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Qm Many Body And Equilibrium Models

1 · Prerequisites

2 · Summary

The local mathematical prerequisites make the realization and integrability scope explicit: CC realization of the bounded spatial operator on a finite-component H² domain treats the bounded spatial operator on its full finite-component H² domain under Countable Choice; Finite spin components of an N-fold spatial tensor product supplies the complete finite spin-component unitary and permutation action; Fixed-particle-number occupation bases and the countable-mode partition criterion treats supplied countably infinite modes at fixed positive particle number; and Spectral first moments and trace products with a density operator separates spectral first moments from identities for literal defined trace-class products. These are local conditional mathematical results; their actual hypotheses remain those of the item Statements.

The declared Coulomb branch uses couplings $g$ and $κ$ in $J m$ . Its one-particle energy includes the bounded plus singular external potential $V_{e x t}^{b} - g / ∣ x ∣$ through the supplied one-particle self-adjoint realization; the pair part is $κ / ∣ x_{i} - x_{j} ∣$ . Non-interacting means $W = 0$ in every realization, with the equivalent condition $κ = 0$ confined to this Coulomb branch. The occupation observables in Thermal occupation means of the declared free-mode models: Bose and Fermi factors, product convergence and dilute remainders have the dimensionless particle-count unit.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The one-body reduction trace law of the finite-mode permutation sectors

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space of finite dimension $d \geq 1$ (Hilbert space), let $n \geq 1$ , let $H^{\otimes n}$ be the $n$ -fold completed tensor power with $H^{\otimes 0} : = C$ , $H^{\otimes 1} : = H$ and $H^{\otimes (k + 1)} : = H^{\otimes k} \hat{\otimes} H$ , let $(U_{π})_{π \in S_{n}}$ be the permutation unitaries and let $P_{σ}^{(n)}$ , $σ \in {+, -}$ , be the sector projections (Permutation sector projections on the n-fold tensor power), with sector space $H_{n}^{σ} : = ran P_{σ}^{(n)} \subseteq H^{\otimes n} .$ For $O \in B (H)$ and $i \in {1, \dots, n}$ let $O^{(i)} \in B (H^{\otimes n})$ be the operator of clause (a) below, and put $Σ (O) : = \sum_{i = 1}^{n} O^{(i)} \in B (H^{\otimes n}) .$ Identify $H^{\otimes n}$ with $H \hat{\otimes} H^{\otimes (n - 1)}$ by the surjective linear isometry $J_{n}$ of clause (b) below, and let ${Tr}_{H^{\otimes (n - 1)}}$ be the partial trace of Finite partial trace, Schmidt decomposition and finite no-signalling clause (a) over the second factor of that identification. Then:

(a) slot operators: for every $O \in B (H)$ and every $i \in {1, \dots, n}$ there is a unique $O^{(i)} \in B (H^{\otimes n})$ with $O^{(i)} (v_{1} \otimes \dots \otimes v_{n}) = v_{1} \otimes \dots \otimes v_{i - 1} \otimes O v_{i} \otimes v_{i + 1} \otimes \dots \otimes v_{n}$ for all $v_{1}, \dots, v_{n} \in H$ ; it satisfies $∥ O^{(i)} ∥ \leq ∥ O ∥$ and $(O^{*})^{(i)} = (O^{(i)})^{*}$ ; in particular $O = O^{*}$ gives $O^{(i)} = (O^{(i)})^{*}$ ;

(b) parenthesisation: the assignment $v_{1} \otimes \dots \otimes v_{n} \mapsto v_{1} \otimes (v_{2} \otimes \dots \otimes v_{n})$ extends to a surjective linear isometry $J_{n} : H^{\otimes n} \to H \hat{\otimes} H^{\otimes (n - 1)}$ ; for $n = 1$ the empty product is $1 \in C$ and $J_{1} u = u \otimes 1$ ;

(c) permutation invariance: $U_{π} Σ (O) U_{π}^{*} = Σ (O)$ for every $π \in S_{n}$ ; consequently $Σ (O)$ preserves both sector spaces and commutes with $P_{σ}^{(n)}$ : $Σ (O) P_{σ}^{(n)} = P_{σ}^{(n)} Σ (O) = P_{σ}^{(n)} Σ (O) P_{σ}^{(n)};$

(d) trace law: for every density operator $ρ$ on the sector space $H_{n}^{σ}$ (Density operators and scalar spectral probability clause (a)), let $j : H_{n}^{σ} ↪ H^{\otimes n}$ be the isometric inclusion and define its zero extension $\tilde{ρ} : = j ρ j^{*} \in B (H^{\otimes n})$ ; then for every $O \in B (H)$ one has $tr (\tilde{ρ} Σ (O)) = n tr (γ O)$ , where $γ : = {Tr}_{H^{\otimes (n - 1)}} (J_{n} \tilde{ρ} J_{n}^{- 1}) \in B (H)$ ;

(e) the reduced operator is a density operator: the operator $γ$ of clause (d) is a density operator on $H$ ; it is independent of the choice of slot in the sense that it equals the partial trace of $\tilde{ρ}$ over the other $n - 1$ slots in the parenthesisation in which the retained slot is the one the display names; for $n = 1$ , $γ = ρ$ . If $H_{n}^{σ} = {0}$ , there is no density operator of trace one on that sector, so clauses (d)-(e) make no assertion about a sector preparation;

(f) scope: clauses (a)-(e) are exact operator identities on the finite-dimensional labelled tensor powers; no physical system, particle species, preparation or measurement is asserted, and the parameter $n$ is a dimension count of the construction, not an observed particle number.

Facts & Assumptions

Slot convention. For $ℓ \geq 1$ , $S_{ℓ}$ acts on ${0, \dots, ℓ - 1}$ as in Permutation sector projections on the n-fold tensor power. On one-based tuple positions use $\hat{π} (k) : = 1 + π (k - 1)$ for $1 \leq k \leq ℓ$ . Conjugation by $k \mapsto k - 1$ gives $\hat{π σ} = \hat{π} \circ \hat{σ}$ and $\hat{π^{- 1}} = {\hat{π}}^{- 1}$ ; thus the supplier's tensor action becomes $U_{π} (v_{1} \otimes \dots \otimes v_{ℓ}) = v_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes v_{{\hat{π}}^{- 1} (ℓ)}$ . The parameters of $U_{π}$ and $sgn (π)$ remain canonical zero-based permutations.

Given: the finite-dimensional complex Hilbert space $H$ of dimension $d \geq 1$ , the integer $n \geq 1$ , the completed tensor powers $H^{\otimes n}$ with permutation unitaries $U_{π}$ and sector projections $P_{σ}^{(n)}$ , the operator $O \in B (H)$ , the slot index $i$ , and the density operator $ρ$ on $H_{n}^{σ}$ with its zero extension $\tilde{ρ}$ .

[F1]

Tensor powers. $H^{\otimes n}$ is a complex Hilbert space; the elementary tensors have dense linear span; $∥ v_{1} \otimes \dots \otimes v_{n} ∥ = ∥ v_{1} ∥ \dots ∥ v_{n} ∥$ and $⟨ v_{1} \otimes \dots \otimes v_{n}, w_{1} \otimes \dots \otimes w_{n} ⟩ = \prod_{k} ⟨ v_{k}, w_{k} ⟩$ ; for orthonormal families spanning $H$ the corresponding elementary tensors form an orthonormal family with dense span (Hilbert tensor construction: inner product, completion and product bases clauses (b), (c), (d)).

[F2]

Parenthesisation. For every parenthesisation the iterated tensor product represents multilinear maps, and different parenthesisations are connected by the unique isomorphism preserving pure tensors (Finite iterated tensor products represent multilinear maps independently of parenthesization).

[F3]

Bounded tensor maps. For complex Hilbert spaces $V, W$ and $S \in B (V)$ , $T \in B (W)$ there is a unique $S \otimes T \in B (V \hat{\otimes} W)$ with $(S \otimes T) (u \otimes v) = S u \otimes T v$ , $∥ S \otimes T ∥ \leq ∥ S ∥ ∥ T ∥$ , $(S \otimes T)^{*} = S^{*} \otimes T^{*}$ , and for endomorphisms $S \otimes I$ and $I \otimes T$ commute (Bounded tensor maps: extension, norm estimate, adjoint and commutation clauses (a), (b), (c)).

[F4]

Permutation unitaries and sectors. $U_{π}$ is unitary with $U_{π} (v_{1} \otimes \dots \otimes v_{n}) = v_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes v_{{\hat{π}}^{- 1} (n)}$ and $U_{π} U_{σ} = U_{π σ}$ ; $U_{π} P_{+}^{(n)} = P_{+}^{(n)} U_{π} = P_{+}^{(n)}$ and $U_{π} P_{-}^{(n)} = P_{-}^{(n)} U_{π} = sgn (π) P_{-}^{(n)}$ ; the range of $P_{+}^{(n)}$ is the subspace of vectors fixed by every $U_{π}$ , and the range of $P_{-}^{(n)}$ is the subspace of vectors transforming by $sgn$ (Permutation sector projections on the n-fold tensor power clauses (a), (b), (c)).

[F5]

Partial trace. For endomorphisms $R$ of $V \otimes W$ with $V, W$ finite-dimensional, there is a unique endomorphism ${Tr}_{W} R$ of $V$ with $tr (({Tr}_{W} R) A) = tr (R (A \otimes I_{W}))$ for every $A \in End (V)$ , independent of the orthonormal basis used in its construction; if $R \geq 0$ and $tr R = 1$ then ${Tr}_{W} R \geq 0$ and $tr {Tr}_{W} R = 1$ (Finite partial trace, Schmidt decomposition and finite no-signalling clauses (a), (b)).

[F6]

Density operators. A density operator on a nonzero complex Hilbert space is a self-adjoint positive trace-class operator of trace one (Density operators and scalar spectral probability clause (a), Trace class operator, Trace of a trace class operator).

[F7]

Trace laws. The trace of an endomorphism of a finite-dimensional space is basis independent, linear, and $tr (X Y) = tr (Y X)$ (The basis-independent trace of an endomorphism of a finite-dimensional vector space, Trace is a linear functional on $M_{n} (F)$ ); for a trace-class operator $T$ and bounded $S$ one has $tr (S T) = tr (T S)$ (Cyclicity of the trace).

[F8]

Agreement on dense sets. Two bounded linear operators from a normed space into a Banach space that agree on a dense subspace are equal (A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm).

[F9]

Unitary conjugation preserves the operator norm. If $U$ is unitary then $∥ U T U^{*} ∥ = ∥ T ∥$ (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces).

Proof

technique · direct; construct the slot operators by conjugating the first-slot tensor map with a permutation unitary, transport the partial trace through the parenthesisation isometry, and reduce every slot to the first by permutation invariance and cyclicity

Given: the data of the statement and [F1]-[F9].

1.1F1F2F8algebra

[The parenthesisation isometry] For $n = 1$ , $J_{1} u = u \otimes 1$ is an isometry by [F1], and every tensor in $H \hat{\otimes} C$ equals $u \otimes 1$ , so it is surjective. For $n \geq 2$ , the map $(v_{1}, \dots, v_{n}) \mapsto v_{1} \otimes (v_{2} \otimes \dots \otimes v_{n})$ is multilinear from $H^{n}$ into $H \hat{\otimes} H^{\otimes (n - 1)}$ , so by [F2] applied to the two parenthesisations of the $n$ -fold algebraic tensor product it defines a linear map on the algebraic tensor product, and this linear map preserves inner products on elementary tensors by [F1]: $⟨ v_{1} \otimes \dots \otimes v_{n}, w_{1} \otimes \dots \otimes w_{n} ⟩ = \prod_{k} ⟨ v_{k}, w_{k} ⟩ = ⟨ v_{1}, w_{1} ⟩ ⟨ v_{2} \otimes \dots \otimes v_{n}, w_{2} \otimes \dots \otimes w_{n} ⟩$ . Since the elementary tensors span a dense subspace and both sides are complete, [F8] and this isometry identity give a linear isometry $J_{n}$ on the completed tensor powers, and the image is closed and contains all elementary tensors $v_{1} \otimes (v_{2} \otimes \dots \otimes v_{n})$ , whose span is dense by [F1]; hence $J_{n}$ is surjective. This proves (b).

1.2F1F3algebra

[The first-slot operator] By [F3] the operator $O \otimes I_{H^{\otimes (n - 1)}} \in B (H \hat{\otimes} H^{\otimes (n - 1)})$ is bounded with $(O \otimes I) (u \otimes w) = O u \otimes w$ . Transporting it through the isometry $J_{n}$ of clause (b) gives a bounded operator $A_{1} : = J_{n}^{- 1} (O \otimes I) J_{n}$ on $H^{\otimes n}$ with $A_{1} (v_{1} \otimes \dots \otimes v_{n}) = J_{n}^{- 1} (O v_{1} \otimes (v_{2} \otimes \dots \otimes v_{n})) = O v_{1} \otimes v_{2} \otimes \dots \otimes v_{n}$ for all $v_{1}, \dots, v_{n} \in H$ .

1.3F4F6F7algebra

[The zero extension and its symmetry] With $j$ the inclusion of the sector, $\tilde{ρ} = j ρ j^{*}$ restricts to $ρ$ on the sector and vanishes on its orthogonal complement. It is self-adjoint and positive because $⟨ \tilde{ρ} x, x ⟩ = ⟨ ρ j^{*} x, j^{*} x ⟩ \geq 0$ ; adjoining an orthonormal basis of the complement to one of the sector gives $tr \tilde{ρ} = tr ρ = 1$ , so it is a density operator by [F6]. Since $j j^{*} = P_{σ}^{(n)}$ and $U_{π} j = {sgn}_{σ} (π) j$ by [F4], $\tilde{ρ} = P_{σ}^{(n)} \tilde{ρ} P_{σ}^{(n)}$ and $U_{π} \tilde{ρ} U_{π}^{*} = \tilde{ρ}$ for every $π$ . A zero sector admits no trace-one endomorphism, so no $ρ$ exists in that case.

2.1F1F3F4F8F9step 1.2algebra

[Slot operators and the permutation invariance of their sum] Fix $i$ , choose $π \in S_{n}$ with $\hat{π} (1) = i$ and set $O^{(i)} : = U_{π} A_{1} U_{π}^{*} \in B (H^{\otimes n})$ ; [F9] and step 1.2 give $∥ O^{(i)} ∥ = ∥ O \otimes I ∥ \leq ∥ O ∥$ , and [F3], [F4] give $(O^{(i)})^{*} = U_{π} J_{n}^{- 1} (O^{*} \otimes I) J_{n} U_{π}^{*} = (O^{*})^{(i)}$ , so $O = O^{*}$ gives $O^{(i)} = (O^{(i)})^{*}$ . On elementary tensors, [F4] gives $U_{π}^{*} (v_{1} \otimes \dots \otimes v_{n}) = v_{\hat{π} (1)} \otimes \dots \otimes v_{\hat{π} (n)}$ ; step 1.2 replaces the first factor by $O v_{\hat{π} (1)} = O v_{i}$ ; and $U_{π}$ then moves that factor to slot $\hat{π} (1) = i$ , leaving every other factor in its original slot, so $O^{(i)} (v_{1} \otimes \dots \otimes v_{n}) = v_{1} \otimes \dots \otimes O v_{i} \otimes \dots \otimes v_{n}$ . If $π^{'}$ is another permutation with $\hat{π^{'}} (1) = i$ , then $τ : = π^{- 1} π^{'}$ has $\hat{τ} (1) = 1$ and $U_{τ} A_{1} U_{τ}^{*}$ agrees with $A_{1}$ on elementary tensors by [F1], [F3], hence equals it by [F8]; thus $U_{π^{'}} A_{1} U_{π^{'}}^{*} = U_{π} U_{τ} A_{1} U_{τ}^{*} U_{π}^{*} = U_{π} A_{1} U_{π}^{*}$ and $O^{(i)}$ is well defined, with the displayed elementary-tensor action. For $τ \in S_{n}$ one then has $U_{τ} O^{(i)} U_{τ}^{*} = U_{τ π} A_{1} U_{τ π}^{*} = O^{(\hat{τ} (i))}$ , and summing over $i$ with $\hat{τ}$ permuting ${1, \dots, n}$ gives $U_{τ} Σ (O) U_{τ}^{*} = Σ (O)$ for $Σ (O) : = \sum_{i = 1}^{n} O^{(i)}$ ; equivalently $Σ (O)$ commutes with every $U_{τ}$ . Since $P_{σ}^{(n)} = \frac{1}{n!} \sum_{π \in S_{n}} {sgn}_{σ} (π) U_{π}$ is a finite linear combination of the $U_{π}$ by [F4], $Σ (O)$ commutes with $P_{σ}^{(n)}$ , so it preserves $H_{n}^{σ}$ and its orthogonal complement and $Σ (O) P_{σ}^{(n)} = P_{σ}^{(n)} Σ (O) = P_{σ}^{(n)} Σ (O) P_{σ}^{(n)}$ ; this proves (a) and (c).

