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Actually constructed Gibbs/KMS preparation — Examples

1 · Prerequisites

2 · Summary

The example places two uncoupled two-level cells at different lapse values in one explicit Rindler stationary domain. It computes each finite Gibbs density, their tensor-product state, and the two proper-time KMS widths from the same Killing-time preparation. The picture shows constant-Rindler-position worldlines at $x = L$ and $x = 2 L$ ; the calculation uses the metric and finite matrices, not the sketch.

\begin{tikzpicture}[>=stealth]
  \draw[->] (0,0) -- (4.4,0) node[right] {$x/L$};
  \draw[->] (0,0) -- (0,3.5) node[above] {$x^0/L$};
  \draw[thick] (1,0) -- (1,3.1);
  \draw[thick] (2,0) -- (2,3.1);
  \fill (1,1.5) circle (2pt);
  \fill (2,2.2) circle (2pt);
  \node[below] at (1,0) {$x=L,\ N_1=1$};
  \node[below] at (2,0) {$x=2L,\ N_2=2$};
  \node[right] at (1,2.7) {$T_1=T_K$};
  \node[right] at (2,2.7) {$T_2=T_K/2$};
\end{tikzpicture}

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: Literature-sourcedVerification: AI-adaptedOpen item page →

Independent redshifted finite cells

Physical scope

An exact two-cell finite-dimensional Gibbs/KMS construction at two points of a Rindler stationary domain, conditional on the adopted bath and the specified neutral whole-shell stationarity input; no measured realization or interacting/infinite-volume state.

Example

Let $L > 0$ be a length in metres and $T_{K} > 0$ a reference temperature in kelvins. On the connected open Rindler domain $O = R_{x^{0}} \times (0, \infty)_{x} \times R_{y} \times R_{z}$ , use length coordinates $x^{0} = c t$ and the metric

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

Choose the future direction of increasing $x^{0}$ and the Killing field $ξ = \partial_{x^{0}}$ . The selected lapse is $N (x) = x / L$ . Define $β_{K} : = (k_{B} T_{K})^{- 1} > 0$ in joules $^{- 1}$ and adopt the stationary-bath temperature field $T (x) = T_{K} L / x$ , so the inverse-temperature vector is $β = β_{K} ξ = ξ / (k_{B} T_{K})$ . 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 $μ = 0$ , global potential $A = 0$ , and compensator $Λ_{ξ} = 0$ . This auxiliary kinetic sector is a supplier hypothesis for the finite-cell construction; it is not identified with the two quantum cells.

Set $x_{1} : = L$ , $x_{2} : = 2 L$ 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 $t_{K}, τ \in R$ , define $γ_{i} (t_{K}) : = (c t_{K}, x_{i}, 0, 0), Γ_{i} (τ) : = (c τ / N_{i}, x_{i}, 0, 0) .$ Here $t_{K}$ is the Killing-flow parameter in seconds and $τ$ is proper time in seconds. Each $γ_{i}$ is a smooth future-directed timelike integral curve of $c ξ$ through $p_{i}$ ; since $N_{i} = x_{i} / L$ is constant on it, $Γ_{i} (τ) = γ_{i} (τ / N_{i})$ and $g (d Γ_{i} / d τ, d Γ_{i} / d τ) = - c^{2}$ . Thus $τ = N_{i} t_{K}$ is the proper-time relation on the two worldlines. For each $i \in {1, 2}$ take $H_{i} = C^{2}$ with an orthonormal basis ${∣ 0_{i} ⟩, ∣ 1_{i} ⟩}$ , let $Q_{i} = 0$ , and set

$H_{i} = Δ_{i} ∣ 1_{i} ⟩ ⟨ 1_{i} ∣, Δ_{i} > 0 joules .$

Set $q = 0$ for the cells and use $μ_{i} = 0$ . Then $N_{1} = 1$ , $N_{2} = 2$ , $T_{1} = T_{K}$ , and $T_{2} = T_{K} / 2$ . With $β_{i} = N_{i} β_{K} = (k_{B} T_{i})^{- 1}$ , the exact local partition functions and states are

$Z_{i} = 1 + e^{- β_{i} Δ_{i}}, ρ_{i} = \frac{∣ 0_{i} ⟩ ⟨ 0_{i} ∣ + e^{- β_{i} Δ_{i}} ∣ 1_{i} ⟩ ⟨ 1_{i} ∣}{Z_{i}} .$

The total state and Killing-time generator are

$ρ = ρ_{1} \otimes ρ_{2} = \frac{e^{- β_{K} (H_{1} + 2 H_{2})}}{Z_{1} Z_{2}}, G_{K} = H_{1} + 2 H_{2} .$

This is a positive normalized finite-dimensional state. It is $β_{K}$ -KMS for the Killing-time dynamics and its $i$ th cell state is $β_{i}$ -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.

Facts & Assumptions

Given: The displayed Rindler metric and stationary bath, $L, T_{K}, Δ_{1}, Δ_{2} > 0$ , the two noninteracting two-level cells, and the auxiliary neutral kinetic-sector inputs.

