Lambda }}_{\xi }$ satisfying ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$, and charged whole-future-mass-shell stationarity hypotheses as in that theorem; no patchwise potential is substituted. Using the length-coordinate convention and $c>0$ in metres per second from Thermodynamics-GR fields, currents, reference normalization and units, define the global gauge-compensated potential scalar ${\mathrm{\Phi }}_{\xi }:O\to \mathbb{R}$ by ${\mathrm{\Phi }}_{\xi }(x):=-c({\iota }_{\xi }A-{\mathrm{\Lambda }}_{\xi })(x)$ in volts, and set ${\mathrm{\Phi }}_i:={\mathrm{\Phi }}_{\xi }(p_i)$. The common parameters satisfy $N_i{T}_i=T_K>0$ and $N_i{\mu }_i+q{\mathrm{\Phi }}_i={\mu }_K$. The theorem's geometric stationarity argument is conditional on the Axiom of Countable Choice, ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no stronger choice axiom is used here.

Put ${\beta }_K=(k_B{T}_K{)}^{-1}$ in joules${}^{-1}$, ${\beta }_i=N_i{\beta }_K=(k_B{T}_i{)}^{-1}$, $H_K={\sum }_i(N_i{H}_i+q{\mathrm{\Phi }}_i{Q}_i)$, $Q={\sum }_i{Q}_i$, and $G_K=H_K-{\mu }_{K}Q={\sum }_i{N}_i(H_i-{\mu }_i{Q}_i)$. The finite-cell theorem constructs

$Z_i={\mathrm{Tr}}_{{\mathcal{H}}_i}{e}^{-{\beta }_i(H_i-{\mu }_i{Q}_i)},Z={\mathrm{Tr}}_{\mathcal{H}}{e}^{-{\beta }_K{G}_K}={\prod }_i{Z}_i,$

${\rho }_i=Z_i^{-1}{e}^{-{\beta }_i(H_i-{\mu }_i{Q}_i)},\rho =Z^{-1}{e}^{-{\beta }_K{G}_K}={⨂}_i{\rho }_i\mathrm{.}$

For $X\in \mathcal{B}(\mathcal{H})$, write ${\omega }_{\rho }(X)=\mathrm{Tr}(\rho X)$ and define the Killing-time grand-generator dynamics, with $t_K$ in seconds, by

${\alpha }_{t_K}^G(X)=e^{i{t}_K{G}_K\mathrm{/}\mathrm{\hslash }}X{e}^{-i{t}_K{G}_K\mathrm{/}\mathrm{\hslash }},\mathrm{\hslash }>0\text{ in joule-seconds}\mathrm{.}$

The referenced finite-dimensional theorem proves that $\rho$ is a normalized positive state and is ${\beta }_K$-KMS for ${\alpha }^G$; the physical-time strip has width $\mathrm{\hslash }{\beta }_K$ seconds under Physical-time KMS strip equilibrium condition. Its restriction to observables commuting with $Q$ is also KMS for the physical $H_K$ dynamics. This postulate adopts that actually constructed finite state as the state assignment for this specified model. It asserts no state on an infinite-volume algebra, no trace-class stress exponential in generic curved quantum field theory, no approach to equilibrium, and no observed preparation or material realization.