3.1F4F5F6F7step 1.1step 1.2step 1.3step 2.1algebra

[Reduction to one slot, the trace law and the reduced operator] By steps 1.3 and 2.1 and [F7], for $\hat{π} (1) = i$ one has $tr (\tilde{ρ} O^{(i)}) = tr (U_{π}^{*} \tilde{ρ} U_{π} A_{1}) = tr (\tilde{ρ} A_{1})$ . Set $R_{n} : = J_{n} \tilde{ρ} J_{n}^{- 1}$ , an endomorphism of $H \hat{\otimes} H^{\otimes (n - 1)}$ . Its partial trace $γ : = {Tr}_{H^{\otimes (n - 1)}} R_{n}$ is an endomorphism of $H$ by [F5]. Step 1.2 and [F5], [F7] give $tr (\tilde{ρ} Σ (O)) = n tr (\tilde{ρ} A_{1}) = n tr (R_{n} (O \otimes I)) = n tr (γ O)$ , proving (d). Unitary transport by $J_{n}$ preserves positivity and trace, so $R_{n}$ is positive of trace one; [F5] then makes $γ$ positive of trace one and hence a density operator on $H$ by [F6]. For the retained slot $i$ , use $J_{n} U_{π}^{*}$ as the parenthesisation map; its transported state is $J_{n} U_{π}^{*} \tilde{ρ} U_{π} J_{n}^{- 1} = R_{n}$ by step 1.3, so its partial trace is the same $γ$ . More generally the pairing $tr (γ O) = n^{- 1} tr (\tilde{ρ} Σ (O))$ for every $O$ uniquely determines $γ$ : testing against $O = T_{u, v}$ gives $tr (γ T_{u, v}) = ⟨ γ u, v ⟩$ by Rank-one operator calculus clause (e), which determines every matrix entry. For $n = 1$ , $J_{1} u = u \otimes 1$ , $P_{σ}^{(1)} = I$ and $Σ (O) = O$ , so this pairing gives $γ = ρ$ . This proves (e), including the one-particle boundary.

4.1step 1.1step 2.1step 3.1∎

[Conclusion and scope] Step 1.1 proves (b), step 2.1 proves (a) and (c), and step 3.1 proves (d) and (e); every argument used only the finite-dimensional tensor construction, the permutation sectors, the partial trace and the trace axioms, and no physical object, preparation or measurement entered, which is clause (f).

Remarks

Reading of the identities. The law $tr (\tilde{ρ} Σ (O)) = n tr (γ O)$ says that the mean of the sum of one-body slot operators in the sector preparation is $n$ times the trace pairing of a single one-particle operator with a reduced operator $γ$ on the one-particle space. Only the identity is asserted here; its physical reading requires the framework postulates and a declared preparation, and is stated separately.
Why the factor $n$ appears. The sum $Σ (O)$ has $n$ slot terms with equal pairings by permutation invariance of the sector state, and $γ$ is normalised once, $tr γ = 1$ ; the factor $n$ is the multiplicity of the sum, not a further normalisation.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Closed formulas for the two-level canonical weights, mean energy, entropy and free energy

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $ε > 0$ and $β > 0$ be real parameters, let $h$ be a two-dimensional complex Hilbert space with orthonormal basis $(u_{0}, u_{1})$ (Hilbert space, Orthonormal families, complete orthonormal systems and Hilbert bases), and let $H : = T_{u_{1}, u_{1}} ε = ε T_{u_{1}, u_{1}}$ be the self-adjoint operator with eigenvalues $0$ on $C u_{0}$ and $ε$ on $C u_{1}$ (Rank-one operator calculus). Let $Z : = \sum_{j = 0}^{1} e^{- β E_{j}}, ρ_{β} : = \frac{1}{Z} \sum_{j = 0}^{1} e^{- β E_{j}} T_{u_{j}, u_{j}}, s (ρ) : = - \sum_{i : w_{i} > 0} w_{i} \log w_{i}$ be the partition sum, canonical operator and entropy of Finite Gibbs free-energy minimum clause (a), with $E_{0} = 0$ , $E_{1} = ε$ and $w_{i}$ the eigenvalues of a density operator $ρ$ on $h$ (The natural logarithm as the inverse of the exponential function). Then:

(a) $Z = 1 + e^{- β ε} \in (1, 2)$ , and $ρ_{β} = \frac{1}{Z} (T_{u_{0}, u_{0}} + e^{- β ε} T_{u_{1}, u_{1}})$ is a density operator on $h$ that is not the maximally mixed state $\frac{1}{2} i d$ ;

(b) the mean energy is the exact identity $tr (ρ_{β} H) = \frac{ε e^{- β ε}}{Z} = \frac{ε}{e^{β ε} + 1}, 0 < tr (ρ_{β} H) < \frac{ε}{2};$

(c) the entropy and free energy satisfy $s (ρ_{β}) = \log Z + \frac{β ε}{e^{β ε} + 1} = \log (1 + e^{- β ε}) + \frac{β ε e^{- β ε}}{1 + e^{- β ε}}, 0 \leq s (ρ_{β}) < \log 2,$ and, with $F : = - β^{- 1} \log Z$ , $tr (ρ_{β} H) - β^{- 1} s (ρ_{β}) = F;$

(d) scope: clauses (a)-(c) are exact algebraic identities for the declared two-dimensional Hilbert space and the two real parameters $ε, β > 0$ ; no temperature, thermodynamic, equilibrium or measurement interpretation is asserted, and the parameter $β$ carries no physical dimension in this lemma.

Facts & Assumptions

Given: the real parameters $ε > 0$ , $β > 0$ , the two-dimensional complex Hilbert space $h$ with orthonormal basis $(u_{0}, u_{1})$ , the operator $H = ε T_{u_{1}, u_{1}}$ , and the data $Z$ , $ρ_{β}$ , $s$ of the statement.

[F1]

Supplied finite Gibbs data. For the finite-dimensional self-adjoint $H$ with real eigenvalues $E_{1}, \dots, E_{d}$ and orthonormal eigenbasis $(v_{1}, \dots, v_{d})$ , and $β > 0$ , one has $Z = \sum_{j} e^{- β E_{j}} \in (0, \infty)$ and $ρ_{β} = Z^{- 1} \sum_{j} e^{- β E_{j}} T_{v_{j}, v_{j}} \in D$ , with $tr (ρ_{β} H) = Z^{- 1} \sum_{j} e^{- β E_{j}} E_{j}$ , $tr (ρ_{β} H) - β^{- 1} s (ρ_{β}) = - β^{- 1} \log Z$ , and for every density operator $ρ$ one has $s (ρ) \leq \log d$ , with equality if and only if $ρ = d^{- 1} i d$ (Finite Gibbs free-energy minimum clauses (a), (b), (c)).

[F2]

Rank-one calculus. $T_{u, v} x = ⟨ x, v ⟩ u$ is bounded with $T_{u, v}^{*} = T_{v, u}$ , $T_{x, x} \geq 0$ , $tr (T_{u, v}) = ⟨ u, v ⟩$ , and for bounded $B$ one has $B T_{u, v} = T_{B u, v}$ and $T_{u, v} B = T_{u, B^{*} v}$ (Rank-one operator calculus clauses (a)-(d)); the trace is linear (Trace is a linear functional on $M_{n} (F)$ ) and $tr (X Y) = tr (Y X)$ (Cyclicity of the trace).

[F3]

Real exponential. $\exp (x) > 0$ , $\exp (- x) = 1 / \exp (x)$ , and $\exp (x + y) = \exp (x) \exp (y)$ for all real $x, y$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ ); $\exp$ is strictly increasing on $R$ (The exponential function is strictly increasing). Hence $\exp (0) = \exp (0)^{2}$ with $\exp (0) > 0$ , so $\exp (0) = 1$ , and $\exp (t) > \exp (0) = 1$ for every $t > 0$ .

[F4]

Logarithm. $\log : (0, \infty) \to R$ is the inverse function of $\exp$ and satisfies $\log (x y) = \log x + \log y$ , $\log (1 / x) = - \log x$ and $\log 1 = 0$ (The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

Proof

technique · direct; instantiate the supplied finite Gibbs formulas at the two eigenvalues $0,\varepsilon$, and control the exponential factor at $\beta\varepsilon>0$ with strict increase, positivity and the reciprocal law

Given: the data of the statement and [F1]-[F4].

1.1F3algebra

[Exponential comparison] Since $β ε > 0$ , strict increase in [F3] gives $\exp (β ε) > \exp (0) = 1$ , and [F3] gives $\exp (- β ε) = 1 / \exp (β ε) \in (0, 1)$ ; consequently $1 < Z = 1 + \exp (- β ε) < 2$ .

2.1F1step 1.1algebra

[The canonical operator] By [F1] with $E_{0} = 0$ , $E_{1} = ε$ and $(v_{1}, v_{2}) = (u_{0}, u_{1})$ , $Z = 1 + e^{- β ε}$ and $ρ_{β} = Z^{- 1} (T_{u_{0}, u_{0}} + e^{- β ε} T_{u_{1}, u_{1}}) \in D$ , so $ρ_{β}$ is a density operator; its two weights are $Z^{- 1}$ and $e^{- β ε} Z^{- 1}$ , which are distinct because $\exp (- β ε) \neq 1$ by step 1.1.

3.1F1F2F4step 1.1step 2.1algebra

[Mean energy, entropy and free energy] Using [F2] and $T_{u_{0}, u_{0}} T_{u_{1}, u_{1}} = ⟨ u_{1}, u_{0} ⟩ T_{u_{0}, u_{1}} = 0$ by orthogonality, and $T_{u_{1}, u_{1}} T_{u_{1}, u_{1}} = T_{u_{1}, u_{1}}$ with $⟨ u_{1}, u_{1} ⟩ = 1$ , the product $ρ_{β} H$ equals $ε e^{- β ε} Z^{- 1} T_{u_{1}, u_{1}}$ , so [F2] gives $tr (ρ_{β} H) = ε e^{- β ε} / Z$ ; by step 1.1, $e^{- β ε} / Z = e^{- β ε} / (1 + e^{- β ε}) = 1 / (e^{β ε} + 1)$ , hence $tr (ρ_{β} H) = ε / (e^{β ε} + 1)$ lies in $(0, ε / 2)$ . By [F1], $tr (ρ_{β} H) - β^{- 1} s (ρ_{β}) = - β^{- 1} \log Z$ , so $s (ρ_{β}) = \log Z + β tr (ρ_{β} H) = \log (1 + e^{- β ε}) + β ε / (e^{β ε} + 1) = \log (1 + e^{- β ε}) + β ε e^{- β ε} / (1 + e^{- β ε})$ ; by [F1] and [F2], $0 \leq s (ρ_{β}) < \log 2$ because $ρ_{β} \neq \frac{1}{2} i d$ by step 2.1. Finally $tr (ρ_{β} H) - β^{- 1} s (ρ_{β}) = - β^{- 1} \log Z = F$ with $F$ as defined. This proves (b) and (c), and step 2.1 proved (a).

4.1step 1.1step 2.1step 3.1∎

[Conclusion] Steps 1.1-2.1 prove (a), step 3.1 proves (b) and (c), and every identity is an algebraic consequence of the supplied finite-dimensional formulas and the real exponential and logarithm laws, which is (d).

Remarks

High- and low-temperature expectations. By [F3], $0 < e^{- β ε} < 1$ and $e^{- β ε}$ decreases strictly as the product $β ε$ grows, so the upper level is suppressed in the regime of large $β ε$ , while the bound $s < \log 2$ of clause (c) shows that the entropy stays below the maximal mixed-state value for every finite $β > 0$ . The lemma asserts only these algebraic facts about the weights; it identifies no physical temperature with $β$ .
Use. The formulas are the exact two-level data used by the examples and the thought experiment of the companion page; the physical reading of $β$ as an inverse temperature, and of $ρ_{β}$ as a thermal preparation, is supplied separately by the framework's equilibrium postulate.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

CC realization of the bounded spatial operator on a finite-component H² domain

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix integers $N, M \geq 1$ , $a > 0$ , and a real measurable function $V : R^{3 N} \to R$ with $v = ess sup ∣ V ∣ < \infty$ . Put $H = L^{2} (R^{3 N}; C^{M}), Y = W^{2, 2} (R^{3 N}; C^{M}) = H^{2} (R^{3 N}; C^{M}),$ where the vector space, pairing and Sobolev norm are the finite component sums of the scalar conventions of Hilbert space, The space $L^{p} (μ)$ as the quotient by null functions and Integer-order Sobolev spaces and their norms. Use the componentwise Plancherel unitary of Plancherel theorem, with the negative-sign $2 π$ convention of Fourier transform on complex L1 classes. Define $w (ξ) = 4 π^{2} a ∣ ξ ∣^{2}, K = F_{2}^{- 1} M_{w} F_{2}, D (K) = {u : w F_{2} u \in H} .$ Then $D (K) = Y$ , and $K u = - a Δ u$ for every $u \in Y$ , with the distributional Laplacian acting componentwise. The operator $H = K + M_{V} = - a Δ + V, D (H) = Y,$ is densely defined and self-adjoint on exactly $Y$ , and $H \geq - v$ . Both $K$ and $H$ have graph norms equivalent to the stated Sobolev norm, and $C_{c}^{\infty} (R^{3 N}; C^{M})$ is a core for both. For $z = \pm i μ$ with $μ > v$ , there is the concrete two-sided inverse $R_{H} (z) = (H - z)^{- 1} = R_{0} (z) \sum_{k \geq 0} (- M_{V} R_{0} (z))^{k}, R_{0} (z) = F_{2}^{- 1} M_{(w - z)^{- 1}} F_{2},$ with range exactly $Y$ and $∥ R_{H} (z) ∥ \leq (μ - v)^{- 1}$ .

This realizes the equal-mass bounded spatial operator used at fixed $N$ by setting $a = ℏ^{2} / (2 m)$ and any finite spin multiplicity $M$ ; for an $N$ -fold spin- $s$ tensor the full multiplicity is $M = (2 s + 1)^{N}$ . The lemma makes no statement about Coulomb potentials, unequal-mass operators, thermodynamic limits, or physical assignments. It uses no Axiom of Choice or Kato–Rellich theorem.

Facts & Assumptions

Given: CC; $N, M, a, V, v, H, Y, w$ as above. Pairings are linear in the first argument.

[F1]

Under CC, the scalar Plancherel transform is a surjective complex-linear isometry preserving the pairing (Plancherel theorem, Statement). Under CC, $W^{2, 2}$ equals the weighted Fourier space, with its Sobolev norm equivalent to $∥ (1 + ∣ ξ ∣^{2}) F_{2} u ∥_{2}$ , and the $2 π$ derivative multiplier is $F_{2} (D^{α} u) = (2 π i ξ)^{α} F_{2} u$ when the weak derivative is in $L^{2}$ (Integer-order W^{k,2} and H^k agree with equivalent norms, Statement (1),(2), Distributional derivatives are polynomial Fourier multipliers, Statement). Finite component sums preserve these identities and norm equivalences.

[F2]

For a densely defined operator, the adjoint domain consists of vectors admitting a representing vector in its pairing identity; a symmetric operator is self-adjoint exactly when its adjoint has the same domain and action (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators). A self-adjoint operator is closed.

[F3]

Under CC, $C_{c}^{\infty} (R^{3 N}; C)$ is dense in $W^{2, 2}$ and in $L^{2}$ (Compactly supported smooth functions are dense in W^{k,p}(R^n), Statement with $(k, p) = (2, 2)$ and $(0, 2)$ ). The same holds for finitely many components. Dominated convergence holds for the squared cutoff tails (Dominated convergence).

Proof

Given: The data of the statement and [F1]–[F3].

1.1givenF2F3algebra

[Concrete multiplier, including the adjoint domain] For any real finite-valued measurable function $b$ on $R^{3 N}$ , multiplication on $D (M_{b}) = {f : b f \in H}$ is densely defined: $f_{L} = 1_{∣ b ∣ \leq L} f$ belongs to this domain and tends to $f$ by dominated convergence. It is symmetric because $b$ is real and the pairings are integrable by Cauchy–Schwarz. If $g \in D (M_{b}^{*})$ with representing vector $h$ , every $f \in H$ supported on $E_{L} = {∣ b ∣ \leq L}$ belongs to the multiplier domain and satisfies $⟨ f, b g - h ⟩ = 0$ on that set. The vector $1_{E_{L}} (b g - h)$ lies in $H$ , so testing with that vector proves $h = b g$ a.e. on $E_{L}$ . Increasing the integer $L$ covers the space up to null sets, proving $b g = h \in H$ and $g \in D (M_{b})$ . Conversely the symmetry identity gives every vector of $D (M_{b})$ in the adjoint domain with adjoint value $b g$ . Thus $M_{b}$ is self-adjoint on the exact maximal domain, including for the weight $w$ . This proof is valid for the finite vector-valued space by its component-sum pairing; it makes no arbitrary family of choices.

