Constructed independent cell Gibbs states

Let $r\ge 1$ be finite. For each cell $i\in \{1,\dots ,r\}$, let ${\mathcal{H}}_i$ be a complex Hilbert space of finite dimension $d_i\ge 1$, let $H_i=H_i^{\ast }$ be a Hermitian proper Hamiltonian in joules, and let $Q_i=Q_i^{\ast }$ be a dimensionless number observable whose eigenvalues lie in ${\mathbb{N}}_0$. Assume $[H_i,Q_i]=0$. Operators $H_i,Q_i$ on the total space $\mathcal{H}:={⨂}_{i=1}^r{\mathcal{H}}_i$ mean their lifts acting on the $i$th tensor factor and as the identity on the other factors; no intercell interaction is included.

Let $(M,g)$ be a smooth time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$, and let $O\subseteq M$ be a connected open set on which a smooth future-directed timelike Killing field $\xi$ is defined. Use length coordinates $x^0=ct$ and the corresponding dimensionless components of $\xi$. Define its dimensionless lapse $N(x):=\sqrt{-g_x({\xi }_x,{\xi }_x)}>0$. Let $c>0$ be the speed of light in $\mathrm{m}{\mathrm{s}}^{-1}$, $k_B>0$ Boltzmann's constant in $\mathrm{J}{\mathrm{K}}^{-1}$, and let $A$ be a global smooth real one-form with SI components in tesla-metres, $F:=dA$ with components in teslas, and ${\mathrm{\Lambda }}_{\xi }:O\to \mathbb{R}$ a global smooth compensator in tesla-metres with ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$. Define ${\mathrm{\Phi }}_{\xi }(x):=-c(({\iota }_{\xi }A{)}_x-{\mathrm{\Lambda }}_{\xi }(x))$ in volts. Let $T:O\to (0,\mathrm{\infty })$ and $\mu :O\to \mathbb{R}$ be smooth fields in kelvins and joules per particle. Fix a common particle rest mass $m>0$ in kilograms and charge $q\in \mathbb{R}$ in coulombs, put $M:=m{c}^2>0$ in joules, and define $u:=\xi \mathrm{/}N$, $\beta :=u\mathrm{/}(k_{B}T)$ in ${\mathrm{J}}^{-1}$, and $\zeta :=\mu \mathrm{/}(k_{B}T)$ dimensionlessly. On the future massive shell ${\mathcal{P}}_M^{+}(O):=\{(x,P)\in TO:g_x(P,P)=-M^2,P\text{ future-directed}\}$, where $P$ has energy-vector units and $g_x(P,P)$ has units ${\mathrm{J}}^2$, let $f(x,P):=\exp (\zeta (x)+g_x(\beta (x),P))$, as defined for the same charged data in Global electrochemical redshift from charged Liouville stationarity, and assume the exact whole-shell stationarity condition ${\mathcal{L}}_{F,q}f(x,P)=0$ for every $(x,P)\in {\mathcal{P}}_M^{+}(O)$, where ${\mathcal{L}}_{F,q}$ is the charged transport field in that supplier's formulation. This is the stationarity premise of Global electrochemical redshift from charged Liouville stationarity. Choose a cell-location map $\chi :\{1,\dots ,r\}\to O$ and set $x_i:=\chi (i)$. Define the lapse, compensated potential, proper temperature, and chemical potential at cell $i$ by evaluation: $N_i:=N(x_i),{\mathrm{\Phi }}_i:={\mathrm{\Phi }}_{\xi }(x_i),T_i:=T(x_i),{\mu }_i:=\mu (x_i)\mathrm{.}$ The same theorem supplies constants $T_K>0$ and ${\mu }_K\in \mathbb{R}$ such that $N_i{T}_i=T_K,N_i{\mu }_i+q{\mathrm{\Phi }}_i={\mu }_K,$ where $q$ is the common particle charge in coulombs. This use of the charged smooth-geometry supplier retains its explicit ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption.

