Thermal occupation means of the declared free-mode models: Bose and Fermi factors, product convergence and dilute remainders

[The single bosonic mode] Apply [F1] with $\omega =\epsilon$: ${\rho }_{\beta }$ is the canonical preparation of the declared energy operator $\epsilon N$, the number readout $N$ is self-adjoint with $N{e}_n=n{e}_n$ by [F4], and its thermal mean in ${\rho }_{\beta }$ equals the absolutely convergent sum ${\sum }_{n}n{q}^n\mathrm{/}{\sum }_n{q}^n=1\mathrm{/}(e^{\beta \epsilon }-1)$ with $q=e^{-\beta \epsilon }$; all higher moments are finite by [F1]. This proves (a).

[The fermionic orbital and the shift] For the stated arbitrary real ${\epsilon }_1,\mu$, direct evaluation in the two occupation vectors gives the unnormalized positive operator $G={\sum }_{k=0}^1{q}^k{T}_{u_k,u_k}$ with $q=e^{-\beta ({\epsilon }_1-\mu )}>0$ and $\mathrm{tr}G=Z=1+q$. Thus $F=G\mathrm{/}Z$ has trace one and occupation mean $q\mathrm{/}(1+q)=1\mathrm{/}(e^{\beta ({\epsilon }_1-\mu )}+1)$ with $q=e^{-\beta ({\epsilon }_1-\mu )}$; for $\mu =0$, ${\epsilon }_1>0$ this is the model's thermal mean at the declared energy ${\epsilon }_1$. This proves (b) and the identities of (d).

[The number-variable space and complete basis] Let $H_0$ be the one-particle Hilbert space with supplied countable orthonormal basis $(e_j{)}_{j\ge 1}$ of [F3]. For either sign define ${\mathfrak{H}}_0^{\sigma }=\mathbb{C}$ and ${\mathcal{F}}_{\sigma }:=\{({\psi }_N{)}_{N\ge 0}:{\psi }_N\in {\mathfrak{H}}_N^{\sigma },{\sum }_N\mathrm{\parallel }{\psi }_N{\mathrm{\parallel }}^2<\mathrm{\infty }\}$ with inner product ${\sum }_N⟨{\psi }_N,{\phi }_N⟩$, whose absolute convergence follows from Cauchy-Schwarz applied to the two square-summable sector norm sequences as in the direct-sum construction of [F5], where the positive-degree sectors are those of [F5]. The plus space is the supplied bosonic direct sum. For the minus space, each coordinate of a Cauchy sequence converges in its closed Hilbert sector; bounded finite partial sums of squared norms put the coordinate limit in ${\mathcal{F}}_{-}$, and taking coordinate limits in the Cauchy bound, then the supremum over finite partial sums, proves convergence in the full norm. Thus it too is Hilbert; this is a local argument, not a fermionic clause attributed to the bosonic supplier. Embed each fixed-$N$ basis of [F5] in coordinate $N$ and add the unit vacuum in coordinate zero. Orthogonality holds between different degrees, and truncating degree tails then approximating in each of finitely many complete sector bases proves completeness of their union. Label this countable basis $(b_n^{\sigma })$ by ${\mathcal{C}}^{\sigma }:=\{n\in {\mathbb{N}}^{{\mathbb{N}}_{>0}}:n\text{ has finite support},n_j\le 1\text{ if }\sigma =-\}$, with $\mathrm{\mid }n\mathrm{\mid }={\sum }_j{n}_j$; for minus use the Slater vector of the occupied subset. Enumerate these labels by finite integer sequences and apply the diagonal lemma of [F5].

[Exact generator and trace] On this basis define the self-adjoint diagonal $H$, $\mathcal{N}$ and $G$ with respective eigenvalues $E_n={\sum }_j{\epsilon }_j{n}_j$, $\mathrm{\mid }n\mathrm{\mid }$ and $g_n={\sum }_j({\epsilon }_j-\mu )n_j$, each on its exact domain $\{{\sum }_n{a}_n{b}_n^{\sigma }:{\sum }_n\mathrm{\mid }{w}_n{\mathrm{\mid }}^2\mathrm{\mid }{a}_n{\mathrm{\mid }}^2<\mathrm{\infty }\}$ for the corresponding weight $w_n$, by [F5]. The self-adjoint $G$ is the closure of the common finite-occupation action $H-\mu \mathcal{N}$, since finite sums are its graph core; it is not an assertion about the domain of a formal unbounded subtraction. In the finite bosonic-product case $g_n\ge 0$. In the finite fermionic-product case summability gives ${\epsilon }_j\to \mathrm{\infty }$, so $g_n\ge {\sum }_{j:{\epsilon }_j<\mu }({\epsilon }_j-\mu )>-\mathrm{\infty }$, the sum having finitely many terms. The coordinate spectral PVM is $E_G(B){\sum }_n{a}_n{b}_n^{\sigma }={\sum }_n{\mathbf{1}}_B(g_n)a_n{b}_n^{\sigma }$, and spectral integration gives the bounded positive $e^{-\beta G}$ with weights ${\prod }_j{q}_j^{n_j}$, where $q_j=e^{-\beta ({\epsilon }_j-\mu )}$. Its extended positive trace is the nonnegative sum of these weights by the ensemble postulate. Restricting support to $\{1,\dots ,M\}$ gives ${\prod }_{j=1}^M(1-q_j{)}^{-1}$ for bosons with all $q_j<1$, and ${\prod }_{j=1}^M(1+q_j)$ for fermions, by geometric summation or expansion of the finite binary product. These support sets exhaust ${\mathcal{C}}^{\sigma }$, so the full sum is the increasing limit of those products; if a bosonic $q_j\ge 1$, its one-mode subseries already diverges. [F2] now proves precisely the finiteness criteria in (c), including the bosonic strict chemical-potential condition. In each finite case the ensemble postulate identifies this sum with the ordinary trace of a trace-class operator and licenses normalization to a density. For each $j$, the same diagonal lemma defines the mode readout $N_j$ with weights $n_j$ on its exact squared-weight domain. The normalized marginal of each mode is proportional to $q_j^k$, $k\ge 0$ for bosons and $k\in \{0,1\}$ for fermions, by separating that mode in the same nonnegative sums; its mean is therefore the corresponding factor of [F2]. This proves (c) on the explicitly realized number-variable spaces.

[Readings] In each case the occupation number is an assigned sharp observable by [F4], so the mean of its outcome law in the canonical preparation is the number computed in steps 1.1-1.3 and 2.1; the identification of the computed parameters with a physical temperature and chemical potential is the adopted assumption of the ensemble postulate, and no convergence statement beyond the displayed criteria is asserted. This gives the readings of (a)-(d).

[Conclusion and scope] Steps 1.1-1.3, 2.1 and 3.1 prove (a)-(d). The argument is a finite-mode or product computation at assigned temperature; no thermodynamic limit, continuum of modes, condensation or measured occupation entered, which is (e).