2.1step 1.1F1F2algebra

[Fourier transfer, exact H² and normalization] Apply the componentwise unitary $F_{2}$ from [F1] to step 1.1 with $b = w$ . For a vector $y$ in the adjoint domain of $K$ , changing variables $f = F_{2} u$ in its representing functional gives $F_{2} y \in D (M_{w}^{*}) = D (M_{w})$ with the corresponding adjoint value; the converse follows from symmetry. Hence $K$ is self-adjoint on $F_{2}^{- 1} D (M_{w})$ . Since $w = 4 π^{2} a ∣ ξ ∣^{2}$ with $a > 0$ , $1 + w$ is bounded above and below by positive multiples of $1 + ∣ ξ ∣^{2}$ . For $u \in L^{2}$ , $w F_{2} u \in L^{2}$ therefore holds exactly when $(1 + ∣ ξ ∣^{2}) F_{2} u \in L^{2}$ , which is $u \in Y$ by [F1]. Moreover $∥ F_{2} u ∥_{2} + ∥ w F_{2} u ∥_{2}$ is equivalent to $∥ (1 + w) F_{2} u ∥_{2}$ by the triangle inequality and nonnegativity of $w$ , and hence to $∥ u ∥_{Y}$ . This proves graph-norm equivalence. The weak-derivative multiplier in [F1] gives $F_{2} (- a Δ u) = 4 π^{2} a ∣ ξ ∣^{2} F_{2} u$ , proving the exact differential action for all $u \in Y$ . Finally $⟨ K u, u ⟩ = \int w ∥ F_{2} u ∥^{2} \geq 0$ , with integrability by Cauchy–Schwarz.

3.1step 2.1F1F2constructalgebra

[Explicit perturbed inverses on the fixed domain] Multiplication by $V$ is bounded and self-adjoint on $H$ , with norm at most $v$ , since $V$ is real and essentially bounded. Thus $H = K + M_{V}$ is symmetric on the dense $Y$ and is closed there: if $u_{l} \to u$ and $H u_{l} \to h$ , boundedness gives $K u_{l} \to h - M_{V} u$ , and closedness of $K$ gives $u \in Y$ with $H u = h$ . For $z = \pm i μ$ , the concrete multiplier $(w - z)^{- 1}$ is bounded by $μ^{- 1}$ and $∣ w / (w - z) ∣ \leq 1$ . Therefore $R_{0} (z)$ maps $H$ into $Y$ and satisfies both inverse equations of $K - z$ , with range exactly $Y$ . On $Y$ the correctly ordered factorization is $H - z = (I + M_{V} R_{0} (z)) (K - z)$ . Since $∥ M_{V} R_{0} (z) ∥ \leq v / μ < 1$ , the geometric series in the statement is the inverse of the first factor: finite-sum multiplication leaves a remainder tending to zero in norm. The bounded operator space is complete, because an operator-norm Cauchy sequence has a bounded pointwise limit on the complete target and the same uniform Cauchy estimate gives operator-norm convergence. Thus $R_{H} (z)$ exists, maps into $Y$ , and its two ordered inverse equations follow from the factorization. Surjectivity of this range onto $Y$ follows from the second inverse equation. Its norm is at most $μ^{- 1} / (1 - v / μ) = (μ - v)^{- 1}$ .

4.1step 2.1step 3.1F2algebra

[Self-adjointness without an AC perturbation theorem] Let $y \in D (H^{*})$ . By step 3.1, $H - i μ : Y \to H$ is onto; solve $(H - i μ) x = (H^{*} - i μ) y$ with its explicit inverse, so $x \in Y$ . Symmetry implies $H^{*} x = H x$ , whence $y - x \in \ker (H^{*} - i μ)$ . For a densely defined operator with first-linear pairing, this kernel is $ran (H + i μ)^{⊥}$ , directly from the adjoint identity. The other inverse from step 3.1 makes that range all of $H$ , so $y = x \in Y$ . Since $H \subseteq H^{*}$ by symmetry, the adjoint domains and actions are equal, proving self-adjointness on exactly $Y$ . Also $⟨ H u, u ⟩ = ⟨ K u, u ⟩ + \int V ∥ u ∥^{2} \geq - v ∥ u ∥^{2}$ for every $u \in Y$ , proving the claimed lower bound. Only the two explicit imaginary resolvents were used.

5.1step 2.1step 3.1step 4.1F1F2F3algebra∎

[Graph norms, cores and scope] For $u \in Y$ , boundedness of $V$ gives $∥ H u ∥ \leq ∥ K u ∥ + v ∥ u ∥$ and $∥ K u ∥ \leq ∥ H u ∥ + v ∥ u ∥$ . Thus the graph norms of $H$ and $K$ are equivalent, and step 2.1 identifies both with the Sobolev norm. The finite-component smooth density from [F3] therefore makes $C_{c}^{\infty} (R^{3 N}; C^{M})$ graph dense in the exact domain of each operator, which is precisely the core claim. Steps 1.1–4.1 supply the self-adjointness, exact differential action, lower bound and explicit inverses. All primitive suppliers used here assume CC; neither an AC multiplier theorem nor Kato–Rellich nor any assertion concerning a singular potential or an arbitrary spin basis was invoked.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Finite spin components of an N-fold spatial tensor product

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $N, r \geq 1$ be integers, $H = L^{2} (R^{3}; C^{r})$ , $S_{N} = (C^{r})^{\otimes N}$ , and $M = r^{N}$ . In the product basis $e_{a} = e_{a_{1}} \otimes \dots \otimes e_{a_{N}}$ indexed by $a \in {1, \dots, r}^{N}$ , identify $S_{N}$ with $C^{M}$ and write $L_{N} = L^{2} (R^{3 N}; C^{M})$ . Then:

(a) There is a unique unitary $Θ_{N} : H^{\otimes N} \to L_{N}$ such that $[Θ_{N} (f_{1} \otimes \dots \otimes f_{N})]_{a} (x_{1}, \dots, x_{N}) = \prod_{j = 1}^{N} [f_{j} (x_{j})]_{a_{j}} .$

(b) For $π \in S_{N}$ , let $U_{π}$ be the permutation of the complete one-particle factors and $R_{π} e_{a} = e_{a_{{\hat{π}}^{- 1} (1)}} \otimes \dots \otimes e_{a_{{\hat{π}}^{- 1} (N)}}$ the permutation of spin factors. With $X^{π} = (x_{\hat{π} (1)}, \dots, x_{\hat{π} (N)})$ , the conjugate permutation is $U_{π} : = Θ_{N} U_{π} Θ_{N}^{- 1}, (U_{π} Ψ) (X) = R_{π} Ψ (X^{π}) .$ In particular $[U_{π} Ψ]_{a} (X) = Ψ_{(a_{\hat{π} (1)}, \dots, a_{\hat{π} (N)})} (X^{π})$ ; averaging this combined action gives $P_{\pm} = Θ_{N} P_{\pm}^{(N)} Θ_{N}^{- 1}$ .

(c) The component domain $D_{N} = W^{2, 2} (R^{3 N}; C^{M})$ is the set of tuples with every scalar component in $W^{2, 2} (R^{3 N}; C)$ . If a scalar self-adjoint $T$ has this scalar domain and its domain is preserved by every spatial block permutation, commuting with that permutation on the domain, its finite direct sum $T_{M} Ψ = (T Ψ_{a})_{a}$ on $D_{N}$ is self-adjoint, retains every scalar lower bound $T \geq c I$ , and commutes with the combined permutations on that domain. The restrictions of $T_{M}$ to $ran P_{\pm}$ , with domain $D_{N} \cap ran P_{\pm}$ , are self-adjoint when the sector is nonzero; on the zero sector the unique zero operator has domain ${0}$ .

(d) For matrices $A_{j} \in B (C^{r})$ , the constant spin action $A_{1} \otimes \dots \otimes A_{N}$ preserves $D_{N}$ and commutes with $T_{M}$ . Its permutation covariance is $R_{π} (A_{1} \otimes \dots \otimes A_{N}) R_{π}^{- 1} = A_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes A_{{\hat{π}}^{- 1} (N)} .$ Thus a common internal unitary $A^{\otimes N}$ commutes with $U_{π}$ , preserves both sectors and commutes with their restricted operators. This is a tensor identity and asserts no physical spin-statistics assignment. For $r = 1$ all spin actions are scalar and $Θ_{N}$ is the iterated scalar product unitary; for $N = 1$ it is the identity component identification.

Facts & Assumptions

Given: $N, r, H, S_{N}, M, L_{N}$ , the product spin basis, and in (c) the scalar self-adjoint $T$ with domain $W^{2, 2}$ and spatial permutation invariance.

[F1]

The scalar Euclidean product map is unitary and permits finite component sums (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}) Statement); iterating it gives the scalar N-factor product map. The component identification is unitary for every Euclidean dimension and finite component count (Unitary identifications of the component-valued wave-function space (a), (b)).

[F2]

Tensor pairings multiply, elementary tensors span densely, and spin product bases are orthonormal; bounded factor maps extend with their tensor action and adjoints (Hilbert tensor construction: inner product, completion and product bases, Bounded tensor maps: extension, norm estimate, adjoint and commutation). Factor permutations and their symmetric/sign averages are unitary and orthogonal projections respectively (Permutation sector projections on the n-fold tensor power (a)-(c)).

[F3]

A self-adjoint operator invariant on its domain under a unitary finite-group permutation representation restricts self-adjointly to each nonzero symmetric/sign sector (Permutation-restricted self-adjointness (a)-(d)); the scalar-only application (e) is not needed for a spin sector.

Proof

Given: The data of the Statement and [F1]-[F3].

1.1F1F2givenalgebra

[Finite component product unitary] Resolve each one-particle vector into its $r$ components by [F1]. On the dense span of vectors $(g_{1} e_{a_{1}}) \otimes \dots \otimes (g_{N} e_{a_{N}})$ , send this vector to the function $(\prod_{j} g_{j} (x_{j})) e_{a}$ . Inner products of such vectors are the product of the scalar pairings times $\prod_{j} δ_{a_{j} b_{j}}$ by [F2]; their image pairings have exactly the same value by the iterated scalar unitary [F1]. Linearity therefore gives a well-defined isometry on this dense span. Its image is dense: in each of the finitely many components the scalar products span densely by [F1]. The isometry extends to the completion; its range is closed by completeness and dense, hence all of $L_{N}$ . Expansion of the finite components gives the formula in (a), and equality on a dense span gives uniqueness.

2.1step 1.1F2algebra

[Combined permutation] On the component product vectors of step 1.1, applying $U_{π}$ replaces the factor in slot $j$ by the former factor in slot ${\hat{π}}^{- 1} (j)$ . Evaluating the product formula gives $(U_{π} Ψ) (X) = R_{π} Ψ (X^{π})$ , including the displayed component-index formula. Both operators are bounded unitaries, so the equality extends by density. Conjugating the finite averages of [F2] gives $P_{\pm}$ ; this simultaneously permutes spatial blocks and spin slots.

3.1givenstep 2.1algebra

[Finite component domain and self-adjointness] Since there are finitely many components, the definition of weak derivatives shows $D_{N} = (W^{2, 2} (R^{3 N}; C))^{\oplus M}$ : derivative testing and integrability hold componentwise in both directions, and all norms are finite sums. The direct sum $T_{M}$ is dense and symmetric. A vector $η$ lies in $D (T_{M}^{*})$ precisely when testing separately in each component puts $η_{a} \in D (T^{*}) = D (T)$ for every $a$ ; conversely these finitely many adjoint values give a vector in $L_{N}$ . Thus $D (T_{M}^{*}) = D_{N}$ and $T_{M}^{*} = T_{M}$ . If $T \geq c I$ , summing the scalar quadratic inequalities gives $T_{M} \geq c I$ . The hypothesis on scalar spatial permutations gives domain preservation and commutation for their componentwise action; $R_{π}$ is a constant permutation of finitely many components, so it preserves the domain and commutes with $T_{M}$ too. Their combined action in step 2.1 has both properties.

4.1step 2.1step 3.1F3algebra

[Sector restrictions] Apply [F3] to the self-adjoint $T_{M}$ of step 3.1 and the combined representation from step 2.1. It gives reduction and self-adjointness on every nonzero $ran P_{\pm}$ with exactly the stated intersection domain. If a sector is zero, its Hilbert space and dense operator domain are both ${0}$ , so its unique operator is its own adjoint.

5.1step 1.1step 2.1step 3.1step 4.1F2algebra∎

[Spin covariance and edge cases] A constant spin matrix makes finite linear combinations of the scalar components, so it preserves $D_{N}$ and commutes with the componentwise scalar $T_{M}$ . On elementary spin tensors the conjugation by $R_{π}$ reorders the matrices to $A_{{\hat{π}}^{- 1} (j)}$ in slot $j$ ; by linearity this is the displayed covariance on all of $S_{N}$ . For a common $A$ , the reordering leaves $A^{\otimes N}$ unchanged, so it commutes with each combined permutation and its averages; if $A$ is unitary the inverse has the same properties, proving onto preservation of both sectors and their domains. At $r = 1$ the sole component is scalar, and at $N = 1$ the product formula is the original vector-valued function. These establish (d) and all stated branches.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Fixed-particle-number occupation bases and the countable-mode partition criterion

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space with a supplied countably infinite orthonormal basis $(e_{j})_{j \geq 1}$ , let $N \geq 1$ be an integer, and let $ε_{j} \in R$ with $ε_{j} \geq b$ for a fixed $b \in R$ . Use the completed tensor power and permutation-sector projections of Hilbert tensor construction: inner product, completion and product bases and Permutation sector projections on the n-fold tensor power, and write $H_{N}^{σ} : = ran P_{σ}^{(N)}$ for $σ \in {+, -}$ . Then:

(a) For every finitely supported sequence $n = (n_{j})_{j \geq 1}$ of nonnegative integers with $\sum_{j} n_{j} = N$ , put $orb (n) : = {(i_{1}, \dots, i_{N}) : # {k : i_{k} = j} = n_{j} for every j}$ and $b_{n}^{+} : = ∣ orb (n) ∣^{- 1 / 2} \sum_{i \in orb (n)} e_{i_{1}} \otimes \dots \otimes e_{i_{N}} .$ For every $N$ -element subset $J = {j_{1} < \dots < j_{N}}$ of the positive integers, put $b_{J}^{-} : = \frac{1}{\sqrt{N!}} \sum_{π \in S_{N}} sgn (π) e_{j_{\hat{π} (1)}} \otimes \dots \otimes e_{j_{\hat{π} (N)}} .$ These are orthonormal bases of the symmetric and antisymmetric sectors respectively. Denote their countable index sets by $C_{N}^{+}$ and $C_{N}^{-}$ , and write $b_{c}^{σ}$ for the corresponding vector. Define $E_{n} : = \sum_{j} n_{j} ε_{j}$ and $E_{J} : = \sum_{j \in J} ε_{j}$ .

(b) The fixed- $N$ free energy operator is the self-adjoint diagonal operator $D (H_{N}^{σ}) : = {ψ = \sum_{c \in C_{N}^{σ}} a_{c} b_{c}^{σ} : \sum_{c} ∣ E_{c} ∣^{2} ∣ a_{c} ∣^{2} < \infty}, H_{N}^{σ} ψ : = \sum_{c} E_{c} a_{c} b_{c}^{σ} .$ It is bounded below by $N b$ , and is the closure of its action on finite occupation sums. On elementary tensors of modes it agrees with the restriction of the labelled energy sum $\sum_{k = 1}^{N} ε_{i_{k}}$ , with the same diagonal-domain closure. Its spectral PVM is $E_{N}^{σ} (B) ψ = \sum_{c} 1_{B} (E_{c}) a_{c} b_{c}^{σ} .$

(c) For $β > 0$ , let $q_{j} : = e^{- β ε_{j}} > 0$ , so $q_{j} \leq e^{- β b}$ . Define the extended positive trace of a bounded positive operator as its nonnegative diagonal sum in the supplied occupation basis; its independence of any other supplied orthonormal basis is proved below. With this trace (allowed to be $+ \infty$ ), $Z_{N}^{+} (β) : = Tr e^{- β H_{N}^{+}} = h_{N} (q) : = \sum_{1 \leq i_{1} \leq \dots \leq i_{N}} q_{i_{1}} \dots q_{i_{N}},$ $Z_{N}^{-} (β) : = Tr e^{- β H_{N}^{-}} = e_{N} (q) : = \sum_{1 \leq i_{1} < \dots < i_{N}} q_{i_{1}} \dots q_{i_{N}} .$ All sums are nonnegative-series sums. For either sign, $(\prod_{k = 1}^{N - 1} q_{k}) \sum_{j = N}^{\infty} q_{j} \leq Z_{N}^{σ} (β) \leq {(\sum_{j = 1}^{\infty} q_{j})}^{N} .$ For $N = 1$ the product is $1$ , the tail starts at $1$ , and both partition sums equal $\sum_{j} q_{j}$ . Consequently, for every fixed $N \geq 1$ and either statistics, $0 < Z_{N}^{σ} (β) < \infty ⟺ \sum_{j} e^{- β ε_{j}} < \infty .$ The bounded positive operator $e^{- β H_{N}^{σ}}$ is trace class exactly in this case, and its ordinary trace then equals the displayed occupation sum. This is a fixed- $N$ criterion, not a trace over all particle numbers.