Set ${\beta }_K=(k_B{T}_K{)}^{-1}>0$ in joules${}^{-1}$, ${\beta }_i=N_i{\beta }_K=(k_B{T}_i{)}^{-1}$, $H_K={\sum }_{i=1}^r(N_i{H}_i+q{\mathrm{\Phi }}_i{Q}_i)$, and $Q={\sum }_{i=1}^r{Q}_i$. Then the grand generator $G_K:=H_K-{\mu }_{K}Q$ is Hermitian and $G_K={\sum }_{i=1}^r{N}_i(H_i-{\mu }_i{Q}_i)\mathrm{.}$ The finite partition sums $Z_i:={\mathrm{Tr}}_{{\mathcal{H}}_i}{e}^{-{\beta }_i(H_i-{\mu }_i{Q}_i)},Z:={\mathrm{Tr}}_{\mathcal{H}}{e}^{-{\beta }_K{G}_K}$ are positive and finite, with $Z={\prod }_{i=1}^r{Z}_i$. The normalized Gibbs density factorizes exactly: $\rho :=Z^{-1}{e}^{-{\beta }_K{G}_K}={⨂}_{i=1}^r{\rho }_i,{\rho }_i:=Z_i^{-1}{e}^{-{\beta }_i(H_i-{\mu }_i{Q}_i)}\mathrm{.}$ It is a ${\beta }_K$-KMS state for the grand-generator dynamics on the full finite matrix algebra. On the neutral subalgebra ${\mathcal{A}}_0:=\{B\in \mathcal{B}(\mathcal{H}):[B,Q]=0\}$, grand-generator and physical $H_K$ dynamics coincide, so the restricted state is ${\beta }_K$-KMS for the physical dynamics there. This constructs only independent finite cells; it does not claim that an interacting system's reduced cell states are Gibbs states.

All deductions are conditional on the actual interfaces in [F1]–[F3]. In particular, the charged redshift relation is used at its stated global-compensator and whole-shell hypotheses; no generic curved-spacetime quantum state is inferred.

Let $\mathcal{A}$ be a unital complex $C^{\ast }$-algebra and let $t\mapsto {\alpha }_t$ be a strongly continuous one-parameter group of $\ast$-automorphisms of $\mathcal{A}$. Let $\omega$ be a ${\beta }_K$-KMS state for $\alpha$ in the physical-time convention of Physical-time KMS strip equilibrium condition, where ${\beta }_K>0$ has units ${\mathrm{J}}^{-1}$ and $\mathrm{\hslash }>0$ has units $\mathrm{J}\mathrm{s}$. For a constant dimensionless $\lambda >0$, define ${\alpha }_{\tau }^{\lambda }:={\alpha }_{\tau \mathrm{/}\lambda },$ where $t$ and $\tau$ are real time parameters in seconds. Then $\omega$ is a $\lambda {\beta }_K$-KMS state for ${\alpha }^{\lambda }$, with strip width $\lambda {\beta }_K\mathrm{\hslash }$ seconds. Conversely, if $\omega$ is a $\widehat{\beta }$-KMS state for ${\alpha }^{\lambda }$, then it is a $(\widehat{\beta }\mathrm{/}\lambda )$-KMS state for $\alpha$.

For the geometric interpretation, let $O$ be a connected open subset of a smooth time-oriented Lorentzian four-manifold $(M,g)$ of signature $(-,+,+,+)$, and let $\xi$ be a smooth future-directed timelike Killing field on $O$ with dimensionless components in length coordinates $x^0=ct$, where $c>0$ is the speed of light in $\mathrm{m}{\mathrm{s}}^{-1}$. Define $N_{\xi }(x):=\sqrt{-g_x({\xi }_x,{\xi }_x)}>0$ and $u:=\xi \mathrm{/}{N}_{\xi }$. For a selected point $x_0\in O$, let $I_K\subseteq \mathbb{R}$ be an interval in seconds containing $0$, and let ${\gamma }_{x_0}:I_K\to O$ be the $C^1$ integral curve of $c\xi$ with ${\gamma }_{x_0}(0)=x_0$, so ${\gamma }_{x_0}^{\mathrm{\prime }}(t_K)=c{\xi }_{{\gamma }_{x_0}(t_K)}$; this is the Killing orbit through $x_0$ with Killing-time parameter $t_K$. For a cell assigned to $x_i$, take $x_0=x_i$ and denote this curve by ${\gamma }_i$. Define its proper-time parameter from the metric by $\tau (t_K):=\frac{1}{c}{\int }_0^{t_K}\sqrt{-g_{{\gamma }_{x_0}(s)}({\gamma }_{x_0}^{\mathrm{\prime }}(s),{\gamma }_{x_0}^{\mathrm{\prime }}(s))}ds,\tau (0)=0.$ By Killing temperature normalization and rigidity, $\xi (N_{\xi })=0$, so $N:=N_{\xi }(x_0)>0$ is constant along this orbit. The metric definition then gives $d\tau =Nd{t}_K$ and, with the chosen origins, $\tau =N{t}_K$; equivalently $d{\gamma }_{x_0}\mathrm{/}d\tau =cu$ for $u=\xi \mathrm{/}{N}_{\xi }$. For $k_B>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$ and reference temperature $T_K>0$ in kelvins, the inverse temperature ${\beta }_K=(k_B{T}_K{)}^{-1}$ rescales orbitwise to ${\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}:=N{\beta }_K$; defining $T_{\mathrm{l}\mathrm{o}\mathrm{c}}:=(k_B{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}{)}^{-1}$ gives $T_{\mathrm{l}\mathrm{o}\mathrm{c}}=T_K\mathrm{/}N$. No spatially varying lapse is treated as one global time-rescaled automorphism group.

