Diagonalisation of the non-interacting finite-mode sector: occupation eigenvalues, partition sum and ground energy

Adopt the state and composition postulate P1 (State and composition postulate P1: Hilbert spaces, preparations and composite systems), the observable and probability postulate P2 (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes), the autonomous-dynamics postulate P4 (Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) and the identical-particle and spin postulate P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations) in the readings of Declared species sectors and their preparations: the framework's reading of P6 and Assigned sharp observables and the outcome law of the framework, and assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Work in the finite-mode realization of the declared fixed-$N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable: one-particle space $K$ of finite dimension $d\ge 1$ with declared modes $(f_1,\dots ,f_d)$, particle number $N\ge 1$, sector space ${\mathfrak{H}}_N^{\sigma }=\mathrm{ran}{P}_{\sigma }^{(N)}$, one-particle self-adjoint operator $h\in \mathcal{B}(K)$, and non-interacting interaction $W=0$, so that $H_{\mathrm{l}\mathrm{a}\mathrm{b}}={\sum }_{i=1}^N{h}^{(i)}$ and $H_N^{\sigma }=H_{\mathrm{l}\mathrm{a}\mathrm{b}}{\mathrm{\mid }}_{{\mathfrak{H}}_N^{\sigma }}$ in the notation of that definition. Let $e_1,\dots ,e_d$ be an orthonormal basis of eigenvectors of $h$ with real eigenvalues ${\epsilon }_1,\dots ,{\epsilon }_d$ and let ${\epsilon }_{(1)}\le \cdots \le {\epsilon }_{(d)}$ be the nondecreasing rearrangement of the list. Then:

(a) the labelled operator is diagonal: $H_{\mathrm{l}\mathrm{a}\mathrm{b}}$ is bounded and self-adjoint with $\mathrm{\parallel }{H}_{\mathrm{l}\mathrm{a}\mathrm{b}}\mathrm{\parallel }\le N\mathrm{\parallel }h\mathrm{\parallel }$, it commutes with every permutation unitary $U_{\pi }$, and for every multi-index $i=(i_1,\dots ,i_N)\in \{1,\dots ,d{\}}^N$, $H_{\mathrm{l}\mathrm{a}\mathrm{b}}(e_{i_1}\otimes \cdots \otimes e_{i_N})=\left({\sum }_{k=1}^N{\epsilon }_{i_k}\right)e_{i_1}\otimes \cdots \otimes e_{i_N};$

(b) occupation eigenvalues and spectrum: for every $N$-element subset $J\subseteq \{1,\dots ,d\}$ the Slater vector $e_J^{\wedge }$ of Slater and symmetric occupation-number bases of the finite-dimensional permutation sectors clause (b) is an eigenvector of $H_N^{-}$ with eigenvalue ${\lambda }_J:={\sum }_{j\in J}{\epsilon }_j$, and for every occupation vector $\lambda \in W(N,d)=\{\lambda \in {\mathbb{N}}^d:{\sum }_j{\lambda }_j=N\}$ the symmetric vector $m_{\lambda }$ of clause (c) is an eigenvector of $H_N^{+}$ with eigenvalue ${\lambda }_{\lambda }:={\sum }_{j=1}^d{\lambda }_j{\epsilon }_j$; the families $(e_J^{\wedge })$ and $(m_{\lambda })$ are orthogonal bases of ${\mathfrak{H}}_N^{-}$ and ${\mathfrak{H}}_N^{+}$ respectively, so the spectrum of $H_N^{\sigma }$ is the finite set of these eigenvalues (with multiplicities given by the number of configurations producing them); for $\sigma =-$ and $N>d$ the sector space is $\{0\}$ and the model has no sector states;