Facts & Assumptions

Given: Countable Choice, the supplied basis, $N \geq 1$ , the lower bound $b$ , real energies $ε_{j} \geq b$ , and $β > 0$ in clause (c).

[F1]

Completed tensor products have dense algebraic tensor products and the elementary-tensor inner-product formula; the permutation projections are bounded orthogonal projections acting by finite permutation sums (Hilbert tensor construction: inner product, completion and product bases, Permutation sector projections on the n-fold tensor power).

[F2]

In each finite prefix $H_{m} : = span {e_{1}, \dots, e_{m}}$ the Slater vectors indexed by $N$ -element subsets form an orthonormal antisymmetric basis, and the orbit sums are a nonzero orthogonal symmetric basis (Slater and symmetric occupation-number bases of the finite-dimensional permutation sectors (a)-(c)).

[F3]

Cauchy-Schwarz controls square-summable coefficient pairings (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs). A complete orthonormal family gives Parseval and norm-convergent coordinate expansion (Parseval equivalences for an orthonormal family). The adjoint of a densely defined operator is characterized by $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ , and equality with the adjoint means self-adjointness (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators).

[F4]

A PVM is strongly countably additive; integration of bounded Borel functions gives the corresponding bounded operator and its scalar spectral integrals (Projection valued measure, Bounded borel pvm integral). The unbounded identity integral has the scalar squared-moment domain and is obtained by bounded coordinate truncations (Integral of a measurable function against a projection-valued measure, The unbounded PVM integral is densely defined, closed and normal).

[F5]

Finite-rank operators are compact and their operator-norm limits are compact (Bounded finite rank operators are compact, Norm limit of compact operators is compact). An absolutely summable nuclear series defines a trace-class operator, with trace equal to the sum of its rank-one traces; a trace-class operator has an absolutely summable diagonal in every supplied orthonormal basis (Nuclear series characterizes trace norm, Trace is absolutely convergent and basis independent).

Proof

technique · direct; exhaust the supplied basis by finite prefixes, compute the unbounded diagonal adjoint on coordinates, and compare nonnegative fixed-N occupation sums

Given: The data of the statement and [F1]-[F5].

1.1F1F2F3algebraconstruct

[Product basis and finite-prefix exhaustion] The tensors $e_{i_{1}} \otimes \dots \otimes e_{i_{N}}$ are orthonormal by [F1]. Their span is dense: approximate each factor of an elementary tensor by its finite basis truncation using [F3]; the tensor norm formula and the telescoping difference of two products show convergence of these approximations, and finite sums of elementary tensors are dense by [F1]. Every finite collection of words is contained in some $H_{m}^{\otimes N}$ . Since each $P_{σ}^{(N)}$ is bounded, its images of this dense word span are dense in its range. In the finite prefix, these images span precisely the finite Slater or orbit-sum basis of [F2]. A word in $orb (n)$ uses only finitely many modes, and the orbit consists of distinct orthonormal product vectors, so its sum has norm squared $∣ orb (n) ∣ > 0$ . Thus its displayed normalization agrees in every larger prefix. The finite-prefix Slater vectors likewise agree in every larger prefix. Any pair of proposed occupation vectors lies in a common prefix, where [F2] gives orthogonality; the normalized vectors have norm one. Their span is dense in the sector by the projection argument, proving (a). The index sets are countable because they are countable unions of finite prefix index sets.

2.1F3step 1.1algebra

[Diagonal domain and self-adjointness] Write either sector in the orthonormal coordinates of step 1.1. The specified domain contains every finite coordinate vector, hence is dense. For a vector in that domain, Parseval makes the defining energy series converge. If $y$ belongs to the adjoint domain, with adjoint value $z$ , testing against each $b_{c}^{σ}$ gives $z_{c} = E_{c} y_{c}$ because $E_{c}$ is real. Since $z$ is square summable, $\sum_{c} ∣ E_{c} y_{c} ∣^{2} < \infty$ , so $y$ belongs to the specified domain. Conversely, if that sum is finite, Cauchy-Schwarz applied to the square-summable coefficient families gives $⟨ H_{N}^{σ} x, y ⟩ = ⟨ x, H_{N}^{σ} y ⟩$ for every domain vector $x$ , so $y$ belongs to the adjoint domain with the stated value. The domains and values agree, proving self-adjointness by [F3]. Since every $E_{c} \geq N b$ , the absolutely convergent energy quadratic sum is at least $N b ∥ x ∥^{2}$ . Finite coordinate truncations converge in both norm and graph norm since both square-sum tails vanish, so finite occupation sums are a core and the operator is their diagonal closure.

3.1F1F3F4step 1.1step 2.1algebra

[Labelled energy and spectral measure] The same coordinate-domain construction on the product-word basis defines a self-adjoint labelled operator with energies $\sum_{k} ε_{i_{k}}$ , agreeing with the free slot-energy sum on finite mode words. Every permutation of a word has the same energy, so its domain and action commute with each permutation unitary and the projections [F1]. Their sector restrictions have exactly the occupation energies and domain of step 2.1. For the displayed $E_{N}^{σ} (B)$ , the projection and intersection identities follow coordinatewise; for disjoint $B_{l}$ , the tails of the square-summable coordinates give strong countable additivity and total mass $I$ , so it is a PVM. The scalar measure of a vector $ψ = \sum_{c} a_{c} b_{c}^{σ}$ is $\sum_{c} ∣ a_{c} ∣^{2} δ_{E_{c}}$ , so [F4] gives the identity-integral domain exactly $\sum_{c} E_{c}^{2} ∣ a_{c} ∣^{2} < \infty$ ; bounded spectral truncations converge to the displayed coordinate action there. This PVM is also uniquely determined by that operator: if another PVM represents it, the zero norm of $(H_{N}^{σ} - E_{c}) b_{c}^{σ}$ forces its scalar measure at $b_{c}^{σ}$ to be supported at $E_{c}$ . Its projections therefore act as $1_{B} (E_{c})$ on every $b_{c}^{σ}$ , and density gives equality on the whole sector. For the exponential, use the bounded Borel function $g (a) = e^{- β \max {a, N b}}$ on $R$ ; it agrees with $e^{- β a}$ on the support of this PVM. [F4] therefore gives the bounded positive operator with eigenvalues $e^{- β E_{c}} \leq e^{- β N b}$ , namely $e^{- β H_{N}^{σ}}$ . This proves (b) and identifies the exponential used in (c).

4.1F3F5step 1.1step 3.1algebra

[Extended trace and occupation sums] Put $r_{c} : = e^{- β E_{c}}$ , a bounded positive family. In the supplied occupation basis the extended positive trace is $\sum_{c} r_{c}$ . This value is basis independent even when infinite: if $(v_{l})$ is any other orthonormal basis, then $\sum_{l} ⟨ e^{- β H_{N}^{σ}} v_{l}, v_{l} ⟩ = \sum_{l} \sum_{c} r_{c} ∣ ⟨ v_{l}, b_{c}^{σ} ⟩ ∣^{2} = \sum_{c} r_{c}$ , since nonnegative-series sums can be interchanged as suprema over finite subsums, and Parseval gives $\sum_{l} ∣ ⟨ v_{l}, b_{c}^{σ} ⟩ ∣^{2} = 1$ . A Bose occupation sequence has a unique sorted word $i_{1} \leq \dots \leq i_{N}$ , and a Fermi subset has a unique increasing word; $e^{- β E_{c}}$ is the product of the corresponding $q_{j}$ . Thus the trace equals $h_{N} (q)$ or $e_{N} (q)$ as stated. If the sum is finite, the series of rank-one operators $\sum_{c} r_{c} ⟨ \cdot, b_{c}^{σ} ⟩ b_{c}^{σ}$ is nuclear and converges in operator norm to the exponential: its norm tail is at most the summable scalar tail. Its finite-rank partial sums and [F5] first give compactness, and the nuclear criterion then gives trace class and the ordinary trace formula. Conversely trace class forces the diagonal sum to be finite by [F5].

5.1step 4.1givenalgebra∎

[Upper and lower bounds and finiteness] Let $Q : = \sum_{j} q_{j} \in (0, \infty]$ . The sum over all ordered words of length $N$ is $Q^{N}$ : on the first $m$ modes it is $(\sum_{j \leq m} q_{j})^{N}$ , and every finite collection of words is contained in one such prefix. The nondecreasing and strictly increasing index sets are subsets of the ordered-word set, giving $Z_{N}^{σ} \leq Q^{N}$ . For $N \geq 2$ , configurations selecting modes $1, \dots, N - 1$ once each and one additional mode $j \geq N$ are valid for both statistics, distinct as $j$ varies, and have weight $(\prod_{k = 1}^{N - 1} q_{k}) q_{j}$ . Their sum gives the lower bound. For $N = 1$ , every configuration is the single mode $j$ , so both traces equal $Q$ ; the empty product is $1$ and the tail starts at $1$ . For any $N$ , the lower-bound coefficient is strictly positive and finite, and the omitted initial segment of $Q$ is finite. Hence $Q = \infty$ forces $Z_{N}^{σ} = \infty$ , whereas $Q < \infty$ gives $Z_{N}^{σ} < \infty$ by the upper bound. The single configuration ${1, \dots, N}$ has positive weight for either sign, so both traces are strictly positive. This proves all of (c). No sum over particle numbers or full-Fock product formula is used.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Spectral first moments and trace products with a density operator

Statement

Assume Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $A$ be a self-adjoint operator on a nonzero complex Hilbert space, let $E_{A}$ be its spectral PVM, let $ρ$ be a density operator, and set $μ_{ρ}^{A} (B) : = tr (ρ E_{A} (B))$ . Then:

(a) The finite first mean is defined exactly under the absolute-first-moment condition $\int_{R} ∣ a ∣ d μ_{ρ}^{A} (a) < \infty,$ and its value is $m : = \int a d μ_{ρ}^{A} (a)$ . For every positive-eigenvalue expansion $ρ = \sum_{k} w_{k} T_{u_{k}, u_{k}}$ of Density operators and scalar spectral probability (a), with $w_{k} > 0$ and $\sum_{k} w_{k} = 1$ , one has $\int ∣ a ∣ d μ_{ρ}^{A} = \sum_{k} w_{k} \int ∣ a ∣ d E_{A, u_{k}}, m = \sum_{k} w_{k} \int a d E_{A, u_{k}} .$ The first identity allows infinity; the second is asserted under the stated integrability condition and converges absolutely. This condition does not require $u_{k} \in D (A)$ .

(b) If $A$ is bounded, the condition in (a) holds automatically and $m = tr (A ρ) = tr (ρ A) .$

(c) Suppose the condition in (a) holds and the literal product $A ρ$ is defined on the whole Hilbert space and is trace class; thus $ρ (H) \subset D (A)$ and $T : = A ρ \in S_{1} (H)$ . Then $m = tr (T) .$ Moreover the densely defined product $ρ A$ has bounded closure $T^{*}$ , that closure is trace class, and $tr (\overline{ρ A}) = m$ . This is a conditional trace identity for defined trace-class products. The existence of the mean alone does not assert any of these unbounded-operator product hypotheses.

Facts & Assumptions

Given: The choice hypotheses, $A, E_{A}, ρ, μ_{ρ}^{A}$ , and in (c) the explicitly stated literal-product assumptions.

[F1]

The density eigenexpansion has positive weights and orthonormal vectors, converges in trace norm, and gives $μ_{ρ}^{A} (B) = \sum_{k} w_{k} E_{A, u_{k}} (B)$ . For $u \in D (A)$ , $\int a d E_{A, u} = ⟨ A u, u ⟩$ ; absolute first integrability alone is equivalent to $u \in D (∣ A ∣^{1 / 2})$ (Density operators and scalar spectral probability (a)-(c)).

[F2]

Nonnegative monotone limits commute with integrals, and integrable domination permits limits of signed integrals (Monotone convergence for the integral, Dominated convergence).

[F3]

Bounded Borel spectral integrals agree with their scalar spectral integrals (Bounded borel pvm integral). Bounded multiplication of a density eigenexpansion gives $tr (A ρ) = \sum_{k} w_{k} ⟨ A u_{k}, u_{k} ⟩$ and bounded cyclicity gives $tr (A ρ) = tr (ρ A)$ (Nuclear bounded cyclicity for trace-class operators).

[F4]

A trace-class operator has its trace as an absolutely convergent diagonal in a supplied Hilbert basis; it also has a nuclear representation whose rank-one traces sum to its trace (Trace is absolutely convergent and basis independent, Nuclear series characterizes trace norm). Under AC, order the set of orthonormal families containing the given family by inclusion. This poset is nonempty because it contains the given family. Every nonempty chain has its orthonormal union as an upper bound in this poset; the empty chain has the given family as an upper bound. Thus Zorn's lemma provides a maximal extension. Its orthogonal complement is zero by maximality, hence it is complete by Parseval equivalences for an orthonormal family. Coordinate expansion is Fourier expansion in a Hilbert space. Finite-rank maps and their operator-norm limits are compact (Bounded finite rank operators are compact, Norm limit of compact operators is compact).

Proof

technique · direct; integrate the density eigenexpansion and compare it with correctly defined trace products

Given: The data of the statement and [F1]-[F4].

1.1F1F2algebra

[First-moment expansion] By [F1] the outcome measure is the countable weighted sum of the scalar measures. This gives $\int g d μ_{ρ}^{A} = \sum_{k} w_{k} \int g d E_{A, u_{k}}$ first for nonnegative simple functions by the measure identity and finite linearity, and then for all nonnegative measurable $g$ by increasing simple approximation and [F2]; the nonnegative sums interchange as increasing finite subsums. Taking $g = ∣ a ∣$ gives the first identity in (a), including infinity. When it is finite, both positive and negative parts are integrable, so subtracting their two finite sums gives $m = \sum_{k} w_{k} \int a d E_{A, u_{k}}$ with $\sum_{k} w_{k} ∣ \int a d E_{A, u_{k}} ∣ \leq \sum_{k} w_{k} \int ∣ a ∣ d E_{A, u_{k}} < \infty$ . For each positive $w_{k}$ , only absolute first integrability is forced; [F1] identifies its form domain, without requiring the operator-square moment. This proves (a).

2.1F1F3step 1.1algebra

[Bounded observable] If $A$ is bounded, every $u_{k}$ is in $D (A)$ , and Cauchy-Schwarz for the scalar probability measure gives $\int ∣ a ∣ d E_{A, u_{k}} \leq (\int a^{2} d E_{A, u_{k}})^{1 / 2} = ∥ A u_{k} ∥ \leq ∥ A ∥$ by [F1] (Cauchy-Schwarz inequality for $L^{2}$ ). Step 1.1 therefore bounds the first absolute moment by $∥ A ∥$ . Each $u_{k}$ is in $D (A)$ and [F1] gives $\int a d E_{A, u_{k}} = ⟨ A u_{k}, u_{k} ⟩$ . Thus step 1.1 and the bounded trace formula [F3] give $m = \sum_{k} w_{k} ⟨ A u_{k}, u_{k} ⟩ = tr (A ρ) = tr (ρ A)$ . This proves (b).

2.2F1F4step 1.1choosealgebra

[Literal trace-class left product] Under (c), $ρ u_{k} = w_{k} u_{k} \in D (A)$ and $w_{k} > 0$ , so each $u_{k} \in D (A)$ and $T u_{k} = w_{k} A u_{k}$ . Complete the orthonormal eigenfamily by an orthonormal basis of its perpendicular subspace using the already assumed AC. The density eigenexpansion shows $ρ = 0$ on that perpendicular subspace, so $T = 0$ there. The trace diagonal formula [F4] therefore gives $tr (T) = \sum_{k} ⟨ T u_{k}, u_{k} ⟩ = \sum_{k} w_{k} ⟨ A u_{k}, u_{k} ⟩$ . Step 1.1 and [F1] identify this absolutely convergent scalar sum with $m$ . This proof uses the literal-product hypothesis to obtain $u_{k} \in D (A)$ ; it does not infer that hypothesis from first-moment existence.