Fix a cell label $i\in \{1,\dots ,r\}$ in Constructed independent cell Gibbs states, let $x_i=\chi (i)$ be its assigned point, and write $N:=N_i=N(x_i)>0$, $H_{\mathrm{l}\mathrm{o}\mathrm{c}}:=H_i$, $Q:=Q_i$, and ${\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}:={\mu }_i$. Let ${\beta }_K>0$ in joules${}^{-1}$ and ${\mu }_K$ in joules be the global parameters supplied there. Put $G_{\mathrm{l}\mathrm{o}\mathrm{c}}:=H_{\mathrm{l}\mathrm{o}\mathrm{c}}-{\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}Q$ and $G_K:=N{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}$. Set ${\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}:=N{\beta }_K$ and define the cell normalizer $Z_{\mathrm{l}\mathrm{o}\mathrm{c}}:=Z_i={\mathrm{Tr}}_{{\mathcal{H}}_i}{e}^{-{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}}={\mathrm{Tr}}_{{\mathcal{H}}_i}{e}^{-{\beta }_K{G}_K}>0$. Let $\sigma$ be the cell Gibbs density supplied by that theorem, so $\sigma =Z_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-1}{e}^{-{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}}=Z_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-1}{e}^{-{\beta }_K{G}_K}$. For neutral Hermitian matrices, define ${\alpha }_{\tau }^{\mathrm{l}\mathrm{o}\mathrm{c}}(C):=e^{i{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}\tau \mathrm{/}\mathrm{\hslash }}C{e}^{-i{G}_{\mathrm{l}\mathrm{o}\mathrm{c}}\tau \mathrm{/}\mathrm{\hslash }},{\alpha }_t^K(C):=e^{i{G}_{K}t\mathrm{/}\mathrm{\hslash }}C{e}^{-i{G}_{K}t\mathrm{/}\mathrm{\hslash }}\mathrm{.}$ For this selected cell, let $q$ be the common charge in coulombs and ${\mathrm{\Phi }}_i:={\mathrm{\Phi }}_{\xi }(x_i)$ its compensated potential in volts from the supplier. Define $H_{K,i}:=N{H}_{\mathrm{l}\mathrm{o}\mathrm{c}}+q{\mathrm{\Phi }}_{i}Q$ in joules, and write $H_K:=H_{K,i}$ in the physical control below. The supplier normalization gives $N{\mu }_{\mathrm{l}\mathrm{o}\mathrm{c}}+q{\mathrm{\Phi }}_i={\mu }_K$, hence $H_K=G_K+{\mu }_{K}Q$.

Let $A=A^{\ast }$ and $B=B^{\ast }$ satisfy $[A,Q]=[B,Q]=0$. Then their local proper-time and Killing-time dynamics agree after $\tau =Nt$.