[F1]

Thermodynamics-GR fields, currents, reference normalization and units fixes the signature $(-, +, +, +)$ , length coordinates $x^{0} = c t$ , and dimensionless lapse for a length-normalized Killing field.

[F2]

Global electrochemical redshift from charged Liouville stationarity: for smooth $T > 0$ , $μ$ , a future timelike Killing field, and the charged exponential ansatz, whole-future-shell stationarity is equivalent to $β$ Killing and $d ζ = - c q ι_{β} F$ ; with a global compensator it yields the required constants. Its proof carries ${A C}_{ω}$ through the smooth-geometry suppliers.

[F3]

Constructed independent cell Gibbs states constructs the finite product density and proves its KMS property when the cells are finite-dimensional, noninteracting, $[H_{i}, Q_{i}] = 0$ , and the common stationary/gauge/whole-shell hypotheses hold.

[F4]

Conditional local KMS clock temperature proves the orbitwise rescaling from the constructed $β_{K}$ -KMS cell state to a $N_{i} β_{K}$ -KMS proper-time state, with width $ℏ N_{i} β_{K}$ seconds.

[A1]

Selected stationary thermal equilibrium is adopted for this selected comoving bath only; the geometry does not imply that every material admits the bath.

[A2]

Actually constructed Gibbs/KMS preparation adopts the constructed finite density as this model's state assignment; it does not claim thermalization or a generic curved-spacetime state.

[A3]

The cells are uncoupled and have $Q_{i} = 0$ . The auxiliary kinetic species is neutral, $m > 0$ , $A = 0$ , $Λ_{ξ} = 0$ , and $μ = 0$ ; it supplies the theorem's stationarity condition and is not a cell degree of freedom.

[A4]

The auxiliary future massive shell, charged transport, and exponential ansatz use the exact formulation in Charged particle and Liouville data on a stationary Lorentzian domain.

[F5]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the inherited ${A C}_{ω}$ assumption of [F2] and [F3].

Verification

technique · Verify the selected Rindler carrier and neutral stationarity input, then apply the exact finite-cell factorization and orbitwise KMS suppliers

1.1givenA1F1algebra

The metric is smooth and has signature $(-, +, +, +)$ on $x > 0$ . The coordinate transformation $X = x \cosh (x^{0} / L)$ , $X^{0} = x \sinh (x^{0} / L)$ gives $- (d X^{0})^{2} + d X^{2} = - (x / L)^{2} (d x^{0})^{2} + d x^{2}$ , so this is the Rindler wedge. The metric coefficients are independent of $x^{0}$ , hence $ξ$ is Killing, and $g (ξ, ξ) = - (x / L)^{2} < 0$ gives $N = x / L$ . Moreover $\partial_{x^{0}} X^{0} = (x / L) \cosh (x^{0} / L) > 0$ , so $ξ$ is future directed.

2.1givenA3A4F2F3step 1.1algebra

With $T = T_{K} L / x$ , $u = ξ / N$ gives $β = u / (k_{B} T) = ξ / (k_{B} T_{K})$ , a Killing field. By [A3]–[A4], the auxiliary inputs have $F = d A = 0$ , $ζ = μ / (k_{B} T) = 0$ , and $q = 0$ , so $d ζ = - c q ι_{β} F = 0$ in the exact transport formulation. The forward direction of [F2] therefore gives whole-future-shell stationarity and the global compensator condition required by [F3].

3.1givenA2A3F2F3F5step 1.1step 2.1algebra

At $x_{1} = L$ and $x_{2} = 2 L$ , step 1.1 gives $N_{1} = 1$ and $N_{2} = 2$ . The adopted field has $T_{i} = T_{K} / N_{i}$ , and the two cells satisfy $[H_{i}, Q_{i}] = 0$ , $Q_{i} = 0$ , and $μ_{i} = q = 0$ . Thus the finite-cell hypotheses of [F3] hold with $G_{K} = N_{1} H_{1} + N_{2} H_{2} = H_{1} + 2 H_{2}$ and $β_{i} = N_{i} β_{K}$ . Applying [F3] gives the positive finite partition sums and the product state in the Example. The construction is conditional on [F5], inherited through [F2].

4.1F3step 3.1algebra

The eigenvalues of $H_{i}$ in the displayed basis are $0$ and $Δ_{i}$ , so their traces are $Z_{i} = 1 + e^{- β_{i} Δ_{i}}$ and the normalized diagonal densities are exactly the displayed $ρ_{i}$ . Since $β_{i} = N_{i} β_{K}$ , their tensor product equals $e^{- β_{K} (H_{1} + 2 H_{2})} / (Z_{1} Z_{2})$ .

5.1A2F4step 3.1step 4.1algebra∎

Apply [F4] separately to the two cells. Their Killing-time KMS parameter is $β_{K}$ ; the constant lapse rescalings give local parameters $N_{1} β_{K}$ and $N_{2} β_{K}$ , equal to $(k_{B} T_{1})^{- 1}$ and $(k_{B} T_{2})^{- 1}$ . This establishes the stated model prediction only and reports no experiment.