3.1F4step 2.2algebra∎

[Closure of the right product] For $x \in D (A)$ and arbitrary $y$ , self-adjointness of $A$ and $ρ$ gives $⟨ ρ A x, y ⟩ = ⟨ A x, ρ y ⟩ = ⟨ x, A ρ y ⟩ = ⟨ x, T y ⟩ = ⟨ T^{*} x, y ⟩$ , so $ρ A$ agrees with the bounded $T^{*}$ on the dense domain $D (A)$ . Its graph closure is all of the graph of $T^{*}$ : approximate any $x$ by $x_{l} \in D (A)$ and use boundedness of $T^{*}$ , while its closed graph contains every graph limit from that restriction. A nuclear representation $T = \sum_{j} T_{v_{j}, z_{j}}$ gives $T^{*} = \sum_{j} T_{z_{j}, v_{j}}$ with the same summable nuclear norms; the operator-norm convergence of these finite-rank partial sums first gives compactness, and then the nuclear criterion gives trace class and [F4] gives $tr (T^{*}) = \overline{tr (T)} = m$ , since the spectral integral $m$ is real. This proves (c), without invoking cyclicity for unbounded multipliers.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable

Physical scope

The declared fixed-N many-body model of the framework: a declared species sector of P6, a declared one-particle energy operator (spatial massive-model or explicitly retained finite-mode realization), a declared pair interaction (bounded or Coulomb), the labelled energy operator and its self-adjoint restriction to the adopted symmetric or antisymmetric N-particle sector, read on nonzero sectors as the model's assigned energy observable and generator of P4 evolution; zero sectors are mathematical bookkeeping with no preparation or physical observable assignment. Finite modes are embedded in the full spatial spin space, while their energy and dynamics are separately adopted effective inputs; full spatial kinematics and spin rotations are not automatically assigned on the retained space. The model is an adopted kinematic-dynamical input of the pair; it asserts no stability of matter, no Hunziker-van Winter-Zhislin threshold statement, no thermodynamic limit, no thermalization and no identification with a laboratory sample, and every interaction constant is an empirical model parameter with uncertainty.

Definition

(a) Adopted readings, units and choice hypotheses. Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1), the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2), the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4), the non-relativistic kinematics postulate Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators (P5) and the identical-particle and spin postulate Identical-particle and spin assignments P6: permutation sectors and finite-spin representations (P6), in the state reading of The state space of the framework: preparations, pure rays and density operators, the observable reading of Assigned sharp observables and the outcome law of the framework, the evolution reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution and the sector vocabulary of Declared species sectors and their preparations: the framework's reading of P6. P5 and the massive spin- $s$ reading below apply to the full spatial space $H_{s} = L^{2} (R^{3}; C^{2 s + 1})$ . In the finite-mode branch, clause (b)(ii) explicitly replaces P5's full-space state and Hamiltonian assignments (iii),(iv) by retained-space assignments: P1, P2 and P4 are read on that retained space, P6(i) gives its permutation sectors, and P6(ii) still specifies spin on $H_{s}$ , with no automatic full spin representation on the retained space. This is an additional model idealization, not a consequence of P5. Let the species parameters mass $m > 0$ (kg), spin label $s \in {0, \frac{1}{2}, 1, \dots}$ and statistics choice $σ \in {+, -}$ be given in the vocabulary of that definition and let $H$ be the one-particle space of the chosen branch below. Use the sector and preparation constructions of that definition (c)-(e) on $H$ ; fix an integer $N \geq 1$ and let $H_{N}^{σ} = ran P_{σ}^{(N)} \subseteq H^{\otimes N}$ be the declared $N$ -particle sector. All quantities below are read in the SI conventions and units of Dimensions, SI unit conventions and quantity types of the framework (energies in joules). Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and the Axiom of Choice (The Axiom of Choice). AC is inherited from the adopted general spectral observable and evolution interfaces of P2 and P4 and the operator suppliers of P5; it is also required by the Coulomb realization in clause (d)(ii). The bounded spatial operator construction in clause (d)(i) itself uses only Countable Choice. Adopt Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions (P7) for the qualified apparatus and parameter identifications below.

(b) Declared one-particle data. The model declares a one-particle energy operator in one of the two following realizations.

(i) Spatial realization. The one-particle space is $H = L^{2} (R^{3}; C^{2 s + 1})$ and, with $ℏ > 0$ the reduced Planck constant, the model declares a kinetic operator $K_{1} : = - \frac{ℏ^{2}}{2 m} Δ$ on $L^{2} (R^{3}; C^{2 s + 1})$ and a real, measurable, bounded external potential $V_{e x t} : R^{3} \to R$ of essential supremum $∥ V_{e x t} ∥_{\infty} < \infty$ ; the declared one-particle operator is $h : = K_{1} + V_{e x t} (Q)$ , self-adjoint by CC realization of the bounded spatial operator on a finite-component H² domain with $N = 1$ , $M = 2 s + 1$ and $a_{1} = ℏ^{2} / (2 m)$ , with domain $W^{2, 2} (R^{3}; C^{2 s + 1})$ . The particular singular external potential $V_{e x t}^{b} - g / ∣ x ∣$ is included in the Coulomb branch of clause (c)(ii), whose one-particle $h$ is specified there using the $N = 1$ instance of clause (d)(ii). Any other more singular declared $V_{e x t}$ requires a separately supplied self-adjoint realization before its operator is read as an observable.

(ii) Finite-mode realization. A finite-dimensional complex Hilbert space $K$ of dimension $d \geq 1$ with a declared orthonormal basis $(f_{1}, \dots, f_{d})$ (Hilbert space) is declared together with a self-adjoint one-particle operator $h \in B (K)$ (A bounded linear operator between normed spaces, Symmetric, self-adjoint and essentially self-adjoint operators). Also declare an orthonormal family $(g_{1}, \dots, g_{d})$ in the full spatial spin space $H_{s}$ of The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (a). Define the retained-mode isometry, its adjoint, and its orthogonal projection by $J : K \to H_{s}, J (\sum_{j = 1}^{d} c_{j} f_{j}) = \sum_{j = 1}^{d} c_{j} g_{j}, J^{*} ψ = \sum_{j = 1}^{d} ⟨ ψ, g_{j} ⟩ f_{j}, R : = J J^{*} .$ Thus $J^{*} J = I_{K}$ , $R^{2} = R = R^{*}$ and $ran R = span {g_{1}, \dots, g_{d}}$ ; inner products are linear in the first variable. Here $H = K$ is the effective one-particle space, and $ψ \in K$ represents the retained full-space vector $J ψ$ . The tensor map $J_{N} : = J^{\otimes N}$ of Bounded tensor maps: extension, norm estimate, adjoint and commutation is an isometry and satisfies $J_{N} U_{π}^{K} = U_{π}^{H_{s}} J_{N}$ : the two sides agree on elementary tensors by the slot convention and hence on the completion. The finite projection sums consequently give $J_{N} P_{σ}^{(N), K} = P_{σ}^{(N), H_{s}} J_{N}$ . The retained sector is therefore isometrically embedded in the full species sector, preserving its statistics choice.

The internal spin matrices and rotations of P6(ii) act on $H_{s}$ . Their restrictions are assigned on $K$ only when they preserve $ran J$ ; for example a full internal rotation action on $K$ requires $(I \otimes D_{s} (A)) ran J = ran J$ for every $A \in S U (2)$ , with $D_{s}$ exactly as in P6(ii), and is then read as $J^{*} (I \otimes D_{s} (A)) J$ . No such invariance is imposed here, so $d$ need not be a multiple of $2 s + 1$ : one retained spinor mode for $s = 1 / 2$ is permitted without identifying $K$ with the full two-dimensional spin factor. P5's full-space position, momentum and commutation relations are not asserted on $K$ . The finite operator $h$ and pair operator $W$ of clause (c)(i) are independently adopted effective energy data; no compression of a spatial Hamiltonian, equality with its evolution, or approximation error bound is asserted. Identifying the retained modes and these energy data with a laboratory preparation is a tested identification with uncertainty under P7, not part of the mathematical embedding.

(c) Declared interaction. The model declares a pair interaction in the matching realization.

(i) Bounded interaction. In the spatial realization let $W : R^{3} \to R$ be real, measurable, bounded with $W (- z) = W (z)$ for almost every $z$ and $∥ W ∥_{\infty} < \infty$ . In the finite-mode realization let $W \in B (K \hat{\otimes} K)$ be self-adjoint with $U_{(0 1)} W U_{(0 1)} = W$ , where $U_{(0 1)}$ is the exchange unitary of Permutation sector projections on the n-fold tensor power clause (a) in the tensor square $\hat{K \otimes K}$ of Hilbert tensor construction: inner product, completion and product bases. The interaction strength constants entering $W$ are empirical model parameters with their uncertainties in the sense of P6 clause (iii).

(ii) Coulomb interaction. In the spatial realization the model may declare $W (z) : = κ / ∣ z ∣$ with a real coupling $κ \in R$ in $J m$ (repulsive for $κ > 0$ ). Retain a real bounded measurable external term $V_{e x t}^{b}$ as in clause (b)(i), and permit the additional singular term $- g / ∣ x ∣$ , where $g \in R$ is also in $J m$ , so each distance denominator in metres gives an energy in joules; the total external potential is $V_{e x t}^{b} (x) - g / ∣ x ∣$ . In this branch the declared one-particle operator is $h = K_{1} + M_{V_{e x t}^{b} - g / ∣ x ∣}$ on $W^{2, 2} (R^{3}; C^{2 s + 1})$ , including the singular external term. Its self-adjointness and lower bound are the $N = 1$ finite-component instance of clause (d)(ii), with no pair term. Thus the bounded one-particle reading in clause (b)(i) is replaced by this supplied Coulomb reading, rather than leaving the singular external term solely in $H_{l a b}$ . The pair interaction still consists only of $κ / ∣ x_{i} - x_{j} ∣$ . The declared scalar potential on $R^{3 N}$ is $V (X) = V_{b} (X) + V_{C} (X), V_{b} (X) = \sum_{i = 1}^{N} V_{e x t}^{b} (x_{i}), V_{C} (X) = - g \sum_{i = 1}^{N} \frac{1}{∣ x_{i} ∣} + κ \sum_{1 \leq i < j \leq N} \frac{1}{∣ x_{i} - x_{j} ∣} .$ The cases $g = 0$ , $κ = 0$ , $V_{e x t}^{b} = 0$ and either sign of either coupling are allowed. Singular values on their null collision sets are irrelevant to the multiplier classes. The singular part is supplied by Fixed-N Coulomb self-adjointness; the bounded external addition is included by the exact-domain argument in clause (d)(ii), rather than attributed to a source statement which contains only the singular sum.

(d) The labelled energy operator and its sector restriction. Fix the particle masses $m_{1} = \dots = m_{N} = m$ of the declared species.

(i) Spatial, bounded case. Put $r_{s} : = 2 s + 1$ , $S_{N} : = (C^{r_{s}})^{\otimes N}$ and $M : = r_{s}^{N}$ . The finite product spin basis identifies $S_{N}$ with $C^{M}$ . Use the full finite-component product unitary $Θ_{N} : H^{\otimes N} ⟶ L^{2} (R^{3 N}; C^{M}), [Θ_{N} (f_{1} \otimes \dots \otimes f_{N})]_{a} (X) = \prod_{j = 1}^{N} [f_{j} (x_{j})]_{a_{j}},$ where $a \in {1, \dots, r_{s}}^{N}$ . Its construction and combined spatial/spin permutation action are proved in Finite spin components of an N-fold spatial tensor product, from the finite-component extension of Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}). For $V (X) = \sum_{i} V_{e x t} (x_{i}) + \sum_{i < j} W (x_{i} - x_{j})$ define $H_{l a b} = - ℏ^{2} Δ_{X} / (2 m) + M_{V}$ componentwise. The CC-only CC realization of the bounded spatial operator on a finite-component H² domain, with $a_{i} = ℏ^{2} / (2 m)$ and component count $M$ , supplies self-adjointness on $D (H_{l a b}) = W^{2, 2} (R^{3 N}; C^{M}) = W^{2, 2} (R^{3 N}; C) \otimes S_{N},$ and the lower bound $- ∥ V ∥_{\infty}$ . Equal masses, identical external potentials and the even pair potential make the scalar domain and operator invariant under spatial block permutations. The finite-component lemma shows that the combined permutations $U_{π} = Θ_{N} U_{π} Θ_{N}^{- 1}$ also permute the spin slots and commute with $H_{l a b}$ on its domain. Thus $P_{σ} : = Θ_{N} P_{σ}^{(N)} Θ_{N}^{- 1}$ reduces the operator by Permutation-restricted self-adjointness (a)-(d), and the transported sector energy is $H_{N}^{σ, c o o r d} : = H_{l a b} ∣_{D (H_{l a b}) \cap ran P_{σ}} .$ It is self-adjoint on $ran P_{σ} = Θ_{N} H_{N}^{σ}$ , with the zero operator on the zero-sector branch. The model's tensor reading is $H_{N}^{σ} : = Θ_{N}^{- 1} H_{N}^{σ, c o o r d} Θ_{N}$ on $H_{N}^{σ}$ . Every constant spin action commutes with the spin-independent labelled operator; a common spin unitary $A^{\otimes N}$ preserves each identical-particle sector by the finite-component lemma. For $s = 0$ , $M = 1$ and this is the scalar product reading; for $N = 1$ , $M = 2 s + 1$ and it is the one-particle component reading.

(ii) Spatial Coulomb case. Put $r = 2 s + 1$ , $S_{N} = (C^{r})^{\otimes N}$ and $M = r^{N}$ . Use the full spin-product unitary $Θ_{N} : (L^{2} (R^{3}; C^{r}))^{\otimes N} \to L^{2} (R^{3 N}; C^{M})$ of Finite spin components of an N-fold spatial tensor product (a),(b), including its combined spatial/internal permutation action. To apply the literal Fourier normalization of Fixed-N Coulomb self-adjointness, choose in that mathematical theorem the masses ${\tilde{m}}_{i} = m / (4 π^{2})$ , its parameter $ℏ$ unchanged, one fixed center at zero, coefficients $z_{i 1} = - g$ and $z_{i j} = κ$ . The mathematical masses here are substitution parameters used to realize the already declared kinetic operator; the physical species mass remains $m$ . The source kinetic symbol becomes $4 π^{2} ℏ^{2} ∣ ξ ∣^{2} / (2 m)$ , so its scalar operator is exactly $T_{C} = - \frac{ℏ^{2}}{2 m} Δ_{X} + V_{C} (X), D (T_{C}) = H^{2} (R^{3 N}; C) .$ By the source's clauses (b),(c), $T_{C}$ and its finite-component extension are self-adjoint on these exact domains and bounded below for all stated real $g, κ$ . The real bounded multiplier $M_{V_{b}}$ has norm at most $N ∥ V_{e x t}^{b} ∥_{\infty}$ , hence relative bound zero with respect to $T_{C}$ and to its finite-component extension. The canonical Kato-Rellich theorem makes $H_{l a b} = - \frac{ℏ^{2}}{2 m} Δ_{X} + V_{C} + V_{b}, D (H_{l a b}) = H^{2} (R^{3 N}; C^{M}),$ self-adjoint on exactly this domain and bounded below; if the singular operator has lower bound $c_{C}$ , the sum has lower bound $c_{C} - N ∥ V_{e x t}^{b} ∥_{\infty}$ . Equal masses, identical external slots and pair-independent signed coefficient $κ$ make the scalar operator invariant under spatial permutations. These permutations preserve $H^{2}$ by permuting the weak-derivative coordinates; the constant internal tensor permutations preserve the finite-component domain too. Consequently the combined projections $P_{σ} = Θ_{N} P_{σ}^{(N)} Θ_{N}^{- 1}$ reduce $H_{l a b}$ , and Finite spin components of an N-fold spatial tensor product (c) gives self-adjointness of its restriction on the exact domain $H^{2} \cap ran P_{σ}$ . The declared sector energy is this restriction transported back by $Θ_{N}^{- 1}$ to $H_{N}^{σ}$ , with the unique zero-space operator when that sector is zero. This Coulomb construction uses AC in addition to the inherited framework choice hypotheses of clause (a).