(c) partition sum and ground energy: write ${\mathcal{C}}_N^{\sigma }:=\{\lambda \in W(N,d):{\lambda }_j\le 1\text{ for every }j\text{ if }\sigma =-\},M_N^{\sigma }:=\mathrm{\mid }{\mathcal{C}}_N^{\sigma }\mathrm{\mid }=\dim {\mathfrak{H}}_N^{\sigma }\mathrm{.}$ The configuration counts are $M_N^{-}=\left(\genfrac{}{}{0px}{}{d}{N}\right)$ and $M_N^{+}=\left(\genfrac{}{}{0px}{}{N+d-1}{N}\right)$. For every inverse-energy parameter $\beta >0$ in ${\mathrm{J}}^{-1}$ (all ${\epsilon }_j$ and configuration energies are read in joules, so each Boltzmann exponent is dimensionless), $Z_N(\beta ):=\mathrm{tr}{e}^{-\beta H_N^{\sigma }}={\sum }_{\lambda \in {\mathcal{C}}_N^{\sigma }}{e}^{-\beta {\lambda }_{\lambda }}$ is finite. If $\sigma =-$ and $N>d$, then ${\mathcal{C}}_N^{-}=\mathrm{\varnothing }$, $Z_N(\beta )=0$, and there is no ground eigenvalue or trace-one density operator on the zero sector; in particular no normalized Gibbs density can be formed there. Otherwise $M_N^{\sigma }>0$ and $0<Z_N(\beta )<\mathrm{\infty }$. In this nonzero-sector case, let $E_0:={\min }_{\lambda \in {\mathcal{C}}_N^{\sigma }}{\lambda }_{\lambda },E_{\max }:={\max }_{\lambda \in {\mathcal{C}}_N^{\sigma }}{\lambda }_{\lambda }\mathrm{.}$ Then the actual Boltzmann bounds are $M_N^{\sigma }{e}^{-\beta E_{\max }}\le Z_N(\beta )\le M_N^{\sigma }{e}^{-\beta E_0}\mathrm{.}$ No nonnegativity hypothesis on the ${\epsilon }_j$ is imposed. For $\sigma =-$ and $N\le d$, $E_0={\sum }_{k=1}^N{\epsilon }_{(k)},E_{\max }={\sum }_{k=d-N+1}^d{\epsilon }_{(k)},$ and for $\sigma =+$, $E_0=N{\epsilon }_{(1)},E_{\max }=N{\epsilon }_{(d)}\mathrm{.}$ For every unit vector $\psi \in {\mathfrak{H}}_N^{\sigma }$ in either nonzero sector, $⟨H_N^{\sigma }\psi ,\psi ⟩\ge E_0$, with equality exactly on the eigenspace of $E_0$;

(d) reading and stationarity: on a nonzero sector the pair $(H_N^{\sigma },\mathrm{J})$ is the model's assigned energy observable; in a declared sector preparation $\rho$ its outcome law is ${\mu }_{\rho }(B)=\mathrm{tr}(\rho E_{H_N^{\sigma }}(B))$, and in the pure preparation of each sector basis vector of clause (b) this law is the point mass at the displayed eigenvalue; each such vector is a stationary state of the model evolution $U(t)=\exp (-it{H}_N^{\sigma }\mathrm{/}\mathrm{\hslash })$, and the energy law of every declared sector preparation is the law of the finite spectral sum in the sector basis;

(e) scope: the theorem is conditional on the finite-mode idealization, on the declared $h$ and on $W=0$; it asserts nothing for interacting models, for spatial realizations, for infinite mode families, about thermal states or equilibration, and about any laboratory system.

• Basis independence. The eigenvalues ${\lambda }_J$ and ${\lambda }_{\lambda }$ depend on the eigenvalue list of $h$, not on the choice of eigenbasis; a degenerate one-particle eigenvalue contributes with its multiplicity through the distinct configurations.
• Fermionic emptiness. For $\sigma =-$ and $N>d$ the antisymmetric sector is zero: the algebraic partition trace is $0$, while there are no sector states, no ground eigenvalue and no normalized density. The observable and stationarity readings of clause (d) apply only to nonzero sectors.
• Energy reference. The bounds in (c) permit arbitrary real one-particle energies. If all configuration energies are nonnegative, then $Z_N\le M_N^{\sigma }$; this count-only upper bound is not valid for unrestricted negative energies. Shifting every one-particle energy by $a\in \mathbb{R}$ multiplies $Z_N$ by $e^{-\beta Na}$ and shifts both extreme energies by $Na$.