Let $h:[0,S]\to \mathbb{R}$ be continuous, with $S>0$ seconds, and let $h(\tau )B$ have units joules. For dimensionless $ϵ\in \mathbb{R}$, the proper-time grand-generator control $G_{\mathrm{l}\mathrm{o}\mathrm{c}}-ϵh(\tau )B$ corresponds in Killing time to $G_K-ϵNh(Nt)B$. Define $E_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )$ and $E_K(t)$ as the finite-time expectations of $A$ from the same initial density $\sigma$ under these controls. Because $A$ and $B$ commute with $Q$, replacing each grand generator by its physical Hamiltonian changes the propagator only by a unitary generated by a constant multiple of $Q$ and leaves the expectation of $A$ unchanged. Thus these expectations also equal those obtained from the respective physical controls $H_{\mathrm{l}\mathrm{o}\mathrm{c}}-ϵh(\tau )B$ and $H_K-ϵNh(Nt)B$. In particular, $E_K(t)=E_{\mathrm{l}\mathrm{o}\mathrm{c}}(Nt)(0\le t\le S\mathrm{/}N)\mathrm{.}$ The finite-time Kubo expansions have identical first-order terms under this rescaling and the same quadratic remainder bound. With $R_{\mathrm{l}\mathrm{o}\mathrm{c}}(u):=\frac{i}{\mathrm{\hslash }}\mathrm{Tr}\text{\hspace{0.17em}⁣}(\sigma [{\alpha }_u^{\mathrm{l}\mathrm{o}\mathrm{c}}(A),B]),C_S:=\frac{2}{\mathrm{\hslash }}{\int }_0^S\mathrm{\mid }h(s)\mathrm{\mid }\mathrm{\parallel }B\mathrm{\parallel }ds,$ the local expansion is $E_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )=\mathrm{Tr}(\sigma {\alpha }_{\tau }^{\mathrm{l}\mathrm{o}\mathrm{c}}(A))+ϵ{\int }_0^{\tau }h(s)R_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau -s)ds+r_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau ),\mathrm{\mid }{r}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }\frac{(\mathrm{\mid }ϵ\mathrm{\mid }{C}_S{)}^2{e}^{\mathrm{\mid }ϵ\mathrm{\mid }{C}_S}}{2}\mathrm{.}$ Use $\mathrm{\hslash }>0$ in joule-seconds, so $C_S$ is dimensionless and $R_{\mathrm{l}\mathrm{o}\mathrm{c}}$ has units $[A][B]\mathrm{/}(\mathrm{J}\mathrm{s})$. The Killing-time response kernel is $R_K(t)=R_{\mathrm{l}\mathrm{o}\mathrm{c}}(Nt)$ and its control amplitude is $Nh(Nt)$.

For the equilibrium correlation spectral measures, use the Fourier convention of Finite spectral balance and distributional fluctuation response. If $D_N(\omega ):=N\omega$, then $S_K=(D_N{)}_{\ast }{S}_{\mathrm{l}\mathrm{o}\mathrm{c}},S_K(\omega )=N^{-1}{S}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\omega \mathrm{/}N)$ in distribution notation; the measure form preserves the weights of delta-function spectral lines. Since ${\beta }_K\mathrm{\hslash }(N\omega )={\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}\mathrm{\hslash }\omega$, detailed balance is unchanged. Any zero-frequency atom remains an atom at zero and is not assigned a value by the coth formula, which remains restricted to nonzero frequencies.

For a static local perturbation $G_{\mathrm{l}\mathrm{o}\mathrm{c}}-\lambda B$, where $\lambda B$ has units joules, the re-prepared Gibbs family satisfies ${\frac{d}{d\lambda }\mathrm{Tr}({\sigma }_{\lambda }B)\mid }_{\lambda =0}={\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}{\mathrm{Cov}}_{\mathrm{K}\mathrm{M},\sigma }(B,B)=N{\beta }_K{\mathrm{Cov}}_{\mathrm{K}\mathrm{M},\sigma }(B,B)\ge 0.$ Here $\delta B:=B-\mathrm{Tr}(\sigma B)I$ and ${\mathrm{Cov}}_{\mathrm{K}\mathrm{M},\sigma }(B,B):={\int }_0^1\mathrm{Tr}\text{\hspace{0.17em}⁣}\left({\sigma }^{1-u}\delta B{\sigma }^u\delta B\right)du\mathrm{.}$ It reduces to the ordinary variance when $B$ commutes with $G_{\mathrm{l}\mathrm{o}\mathrm{c}}$; no such reduction is asserted otherwise. These are exact finite-dimensional deductions for the constructed neutral cell. They imply no infinite-time relaxation, dc transport coefficient, or response law for arbitrary interacting reduced states.