(iii) Finite-mode case. On $K^{\otimes N}$ put $H_{l a b} : = \sum_{i = 1}^{N} h^{(i)} + \sum_{1 \leq i < j \leq N} W^{(i j)},$ where $h^{(i)}$ is the slot operator of Bounded tensor maps: extension, norm estimate, adjoint and commutation (the identity on all factors but the $i$ -th, $h$ on the $i$ -th) and $W^{(i j)}$ is the transport of $W \in B (K \hat{\otimes} K)$ to the pair of slots $(i, j)$ by any canonical permutation $π$ whose transported map $\hat{π}$ moves that pair to the first two slots; the symmetry $U_{(0 1)} W U_{(0 1)} = W$ makes the order of the pair immaterial and $W \otimes I$ commutes with permutations fixing the first two slots, so the transported operator is independent of the permutation chosen. Then $H_{l a b}$ is bounded and self-adjoint, it commutes with every permutation unitary, and the model's sector energy operator is $H_{N}^{σ} : = H_{l a b} ∣_{H_{N}^{σ}} = P_{σ}^{(N)} H_{l a b} P_{σ}^{(N)} ∣_{H_{N}^{σ}}$ .

(e) The model's energy reading and evolution. In every realization with $H_{N}^{σ} \neq {0}$ , the pair $(H_{N}^{σ}, J)$ is declared to be the assigned energy observable of the $N$ -particle sector in the sense of Assigned sharp observables and the outcome law of the framework clause (a): its outcome law in a declared sector preparation $ρ$ is the law $μ_{ρ}^{H_{N}^{σ}} (B) = tr (ρ E_{H_{N}^{σ}} (B))$ of that definition clause (b), and its spectral data are those of the self-adjoint operator constructed in clause (d). The model evolution is the unitary flow $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ of Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b) with this generator. If $H_{N}^{σ} = {0}$ , clause (d) retains the unique zero-space operator only as mathematical bookkeeping: every operator on this space has trace zero, so there is no trace-one preparation, no assigned physical energy observable and no preparation evolution under P1-P4. The identification of the operator with the energy of a concrete sample, and the assignment of numerical masses, couplings and potentials to a laboratory species, are tested identifications with uncertainty under P7; they are not consequences of P1-P6.

(f) The non-interacting reading. The model is called non-interacting when $W = 0$ (equivalently $κ = 0$ only in the Coulomb branch of clause (c)(ii)), so that $H_{l a b} = \sum_{i = 1}^{N} h^{(i)}$ ; in the spatial realization this is the case of the external potential summed over slots of the stated bounded or Coulomb realization.

(g) Scope. The declared model is a kinematic and dynamical input of this pair: on a nonzero sector it fixes a state space, a sector, an energy operator and an evolution; a zero sector carries only the mathematical convention of clause (e). It asserts no stability of matter, no Hunziker-van Winter-Zhislin threshold statement, no statement about excited threshold structure, no thermodynamic limit, no equilibration or thermalization, no measured energy and no identification of its parameters with a particular laboratory sample beyond the tested identifications of P7. The finite-mode realization is the retained-space idealization of clause (b)(ii), with independently assigned finite energy data and no claimed accuracy relative to spatial evolution; the spatial realization is an idealization of an $N$ -particle non-relativistic sample with the declared external field and pair interaction.

Remarks

Why the sector restriction is legitimate. In the spatial realizations the permutation unitaries preserve the domain $W^{2, 2}$ and commute with $H_{l a b}$ , which is exactly the hypothesis of the permutation-restricted self-adjointness theorem; in the finite-mode realization the bounded operator $H_{l a b}$ commutes with the $U_{π}$ by construction, and a bounded operator commuting with a projection restricts to a self-adjoint operator on its range.
Units. $m$ is in kg, $ℏ$ in J s, $V_{e x t}$ and $∥ V ∥_{\infty}$ in J, $W$ in J and $g, κ$ in J m; in the finite-mode realization $h$ and $W$ are declared directly in joules. No magnetic coupling, spin-orbit term or velocity-dependent interaction is part of this model.

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The declared one-body reduced operator and the one-body readout of a sector preparation

Physical scope

The declared one-body reduction of a sector preparation in the framework's finite-mode realization: the reduced preparation on the one-particle mode space, the sector operator that sums a declared bounded one-particle quantity over the labels, and the reading of that sum as an assigned sharp observable. The reduction is a mathematical partial trace of a sector state; it asserts no physical single-particle state, no separability of the sector preparation, no measurement of an individual particle and no fact about a laboratory sample.

Definition

(a) Adopted readings. Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1) and the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) in the state reading of The state space of the framework: preparations, pure rays and density operators and the observable reading of Assigned sharp observables and the outcome law of the framework. Work in the finite-mode realization of a declared species sector of Declared species sectors and their preparations: the framework's reading of P6: the one-particle space is a finite-dimensional complex Hilbert space $K$ of dimension $d \geq 1$ with its declared orthonormal modes $(f_{1}, \dots, f_{d})$ , and the $N$ -particle sector space is $H_{N}^{σ} = ran P_{σ}^{(N)} \subseteq K^{\otimes N}$ with the declared sector preparations of The declared finite-mode model and its occupation readouts. Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

(b) The one-body reduced operator. Let $ρ$ be a declared sector preparation, that is, a density operator on $H_{N}^{σ}$ (Density operators and scalar spectral probability clause (a)). The one-body reduced preparation of $ρ$ is the density operator $γ_{ρ}$ on $K$ of clause (e) of The one-body reduction trace law of the finite-mode permutation sectors (with $H : = K$ , $n : = N$ and the same sector projections). Explicitly, for the sector inclusion $j$ put $\tilde{ρ} = j ρ j^{*}$ on $K^{\otimes N}$ , and for the regrouping unitary $J_{N} : K^{\otimes N} \to K \hat{\otimes} K^{\otimes (N - 1)}$ put $γ_{ρ} = {Tr}_{K^{\otimes (N - 1)}} (J_{N} \tilde{ρ} J_{N}^{- 1})$ . Here $K^{\otimes 0} = C$ , so $N = 1$ gives $γ_{ρ} = ρ$ . A zero sector has no preparation and makes this clause vacuous. By that lemma $γ_{ρ}$ is a density operator on $K$ independent of which factor is traced, and for every $O \in B (K)$ the pairings are related by $tr (\tilde{ρ} Σ (O)) = N tr (γ_{ρ} O),$ where $\tilde{ρ}$ is the zero extension of $ρ$ and $Σ (O)$ is the sum over the $N$ labelled slots of the slot operator $O^{(i)}$ .

(c) The one-body readout. For a declared bounded self-adjoint $O \in B (K)$ with a declared unit $u_{O}$ in the sense of Dimensions, SI unit conventions and quantity types of the framework, the model reads the sector operator $A_{O} : = Σ (O) ∣_{H_{N}^{σ}} = P_{σ}^{(N)} Σ (O) P_{σ}^{(N)} ∣_{H_{N}^{σ}}$ as an assigned sharp observable $(A_{O}, u_{O})$ in the sense of Assigned sharp observables and the outcome law of the framework clause (a): it is bounded and self-adjoint on the nonzero sector space and its outcome law in a declared sector preparation $ρ$ is the law $μ_{ρ}^{A_{O}} (B) = tr (ρ E_{A_{O}} (B))$ of clause (b) of that definition. The one-body readout of $O$ in $ρ$ is the mean of that law when it exists.

(d) Scope. The reduced preparation $γ_{ρ}$ is a mathematical partial trace of the sector state; it is not asserted to be the state of any individual particle. The reduction need not determine the sector preparation in higher-particle sectors; no universal noninjectivity is asserted. For $N = 1$ it is exactly $γ_{ρ} = ρ$ by clause (b), and a one-dimensional sector has only one density preparation. The definition asserts no separability or factorization of $ρ$ , no result for the spatial (infinite-dimensional) realization, whose reduction would require the separable partial trace, and no measurement of individual particle labels, which are mathematical factor labels by P6 clause (i). Whether a declared laboratory preparation realizes a sector preparation $ρ$ and a quantity realizes the pair $(A_{O}, u_{O})$ are tested identifications with uncertainty under P7, not consequences of this definition.

Remarks

Normalization. The one-body reduced preparation is normalized as a probability state, $tr γ_{ρ} = 1$ , while the mean of the sum $A_{O}$ carries the factor $N$ ; the sum $Σ (O)$ is a many-body quantity, not a single-particle quantity, and its mean is $N$ times the single-particle pairing with $γ_{ρ}$ .
Choice of retained factor. Lemma clause (e) shows that reducing over any $N - 1$ of the labelled factors gives the same operator on $K$ ; this is the precise sense in which the reduction is label independent inside the model.

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Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model

Physical scope

The framework's reading of entropy and free energy for a declared fixed-N model whose sector space is finite-dimensional (the finite-mode realization): the dimensionless von Neumann entropy of a declared preparation, the assigned Boltzmann constant with its exact SI value, an assigned temperature and inverse temperature, the physical entropy in J K^-1 and the Helmholtz free-energy functional in joules. The identification beta = 1/(k_B T) and the constancy of k_B are adopted model and unit-convention assumptions; the definition asserts no increase of entropy, no equilibration, no thermodynamic limit and no measured temperature or entropy.

Definition

(a) Adopted readings and restricted sector class. Adopt the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) and the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4), in the readings of Assigned sharp observables and the outcome law of the framework and Assigned energy observable, unitary flow and the two pictures of autonomous evolution, for the declared fixed- $N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable. This definition applies to the class of declared models whose sector space $H_{N}^{σ}$ is finite-dimensional, in particular to every finite-mode realization (The declared finite-mode model and its occupation readouts), where $D : = \dim H_{N}^{σ}$ is finite (and may be zero) by Finite-mode occupation configurations: fermionic exclusion counts, bosonic counts and truncated dimensions clauses (a), (b). Write $D (H_{N}^{σ})$ for the set of declared sector preparations, that is, the density operators on $H_{N}^{σ}$ (Density operators and scalar spectral probability clause (a)). If $D = 0$ , the only operator has trace zero, so $D (H_{N}^{σ}) = \emptyset$ . The partition trace is $Z_{N} = 0$ , and there is no normalized canonical density or equilibrium free energy; $\log D$ , $D^{- 1} i d$ and $\log Z_{N}$ are not used in this branch. Clauses (b), (d) and (e) below apply when $D \geq 1$ ; the assigned parameters of (c) can still be declared when $D = 0$ . Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

(b) Dimensionless entropy. For $ρ \in D (H_{N}^{σ})$ with eigenvalues $w_{1}, \dots, w_{D} \geq 0$ (counted with multiplicity, in any order), the entropy of $ρ$ is $s (ρ) : = - \sum_{i : w_{i} > 0} w_{i} \log w_{i} \in [0, \log D],$ the quantity of Finite Gibbs free-energy minimum clause (a) with $\log$ the natural logarithm (The natural logarithm as the inverse of the exponential function). It is dimensionless, is a function of the eigenvalue multiset of $ρ$ alone, and equals zero exactly for pure preparations; by clause (c) of the cited theorem $s (ρ) = \log D$ holds exactly when $ρ = D^{- 1} i d$ .

(c) Boltzmann constant, temperature and inverse temperature. The model assigns a positive Boltzmann constant $k_{B}$ and a positive temperature $T$ ; in the SI conventions of Dimensions, SI unit conventions and quantity types of the framework the unit of $k_{B}$ is $J K^{- 1}$ , the unit of $T$ is the kelvin, and $k_{B} = 1.380 649 \times 10^{- 23} J K^{- 1}$ fixed exactly by the BIPM SI definition. The model declares the inverse temperature $β : = 1 / (k_{B} T)$ , a positive real number in $J^{- 1}$ . The value and exactness of $k_{B}$ are unit-convention facts; the assignment of the parameter $T$ to a physical sample, and the identification of $β = 1 / (k_{B} T)$ with an inverse thermodynamic temperature, are adopted assumptions of the ensemble model of this pair, not consequences of P1-P4, and are tested identifications with uncertainty under the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions (P7), whose clauses (i), (iii) and (iv) require a declared comparison and calibration model and qualifications for measured inputs. No thermometer, calibration or measured temperature is supplied here.

(d) Physical entropy. The physical entropy of $ρ$ is $S (ρ) : = k_{B} s (ρ),$ read in $J K^{- 1}$ . It is nonnegative, is at most $k_{B} \log D$ , and vanishes exactly for pure preparations.

(e) Free-energy functional. For $ρ \in D (H_{N}^{σ})$ , let $tr (ρ H_{N}^{σ}) \in R$ be the mean of the declared energy observable in $ρ$ (finite because $D < \infty$ , in the moment reading of Assigned sharp observables and the outcome law of the framework clause (e)). The Helmholtz free-energy reading of $ρ$ is $F (ρ) : = tr (ρ H_{N}^{σ}) - k_{B} T s (ρ),$ read in joules; it is the quantity $tr (ρ H) - β^{- 1} s (ρ)$ of Finite Gibbs free-energy minimum clause (a) at the identified $β$ . For the canonical operator $ρ_{β} : = e^{- β H_{N}^{σ}} / Z_{N}$ and $Z_{N} : = tr e^{- β H_{N}^{σ}} \in (0, \infty)$ (positive finite sum in the nonzero finite-dimensional sector), the number $F_{e q} : = - β^{- 1} \log Z_{N}$ is called the equilibrium free energy; it is the value of the functional at $ρ_{β}$ by the cited theorem.

(f) Scope. This definition covers only declared models with finite-dimensional sector space, with the zero-sector branch of (a), and preparations in $D (H_{N}^{σ})$ . For a declared spatial realization, whose sector space is infinite-dimensional, the operator $ρ \log ρ$ need not be trace class; no entropy, physical entropy or free energy is defined here for that case, and a separate trace-class statement would be required. The definition asserts no increase of entropy in time, no equilibration to $ρ_{β}$ , no thermodynamic limit, no pressure or volume variable, and no measured temperature, entropy or free energy.

Remarks

Why the finite case first. In the finite-dimensional sector the entropy is an ordinary finite sum over $D$ eigenvalues, and the comparison theorem of Finite Gibbs free-energy minimum is available without any trace-class convergence hypothesis on $ρ \log ρ$ . The infinite-dimensional questions are deferred rather than assumed.
Units of $β$ . $β = 1 / (k_{B} T)$ has the unit $J^{- 1} K K^{- 1} = J^{- 1}$ , so the exponent $β H_{N}^{σ}$ is dimensionless, as the spectral calculus requires.

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The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature

Physical scope

The adopted equilibrium-ensemble assumption of the many-body pair: for a declared fixed-N model or a separately declared number-variable mode model with self-adjoint semibounded Gibbs generator G (including a declared effective generator H-mu N), assigned temperature and strictly positive finite partition trace, the Gibbs preparation rho_beta = e^{-beta G}/Z, its thermal means, and the identifications of beta = 1/(k_B T) and of rho_beta with the model's description of thermal equilibrium. It is an adopted statistical-mechanical assumption motivated by maximum-entropy reasoning, not a consequence of the framework's state, probability or dynamics postulates; it asserts no thermalization, no equilibration rate, no thermodynamic limit and no fact about a laboratory sample except through the tested identifications of P7.

Postulate

(a) Adopted readings and framework data. Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1), the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2), the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4) and the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions (P7), together with the Boltzmann-constant and temperature conventions of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model. Declare one of the following model readings of the same adopted Gibbs assignment:

(i) a fixed- $N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable, on its sector space, with Gibbs generator $G : = H_{N}^{σ}$ ; or

(ii) a separately declared number-variable mode model, such as a one-mode bosonic Fock space, a fermionic occupation space, or a declared countable independent-mode space, with its declared assigned observables and energy operator $H$ . The model declares a self-adjoint Gibbs generator $G$ on that space: either $G = H$ , or a separately supplied self-adjoint realization of the effective generator $G = H - μ N$ , where $N$ is the declared particle-number operator and $μ \in R$ is an assigned chemical potential in joules. The subtraction is an operator with its supplied domain, not an assertion that a formal unbounded difference is self-adjoint.

In either reading assume $G$ is bounded below and give its spectral PVM $E_{G}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure). Let $T > 0$ be an assigned temperature in kelvin and let $β : = 1 / (k_{B} T)$ in $J^{- 1}$ with the assigned Boltzmann constant $k_{B}$ . The Gibbs assignment to either model, and the chemical-potential identification in (ii), are additional physical preparation assumptions made here; they are not derived from P1-P4 or from the fixed- $N$ construction. In (ii) no restriction on the assigned real $μ$ is imposed beyond the declared generator and trace existence conditions; the mode-specific convergence conditions are proved separately. For reading (ii), $G$ specifies the Gibbs weights. If the assigned autonomous dynamics is generated by $H$ , stationarity of the Gibbs density requires its separate dynamical or commutation hypotheses; this assignment does not identify $H - μ N$ with the physical evolution generator. The physical Helmholtz free-energy vocabulary of the entropy definition continues to apply only to its stated nonzero finite-sector class and is not extended to every effective generator by this postulate. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the choice hypotheses of the spectral calculus.

