Independent redshifted finite cells

Let $L>0$ be a length in metres and $T_K>0$ a reference temperature in kelvins. On the connected open Rindler domain $O={\mathbb{R}}_{x^0}\times (0,\mathrm{\infty }{)}_x\times {\mathbb{R}}_y\times {\mathbb{R}}_z$, use length coordinates $x^0=ct$ and the metric

$d{s}^2=-{\left(\frac{x}{L}\right)}^2(d{x}^0{)}^2+d{x}^2+d{y}^2+d{z}^2\mathrm{.}$

Choose the future direction of increasing $x^0$ and the Killing field $\xi ={\mathrm{\partial }}_{x^0}$. The selected lapse is $N(x)=x\mathrm{/}L$. Define ${\beta }_K:=(k_B{T}_K{)}^{-1}>0$ in joules${}^{-1}$ and adopt the stationary-bath temperature field $T(x)=T_{K}L\mathrm{/}x$, so the inverse-temperature vector is $\beta ={\beta }_K\xi =\xi \mathrm{/}(k_B{T}_K)$. For the global-stationarity input of Constructed independent cell Gibbs states, use an auxiliary neutral species with rest mass $m>0$, charge $q=0$, chemical potential $\mu =0$, global potential $A=0$, and compensator ${\mathrm{\Lambda }}_{\xi }=0$. This auxiliary kinetic sector is a supplier hypothesis for the finite-cell construction; it is not identified with the two quantum cells.

Set $x_1:=L$, $x_2:=2L$ and place the cells at $p_i:=(x^0=0,x=x_i,y=0,z=0)\in O$, as in the placement map of Actually constructed Gibbs/KMS preparation. For $t_K,\tau \in \mathbb{R}$, define ${\gamma }_i(t_K):=(c{t}_K,x_i,0,0),{\mathrm{\Gamma }}_i(\tau ):=(c\tau \mathrm{/}{N}_i,x_i,0,0)\mathrm{.}$ Here $t_K$ is the Killing-flow parameter in seconds and $\tau$ is proper time in seconds. Each ${\gamma }_i$ is a smooth future-directed timelike integral curve of $c\xi$ through $p_i$; since $N_i=x_i\mathrm{/}L$ is constant on it, ${\mathrm{\Gamma }}_i(\tau )={\gamma }_i(\tau \mathrm{/}{N}_i)$ and $g(d{\mathrm{\Gamma }}_i\mathrm{/}d\tau ,d{\mathrm{\Gamma }}_i\mathrm{/}d\tau )=-c^2$. Thus $\tau =N_i{t}_K$ is the proper-time relation on the two worldlines. For each $i\in \{1,2\}$ take ${\mathcal{H}}_i={\mathbb{C}}^2$ with an orthonormal basis $\{\mathrm{\mid }{0}_i⟩,\mathrm{\mid }{1}_i⟩\}$, let $Q_i=0$, and set

$H_i={\mathrm{\Delta }}_i\mathrm{\mid }{1}_i⟩⟨1_i\mathrm{\mid },{\mathrm{\Delta }}_i>0\text{ joules}\mathrm{.}$

Set $q=0$ for the cells and use ${\mu }_i=0$. Then $N_1=1$, $N_2=2$, $T_1=T_K$, and $T_2=T_K\mathrm{/}2$. With ${\beta }_i=N_i{\beta }_K=(k_B{T}_i{)}^{-1}$, the exact local partition functions and states are

$Z_i=1+e^{-{\beta }_i{\mathrm{\Delta }}_i},{\rho }_i=\frac{\mathrm{\mid }{0}_i⟩⟨0_i\mathrm{\mid }+e^{-{\beta }_i{\mathrm{\Delta }}_i}\mathrm{\mid }{1}_i⟩⟨1_i\mathrm{\mid }}{Z_i}\mathrm{.}$

The total state and Killing-time generator are

$\rho ={\rho }_1\otimes {\rho }_2=\frac{e^{-{\beta }_K(H_1+2H_2)}}{Z_1{Z}_2},G_K=H_1+2H_2\mathrm{.}$

This is a positive normalized finite-dimensional state. It is ${\beta }_K$-KMS for the Killing-time dynamics and its $i$th cell state is ${\beta }_i$-KMS for the local proper-time dynamics of Conditional local KMS clock temperature. The equality $N_i{T}_i=T_K$ is exact within this constructed model; no observation, material-realization claim, infinite-volume limit, or interaction between cells is asserted.