The response variables are finite matrices and finite-time controls, not measured apparatus data. The exact Kubo expansion and the static derivative concern different preparations: the former fixes $\sigma$, while the latter re-normalizes the Gibbs family for each $\lambda$.

With the inner product and norm of Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$, the complex sequence space ${\mathrm{\ell }}^2({\mathbb{N}}_0,\mathbb{C}):=\{x=(x_n{)}_{n\ge 0}:{\sum }_{n\ge 0}\mathrm{\mid }{x}_n{\mathrm{\mid }}^2<\mathrm{\infty }\}$ is complete. Hence it is a complex Hilbert space in the sense of Hilbert space.

Let ${\mathbb{N}}_0=\{0,1,2,\dots \}$ and let $\mathcal{H}:={\mathrm{\ell }}^2({\mathbb{N}}_0,\mathbb{C})$ be the complex square-summable sequences with inner product $⟨x,y⟩={\sum }_{n\ge 0}{x}_n\overline{y_n}$, linear in its first argument. Let $e_n$ be the sequence with value $1$ at $n$ and $0$ elsewhere. Write $\mathcal{B}(\mathcal{H})$ for the bounded complex-linear operators on $\mathcal{H}$, with operator norm $\mathrm{\parallel }B\mathrm{\parallel }:={\sup }_{\mathrm{\parallel }x{\mathrm{\parallel }}_2\le 1}\mathrm{\parallel }Bx{\mathrm{\parallel }}_2$; for $C\in \mathcal{B}(\mathcal{H})$, $C^{\ast }$ is its bounded Hilbert-space adjoint, characterized by $⟨Cx,y⟩=⟨x,C^{\ast }y⟩$. A state on this operator algebra means a bounded complex-linear functional $\omega$ with $\omega (I)=1$ and $\omega (C^{\ast }C)\ge 0$ for every $C\in \mathcal{B}(\mathcal{H})$. Define the occupation-number operator $D(\mathcal{N}):=\{x\in \mathcal{H}:{\sum }_{n\ge 0}{n}^2\mathrm{\mid }{x}_n{\mathrm{\mid }}^2<\mathrm{\infty }\},(\mathcal{N}x{)}_n:=n{x}_n\mathrm{.}$ For a densely defined operator $T$ on this space, use the adjoint relation $D(T^{†}):=\{y\in \mathcal{H}:\mathrm{\exists }z\in \mathcal{H}\mathrm{\forall }x\in D(T),⟨Tx,y⟩=⟨x,z⟩\},T^{†}y:=z\mathrm{.}$ The representing $z$ is unique when $D(T)$ is dense; here “self-adjoint” means $D(T^{†})=D(T)$ and $T^{†}x=Tx$ on that domain.

Let $ϵ>0$ be in joules, $m\in \mathbb{Z}$ a dimensionless azimuthal mode number, $\mathrm{\Omega }\in \mathbb{R}$ an angular velocity in ${\mathrm{s}}^{-1}$, $\mathrm{\hslash }>0$ in joule-seconds, and ${\beta }_K>0$ in joules${}^{-1}$. Set $H:=ϵ\mathcal{N},J:=\mathrm{\hslash }m\mathcal{N},a:=ϵ-\mathrm{\hslash }m\mathrm{\Omega }\mathrm{.}$ For $m\mathrm{\ne }0$, $J$ has domain $D(\mathcal{N})$; for $m=0$, $J$ is the zero operator on all of $\mathcal{H}$. The operator $H$ has energy units J, $J$ has angular-momentum units J s, and $a$ has energy units J, so $\mathrm{\Omega }J$ also has units J. Define $K_0:=H-\mathrm{\Omega }J$ on the common domain $D(\mathcal{N})$ and let $K:=\overline{K_0}$ be its operator closure. For $a\mathrm{\ne }0$, $K=a\mathcal{N}$ has domain $D(\mathcal{N})$; for $a=0$, $K$ is the zero operator on all of $\mathcal{H}$. Thus $K$ is the self-adjoint closure of the rotating generator $H-\mathrm{\Omega }J$, including at the zero threshold. Then $\mathcal{N}$, $H$, $J$ and $K$ are self-adjoint on the stated domains.