(b) Existence hypothesis and the Gibbs preparation. The model assumes that the partition trace is strictly positive and finite: $0 < Z : = Tr e^{- β G} < \infty,$ where $Tr B : = \sum_{f \in B} ⟨ B f, f ⟩$ for a bounded positive $B$ and an orthonormal basis $B$ , with value allowed to be $+ \infty$ . Such a basis exists under the declared AC by the maximal orthonormal-family argument using Zorn's lemma and the zero-complement criterion of Parseval equivalences for an orthonormal family. This trace is basis independent: for $C : = B^{1 / 2}$ of Positive square root, the defining sum is $\sum_{f \in B} ∥ C f ∥^{2}$ , whose invariance is The Hilbert–Schmidt norm is basis independent. When the sum is finite, $C$ is Hilbert-Schmidt (Hilbert–Schmidt operator and Hilbert–Schmidt norm), so $B = C^{2}$ is trace class by Trace class iff product of two Hilbert Schmidt operators, and its ordinary trace is this same diagonal sum by Trace is absolutely convergent and basis independent. Apply this to the bounded positive operator $e^{- β G}$ , since the declared Gibbs generator $G$ is bounded below. The model adopts as the canonical or Gibbs preparation of the declared model at temperature $T$ the density operator $ρ_{β} : = Z^{- 1} e^{- β G} .$ If the sector is zero-dimensional, its partition trace is $Z = 0$ , no trace-one density exists, and no canonical density or equilibrium free energy is assigned. On a nonzero finite-dimensional model space, $Z$ is a positive finite sum $\sum_{j} e^{- β E_{j}}$ over the eigenvalues of $G$ ; no energy sign restriction is needed. There $ρ_{β}$ is the canonical operator of Finite Gibbs free-energy minimum, and the strict positivity and finiteness hypothesis is automatic. In reading (i), $Z = Z_{N}$ and this is precisely the finite-sector Gibbs assignment.

Outside the finite-dimensional class the existence hypothesis is a genuine restriction. For a declared non-interacting fixed- $N$ model with a supplied countably infinite orthonormal eigenmode basis and real mode energies $ε_{j}$ bounded below, the complete construction and trace criterion of Fixed-particle-number occupation bases and the countable-mode partition criterion give $Z_{N} = h_{N} (q)$ for bosonic statistics or $Z_{N} = e_{N} (q)$ for fermionic statistics, where $q_{j} = e^{- β ε_{j}}$ and $h_{N}, e_{N}$ are the corresponding nonnegative occupation-series sums. For every fixed $N \geq 1$ , either trace is finite exactly when $\sum_{j} q_{j} < \infty$ . This is a trace on the fixed- $N$ sector and permits negative mode energies.

For the separately declared full bosonic Fock models of reading (ii), the single-mode calculation of Single-mode thermal trace and the product criterion of Countable-mode product criterion provide the separate mathematical full-Fock traces, with their original positive mode-energy hypotheses: they sum over every particle number, including the vacuum, giving $(1 - e^{- β ε})^{- 1}$ for one mode and $\prod_{j} (1 - e^{- β ε_{j}})^{- 1}$ for countably many modes. Their physical Gibbs reading is the additional assignment of (a)(ii), not a consequence of a mathematical trace formula. Those full-Fock traces are not substituted for the fixed- $N$ partition trace of reading (i).

(c) The equilibrium assumption and thermal means. The model assumes that, for a declared sample of the species at the assigned temperature $T$ and in the absence of further preparation information, the preparation description of the sample is $ρ_{β}$ in the sense of P1. For an assigned sharp observable $(A, u_{A})$ with spectral PVM $E_{A}$ , define its outcome law $μ_{ρ_{β}}^{A} (B) : = tr (ρ_{β} E_{A} (B))$ as in Assigned sharp observables and the outcome law of the framework clause (b). If and only if the absolute first moment is finite, $\int_{R} ∣ a ∣ d μ_{ρ_{β}}^{A} (a) < \infty,$ the model defines the thermal mean at temperature $T$ by $⟨ A ⟩_{β} : = \int_{R} a d μ_{ρ_{β}}^{A} (a),$ read in the assigned unit $u_{A}$ . For bounded $A$ this equals $tr (ρ_{β} A) = tr (A ρ_{β})$ . For unbounded $A$ , if the mean exists and the literal product $A ρ_{β}$ is defined on the whole Hilbert space and is trace class, the mean also equals $tr (A ρ_{β})$ and $tr (\overline{ρ_{β} A})$ ; these conditional product identities are proved by the density eigenexpansion in Spectral first moments and trace products with a density operator. First-moment existence alone imposes no such product or operator-square-domain condition. When the absolute first moment is infinite the model asserts no finite thermal mean. The identification of the parameter $T$ with the thermodynamic temperature of a sample, and of $β$ with its inverse temperature, is part of this assumption; the stated conservation and equilibrium properties of $ρ_{β}$ under the flow of P4, and the extremal property of $ρ_{β}$ among preparations, are separate conditional results of this pair.

(d) Motivation and status. The assumption is motivated, but not established, by the maximum-entropy reasoning of the cited sources: among preparations with a prescribed mean energy, the Gibbs weights maximize the entropy functional, and the finite-dimensional comparison theorem makes $ρ_{β}$ the minimizer of the corresponding functional built from $G$ . Its physical Helmholtz free-energy reading applies in the stated finite fixed-sector class; an effective-generator interpretation in a different mode model is a separate declared reading. That reasoning is an argument inside mathematical statistics and thermodynamics; it does not derive the physical assumption that a sample is described by $ρ_{β}$ . The postulate is an adopted statistical-mechanical assumption of the model, in the sense of the content model's distinction between adopted assumptions and derived consequences.

(e) Scope. The postulate asserts nothing about how a sample reaches the Gibbs preparation; in particular it asserts no thermalization, no equilibration time, no ergodicity, no master-equation or Markov approximation, no thermodynamic limit, no pressure, volume or phase-transition structure, and no measured temperature or expectation. The requirement $0 < Z < \infty$ is part of the assumption; the model does not define a canonical preparation when $Z = 0$ or the partition trace diverges. Any comparison with a laboratory sample requires the declared preparation, isolation, detection and sampling models of P7, and agreement of a thermal prediction with data is evidence about the combined model and apparatus assumptions, not a proof of the framework or of this postulate.

Remarks

The finite-sector case. For a nonzero finite-dimensional sector the existence hypothesis is a positive finite sum and hence automatic (and likewise for any separately declared nonzero finite-dimensional mode space); the only content of clause (b) there is the assignment of the Gibbs weights to the declared energies.
Relation to P4. The postulate adds a preparation assumption to the dynamics; by itself P4 describes the unitary evolution of whatever preparation is declared, and does not single out $ρ_{β}$ . The precise sense in which unitary evolution cannot produce $ρ_{β}$ from a different preparation is a separate theorem of this pair.

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

Physical scope

The conditional diagonalisation of the non-interacting finite-mode many-body model: the labelled energy sum is diagonal on the product basis of one-particle eigenmodes, the sector energy operator has the occupation-number eigenvalues on the Slater and symmetric sector bases, the finite partition sum and the ground-state occupation energies are computed, and the sector basis vectors are stationary states of the model evolution. The theorem is conditional on P1, P2, P4 and P6, on the adopted finite-mode idealization and on the declared one-particle Hamiltonian; it asserts no interaction, no spatial or infinite-mode result, no thermal state, no thermodynamic limit and no laboratory identification.

Statement

(a) the labelled operator is diagonal: $H_{l a b}$ is bounded and self-adjoint with $∥ H_{l a b} ∥ \leq N ∥ h ∥$ , it commutes with every permutation unitary $U_{π}$ , and for every multi-index $i = (i_{1}, \dots, i_{N}) \in {1, \dots, d}^{N}$ , $H_{l a b} (e_{i_{1}} \otimes \dots \otimes e_{i_{N}}) = (\sum_{k = 1}^{N} ε_{i_{k}}) e_{i_{1}} \otimes \dots \otimes e_{i_{N}};$

(b) occupation eigenvalues and spectrum: for every $N$ -element subset $J \subseteq {1, \dots, d}$ the Slater vector $e_{J}^{\land}$ of Slater and symmetric occupation-number bases of the finite-dimensional permutation sectors clause (b) is an eigenvector of $H_{N}^{-}$ with eigenvalue $λ_{J} : = \sum_{j \in J} ε_{j}$ , and for every occupation vector $λ \in W (N, d) = {λ \in N^{d} : \sum_{j} λ_{j} = N}$ the symmetric vector $m_{λ}$ of clause (c) is an eigenvector of $H_{N}^{+}$ with eigenvalue $λ_{λ} : = \sum_{j = 1}^{d} λ_{j} ε_{j}$ ; the families $(e_{J}^{\land})$ and $(m_{λ})$ are orthogonal bases of $H_{N}^{-}$ and $H_{N}^{+}$ respectively, so the spectrum of $H_{N}^{σ}$ is the finite set of these eigenvalues (with multiplicities given by the number of configurations producing them); for $σ = -$ and $N > d$ the sector space is ${0}$ and the model has no sector states;

(c) partition sum and ground energy: write $C_{N}^{σ} : = {λ \in W (N, d) : λ_{j} \leq 1 for every j if σ = -}, M_{N}^{σ} : = ∣ C_{N}^{σ} ∣ = \dim H_{N}^{σ} .$ The configuration counts are $M_{N}^{-} = (\binom{d}{N})$ and $M_{N}^{+} = (\binom{N + d - 1}{N})$ . For every inverse-energy parameter $β > 0$ in $J^{- 1}$ (all $ε_{j}$ and configuration energies are read in joules, so each Boltzmann exponent is dimensionless), $Z_{N} (β) : = tr e^{- β H_{N}^{σ}} = \sum_{λ \in C_{N}^{σ}} e^{- β λ_{λ}}$ is finite. If $σ = -$ and $N > d$ , then $C_{N}^{-} = \emptyset$ , $Z_{N} (β) = 0$ , 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 $M_{N}^{σ} > 0$ and $0 < Z_{N} (β) < \infty$ . In this nonzero-sector case, let $E_{0} : = \min_{λ \in C_{N}^{σ}} λ_{λ}, E_{\max} : = \max_{λ \in C_{N}^{σ}} λ_{λ} .$ Then the actual Boltzmann bounds are $M_{N}^{σ} e^{- β E_{\max}} \leq Z_{N} (β) \leq M_{N}^{σ} e^{- β E_{0}} .$ No nonnegativity hypothesis on the $ε_{j}$ is imposed. For $σ = -$ and $N \leq d$ , $E_{0} = \sum_{k = 1}^{N} ε_{(k)}, E_{\max} = \sum_{k = d - N + 1}^{d} ε_{(k)},$ and for $σ = +$ , $E_{0} = N ε_{(1)}, E_{\max} = N ε_{(d)} .$ For every unit vector $ψ \in H_{N}^{σ}$ in either nonzero sector, $⟨ H_{N}^{σ} ψ, ψ ⟩ \geq E_{0}$ , with equality exactly on the eigenspace of $E_{0}$ ;

(d) reading and stationarity: on a nonzero sector the pair $(H_{N}^{σ}, J)$ is the model's assigned energy observable; in a declared sector preparation $ρ$ its outcome law is $μ_{ρ} (B) = tr (ρ E_{H_{N}^{σ}} (B))$ , 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 $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ , 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.

Facts & Assumptions

Given: the finite-mode non-interacting model data of the statement with its eigenbasis $(e_{1}, \dots, e_{d})$ , eigenvalues $ε_{j}$ , sector projections $P_{σ}^{(N)}$ and sector space $H_{N}^{σ}$ .

[F1]

Finite spectral theorem. A self-adjoint endomorphism $h$ of the finite-dimensional complex inner product space $K$ is normal, so $K$ has an orthonormal basis of eigenvectors of $h$ with real eigenvalues (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, Hilbert space).

[F2]

Slot operators. For $h \in B (K)$ the slot operator $h^{(i)}$ acts on elementary tensors by $h^{(i)} (v_{1} \otimes \dots \otimes v_{N}) = v_{1} \otimes \dots \otimes h v_{i} \otimes \dots \otimes v_{N}$ , is bounded with $∥ h^{(i)} ∥ \leq ∥ h ∥$ , and $(h^{(i)})^{*} = (h^{*})^{(i)}$ (Bounded tensor maps: extension, norm estimate, adjoint and commutation, A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F3]

Product basis and occupation bases. The elementary tensors $e_{i} = e_{i_{1}} \otimes \dots \otimes e_{i_{N}}$ over all multi-indices form an orthonormal basis of $K^{\otimes N}$ ; the Slater vectors $e_{J}^{\land} = \sqrt{N!} P_{-}^{(N)} (e_{j_{1}} \otimes \dots \otimes e_{j_{N}})$ over $N$ -element subsets $J = {j_{1} < \dots < j_{N}}$ form an orthonormal basis of $ran P_{-}^{(N)}$ with $e_{J}^{\land}$ zero-free for distinct indices and vanishing on repeated indices; the vectors $m_{λ} = \sum_{c \in orb (λ)} e_{c}$ over $λ \in W (N, d)$ form an orthogonal basis of $ran P_{+}^{(N)}$ , and $N_{j} e_{J}^{\land} = 1_{J} (j) e_{J}^{\land}$ , $N_{j} m_{λ} = λ_{j} m_{λ}$ (Slater and symmetric occupation-number bases of the finite-dimensional permutation sectors clauses (a), (b), (c), (d), Permutation sector projections on the n-fold tensor power, Hilbert tensor construction: inner product, completion and product bases).

[F4]

Configuration counts. The map $J \mapsto 1_{J}$ is a bijection from $N$ -element subsets onto ${λ \in {0, 1}^{d} : \sum_{j} λ_{j} = N}$ , the sector dimensions are $(\binom{d}{N})$ and $(\binom{N + d - 1}{N})$ , and for $N > d$ the antisymmetric sector is ${0}$ (Finite-mode occupation configurations: fermionic exclusion counts, bosonic counts and truncated dimensions clauses (a), (b), (c), Slater and symmetric occupation-number bases of the finite-dimensional permutation sectors clause (b)).

[F5]

Stationary states. For the assigned energy observable $(H_{o p}, u_{H})$ with flow $U (t) = \exp (- i t H_{o p} / ℏ)$ , a nonzero $ψ$ is an eigenvector of $H_{o p}$ if and only if its pure preparation is stationary, and then $ψ (t) = e^{- i t λ / ℏ} ψ$ (Energy conservation, conserved sharp observables and the stationary eigenrays clause (c)); the outcome law of an assigned observable in a preparation is the trace law of Assigned sharp observables and the outcome law of the framework clause (b), and the trace is basis independent (Trace is absolutely convergent and basis independent).

Proof

technique · direct; expand in the product basis of one-particle eigenmodes, transport the diagonal action to the two sector bases, and read the finite spectrum and trace

Given: the data of the statement and [F1]-[F5].

1.1F1F2F3algebra

[The labelled operator on the product basis] By [F2] each $h^{(i)}$ is bounded and self-adjoint, so $H_{l a b} = \sum_{i} h^{(i)}$ is bounded and self-adjoint with $∥ H_{l a b} ∥ \leq N ∥ h ∥$ . On the product basis vector $e_{i} = e_{i_{1}} \otimes \dots \otimes e_{i_{N}}$ the $i$ -th slot operator acts as $h$ on the factor $e_{i_{k}}$ , giving $h^{(k)} e_{i} = ε_{i_{k}} e_{i}$ ; hence $H_{l a b} e_{i} = (\sum_{k} ε_{i_{k}}) e_{i}$ . Since every $U_{π}$ permutes the slots and $H_{l a b}$ is the sum over all slots, $U_{π} H_{l a b} U_{π}^{*} = H_{l a b}$ . This proves (a).

2.1F3F4step 1.1algebra

[Sector eigenvectors] Let $J = {j_{1} < \dots < j_{N}}$ and $i = (j_{1}, \dots, j_{N})$ . By [F3], $e_{J}^{\land} = \sqrt{N!} P_{-}^{(N)} e_{i}$ ; since $H_{l a b}$ commutes with $P_{-}^{(N)}$ by step 1.1 and [F3], $H_{l a b} e_{J}^{\land} = \sqrt{N!} P_{-}^{(N)} H_{l a b} e_{i} = (\sum_{j \in J} ε_{j}) \sqrt{N!} P_{-}^{(N)} e_{i} = λ_{J} e_{J}^{\land}$ . Likewise, for $λ \in W (N, d)$ and any multi-index $c$ with $∣ c ∣ = λ$ one has $P_{+}^{(N)} e_{c} = γ_{c} m_{λ}$ with $γ_{c} > 0$ by [F3], so $H_{l a b} m_{λ} = (\sum_{j} λ_{j} ε_{j}) m_{λ}$ ; because the $e_{J}^{\land}$ and the $m_{λ}$ are orthogonal bases of the respective sectors by [F3], [F4], the spectrum of $H_{N}^{σ}$ is the finite set of the displayed eigenvalues with the configurations as multiplicities, and for $σ = -$ , $N > d$ the sector is ${0}$ . This proves (b).

3.1F3F4F5step 2.1algebra

[Partition sum and the zero sector] Normalize the symmetric vectors to ${\tilde{m}}_{λ} : = m_{λ} / ∥ m_{λ} ∥$ ; by [F3] these form an orthonormal eigenbasis, as do the Slater vectors in the fermionic sector. The trace is therefore $Z_{N} (β) = \sum_{λ \in C_{N}^{σ}} e^{- β λ_{λ}}$ , with the configuration counts $M_{N}^{-} = (\binom{d}{N})$ and $M_{N}^{+} = (\binom{N + d - 1}{N})$ from [F4]. For $σ = -$ , $N > d$ , the index set is empty and the trace is the empty sum $0$ . There is no eigenvector on ${0}$ and hence no ground eigenvalue, and its only operator has trace $0$ , so no trace-one density or normalized Gibbs density exists there. In every other case [F4] gives a finite nonempty index set, so its finite sum of positive Boltzmann weights has $0 < Z_{N} (β) < \infty$ . For that case finite minima and maxima $E_{0}, E_{\max}$ exist, and for each configuration $e^{- β E_{\max}} \leq e^{- β λ_{λ}} \leq e^{- β E_{0}}$ , because $β > 0$ ; summing the $M_{N}^{σ}$ inequalities proves the two bounds. This argument allows negative configuration energies.

3.2F5step 2.1algebra

[Reading and stationarity] The pair $(H_{N}^{σ}, J)$ is the model's assigned energy observable of clause (e) of the declared-model definition, so by [F5] its outcome law in a declared sector preparation $ρ$ is $μ_{ρ} (B) = tr (ρ E_{H_{N}^{σ}} (B))$ ; for the pure preparation of a sector basis vector $Ψ$ of step 2.1, $E ({λ}) Ψ = Ψ$ and hence $μ_{Ψ}$ is the point mass at $λ$ . By [F5] and step 2.1 each such $Ψ$ is an eigenvector of the generator, so its pure preparation is stationary under the flow $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ with $Ψ (t) = e^{- i t λ / ℏ} Ψ$ , and the energy law of any preparation is the finite spectral law in the sector basis. This proves (d).

4.1F3F4step 2.1step 3.1algebra

[Extreme energies and ground eigenspace] Work on a nonzero sector. For $σ = -$ , each configuration selects $N$ distinct eigenvalues, so its energy is at least the sum of the $N$ smallest and at most the sum of the $N$ largest; these bounds are attained by the corresponding subsets. Thus $E_{0} = \sum_{k = 1}^{N} ε_{(k)}$ and $E_{\max} = \sum_{k = d - N + 1}^{d} ε_{(k)}$ . A minimizing subset contains every mode with energy strictly below $ε_{(N)}$ , contains no mode with energy strictly above $ε_{(N)}$ , and fills its remaining places with modes at that boundary energy; this includes all choices allowed by degeneracy. For $σ = +$ , the identity $\sum_{j} λ_{j} = N$ gives $N ε_{(1)} \leq \sum_{j} λ_{j} ε_{j} \leq N ε_{(d)}$ ; the two bounds are attained by putting all particles in a minimum-energy or maximum-energy mode respectively. Equality at the lower bound holds exactly when every occupied mode has energy $ε_{(1)}$ . Finally expand a unit vector in the orthonormal sector basis of step 3.1: $ψ = \sum_{λ} c_{λ} b_{λ}$ , $\sum_{λ} ∣ c_{λ} ∣^{2} = 1$ . Then $⟨ H_{N}^{σ} ψ, ψ ⟩ - E_{0} = \sum_{λ} ∣ c_{λ} ∣^{2} (λ_{λ} - E_{0}) \geq 0$ , with equality exactly when all coefficients at energies greater than $E_{0}$ vanish, that is, exactly on the ground eigenspace. Together with step 3.1 this proves (c).

5.1step 1.1step 2.1step 3.1step 4.1step 3.2∎

[Conclusion and scope] Steps 1.1 and 2.1 prove (a) and (b), steps 3.1 and 4.1 prove (c) and step 3.2 proves (d). The argument used the finite-mode idealization, the declared non-interacting $h$ and the finite-dimensional spectral and tensor calculus; no interaction, spatial realization, infinite mode family, thermal state, equilibration or laboratory identification entered, which is (e).

Remarks

Basis independence. The eigenvalues $λ_{J}$ and $λ_{λ}$ 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 $σ = -$ 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 $Z_{N} \leq M_{N}^{σ}$ ; this count-only upper bound is not valid for unrestricted negative energies. Shifting every one-particle energy by $a \in R$ multiplies $Z_{N}$ by $e^{- β N a}$ and shifts both extreme energies by $N a$ .

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The one-body readout law: sector means of summed bounded one-particle quantities and their reduced forms

Physical scope

The conditional one-body readout law of the declared finite-mode many-body model: for a bounded self-adjoint one-particle quantity the summed sector operator is an assigned sharp observable, its mean in every declared sector preparation exists and is N times the pairing of the one-body reduced preparation with the quantity, and for declared Slater and orthonormal-mode symmetric preparations the reduced preparation is the equal-weight mixture of the occupied modes. The theorem is conditional on P1, P2 and P6, the adopted finite-mode idealization and the declared preparation; it asserts no measured mean, no individual-particle property, no separability of the sector state and no spatial (infinite-mode) reduction.

Statement

(a) the readout is an assigned observable: the sector operator $A_{O} = Σ (O) ∣_{H_{N}^{σ}}$ is bounded and self-adjoint, so the pair $(A_{O}, u_{O})$ is an assigned sharp observable of the sector in the sense of Assigned sharp observables and the outcome law of the framework clause (a);

(b) the mean law: for every declared sector preparation $ρ$ the mean of the outcome law of $(A_{O}, u_{O})$ in $ρ$ exists and equals $m_{ρ} (A_{O}) = N tr (γ_{ρ} O);$

(c) product-type preparations: let $u_{1}, \dots, u_{N} \in K$ be pairwise orthonormal. For $σ = -$ and $N \geq 2$ , let $Ψ : = S (u_{1}, \dots, u_{N})$ be the Slater vector of Slater direct and exchange calculation clause (a); for $σ = -$ and $N = 1$ , put $S (u_{1}) : = u_{1}$ and $Ψ : = u_{1}$ . For $σ = +$ , put $Ψ : = \sqrt{N!} P_{+}^{(N)} (u_{1} \otimes \dots \otimes u_{N})$ . In either case $Ψ$ is a unit vector of the sector. For $ρ : = T_{Ψ, Ψ}$ one has $γ_{ρ} = N^{- 1} \sum_{j = 1}^{N} T_{u_{j}, u_{j}}$ and $m_{ρ} (A_{O}) = \sum_{j = 1}^{N} ⟨ O u_{j}, u_{j} ⟩ .$

(d) scope: the theorem is conditional on the adopted sector of P6, on the declared finite-mode idealization and on the declared preparation; it asserts no measured mean, no property of an individual labelled particle, no separability of $ρ$ , no result for spatial infinite-mode models, and no sampling or detection model. The equality $m_{ρ} (A_{O}) = N tr (γ_{ρ} O)$ is exact within the model.

Facts & Assumptions

Given: the finite-mode sector data of the statement, the self-adjoint $O \in B (K)$ with unit $u_{O}$ , and a declared sector preparation $ρ$ .

[F1]

Sector vocabulary. $H_{N}^{σ} = ran P_{σ}^{(N)}$ is a closed subspace of $K^{\otimes N}$ ; every permutation unitary $U_{π}$ preserves it and acts there as the scalar ${sgn}_{σ} (π)$ ; a declared sector preparation is a density operator on $H_{N}^{σ}$ (Declared species sectors and their preparations: the framework's reading of P6 clauses (c)-(d), Permutation sector projections on the n-fold tensor power clause (c)).

[F2]

Slot operators and reduction. For $O \in B (K)$ and $i \in {1, \dots, N}$ the slot operator $O^{(i)}$ satisfies $O^{(i)} (v_{1} \otimes \dots \otimes v_{N}) = v_{1} \otimes \dots \otimes O v_{i} \otimes \dots \otimes v_{N}$ , is bounded with $∥ O^{(i)} ∥ \leq ∥ O ∥$ , and $(O^{(i)})^{*} = (O^{*})^{(i)}$ ; $Σ (O) = \sum_{i} O^{(i)}$ commutes with every $U_{π}$ and with $P_{σ}^{(N)}$ , preserves the sectors, and for every sector preparation $ρ$ with zero extension $\tilde{ρ}$ one has $tr (\tilde{ρ} Σ (O)) = N tr (γ_{ρ} O)$ , where $γ_{ρ}$ is a density operator on $K$ (The one-body reduction trace law of the finite-mode permutation sectors clauses (a), (c), (d), (e)).

[F3]

Observable reading and moments. An assigned sharp observable $(A, u_{A})$ has outcome law $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ ; for a bounded self-adjoint $A$ and every preparation $ρ$ the mean exists and equals $tr (ρ A)$ , and for a pure preparation of a unit vector $ψ$ it equals $⟨ A ψ, ψ ⟩$ (Assigned sharp observables and the outcome law of the framework clauses (a), (b), Mean, second moment and variance of an assigned observable in a preparation clauses (a), (d), Trace is a linear functional on $M_{n} (F)$ , Cyclicity of the trace).

[F4]

Slater and symmetric product vectors. For $N \geq 2$ and pairwise orthonormal $u_{1}, \dots, u_{N}$ the Slater vector $S (u_{1}, \dots, u_{N}) = N!^{- 1 / 2} \sum_{σ \in S_{N}} sgn (σ) u_{\hat{σ} (1)} \otimes \dots \otimes u_{\hat{σ} (N)}$ has norm one and equals $\sqrt{N!} P_{-}^{(N)} (u_{1} \otimes \dots \otimes u_{N})$ (Slater direct and exchange calculation clauses (a), (b)). For $N = 1$ , $S (u_{1}) : = u_{1}$ and $P_{\pm}^{(1)} = I$ give the same norm and projection identities directly. Setting $x : = u_{1} \otimes \dots \otimes u_{N}$ , orthonormality gives $⟨ P_{+}^{(N)} x, x ⟩ = \frac{1}{N!} \sum_{σ \in S_{N}} ⟨ U_{σ} x, x ⟩ = \frac{1}{N!} ⟨ x, x ⟩ = \frac{1}{N!}$ , so $Ψ : = \sqrt{N!} P_{+}^{(N)} x$ is a unit vector of the symmetric sector with $P_{+}^{(N)} Ψ = Ψ$ (Permutation sector projections on the n-fold tensor power clauses (b), (c), Hilbert tensor construction: inner product, completion and product bases).

[F5]

Rank-one pairing. $T_{u, v} x = ⟨ x, v ⟩ u$ is bounded with $tr (T_{u, v}) = ⟨ u, v ⟩$ and $T_{u, v}^{*} = T_{v, u}$ ; the map $(u, v) \mapsto T_{u, v}$ is linear in $u$ and conjugate-linear in $v$ (Rank-one operator calculus clauses (a), (b), (c)).

Proof

technique · direct; read the sector sum as a bounded observable, apply the supplied reduction identity, and evaluate the product-type cases by orthonormality of the mode family

Given: the data of the statement and [F1]-[F5].

1.1F2F3def. boundedalgebra

[The readout is a bounded self-adjoint sector observable] By [F2] each slot operator is bounded with $∥ O^{(i)} ∥ \leq ∥ O ∥$ , so $Σ (O)$ is bounded with $∥ Σ (O) ∥ \leq N ∥ O ∥$ ; since $O = O^{*}$ , [F2] gives $O^{(i) *} = O^{(i)}$ for every $i$ and $Σ (O)^{*} = Σ (O)$ . By [F2] $Σ (O)$ commutes with $P_{σ}^{(N)}$ , hence preserves $H_{N}^{σ}$ and its orthogonal complement, and the restriction $A_{O} = Σ (O) ∣_{H_{N}^{σ}}$ is a bounded self-adjoint operator on the sector space; with the unit $u_{O}$ the pair $(A_{O}, u_{O})$ is therefore an assigned sharp observable in the sense of [F3].

1.2F1F2F4algebra

[Reduced preparation of product-type states] Let $u_{1}, \dots, u_{N}$ be pairwise orthonormal, put $x : = u_{1} \otimes \dots \otimes u_{N}$ and let $Ψ$ be the unit vector of clause (c) for $σ = -$ or $σ = +$ ; both satisfy $P_{σ}^{(N)} Ψ = Ψ$ by [F4], and both equal $\sqrt{N!} P_{σ}^{(N)} x$ . Since $Σ (O)$ commutes with $P_{σ}^{(N)}$ and $P_{σ}^{(N)}$ is self-adjoint, $⟨ Σ (O) Ψ, Ψ ⟩ = N! ⟨ Σ (O) P_{σ}^{(N)} x, P_{σ}^{(N)} x ⟩ = N! ⟨ P_{σ}^{(N)} Σ (O) P_{σ}^{(N)} x, x ⟩ = N! \cdot \frac{1}{N!} \sum_{τ \in S_{N}} {sgn}_{σ} (τ) ⟨ Σ (O) U_{τ} x, x ⟩$ , with ${sgn}_{+} = 1$ and ${sgn}_{-} = sgn$ . For fixed $τ$ , $⟨ Σ (O) U_{τ} x, x ⟩ = \sum_{i = 1}^{N} ⟨ O u_{{\hat{τ}}^{- 1} (i)}, u_{i} ⟩ \prod_{k \neq i} ⟨ u_{{\hat{τ}}^{- 1} (k)}, u_{k} ⟩$ by [F1] and the elementary-tensor pairing, and orthonormality makes every factor $δ_{{\hat{τ}}^{- 1} (k), k}$ vanish unless $τ = i d$ ; hence only the identity permutation contributes and $⟨ Σ (O) Ψ, Ψ ⟩ = \sum_{i = 1}^{N} ⟨ O u_{i}, u_{i} ⟩$ .

2.1F2F3step 1.1algebra

[The mean law] Let $ρ$ be a declared sector preparation with zero extension $\tilde{ρ}$ and let $A_{O}$ be the readout of step 1.1. Since $A_{O}$ is bounded, [F3] gives $m_{ρ} (A_{O}) = tr (ρ A_{O})$ , and by steps 1.1 and [F2], $tr (ρ A_{O}) = tr (\tilde{ρ} Σ (O)) = N tr (γ_{ρ} O)$ . This proves (a) and (b).

3.1F5step 1.2step 2.1algebra

[Product-type means and the reduced preparation] In the situation of step 1.2, the mean is $m_{ρ} (A_{O}) = ⟨ Σ (O) Ψ, Ψ ⟩ = \sum_{j = 1}^{N} ⟨ O u_{j}, u_{j} ⟩$ , while the mean law of step 2.1 gives $m_{ρ} (A_{O}) = N tr (γ_{ρ} O)$ . The identity extends from self-adjoint $O$ to every $O$ by writing $O = (O + O^{*}) / 2 + i (O - O^{*}) / (2 i)$ and using linearity of both pairings. Testing it against $O = T_{u, v}$ and using [F5], $tr (γ_{ρ} T_{u, v}) = ⟨ γ_{ρ} u, v ⟩$ and $N^{- 1} \sum_{j} ⟨ T_{u, v} u_{j}, u_{j} ⟩ = N^{- 1} \sum_{j} ⟨ u_{j}, v ⟩ ⟨ u, u_{j} ⟩$ ; the identity for all $u, v$ therefore gives $γ_{ρ} = N^{- 1} \sum_{j = 1}^{N} T_{u_{j}, u_{j}}$ , since two endomorphisms of a finite-dimensional space with equal pairings against every rank-one operator are equal. This proves (c).

4.1step 1.1step 2.1step 3.1∎

[Conclusion and scope] Steps 1.1-2.1 prove (a) and (b) and step 3.1 proves (c). Every quantity is a spectral datum of the declared finite-mode sector model and the argument used only P1, P2, P6 and the finite-dimensional tensor and trace calculus; no measured mean, individual-particle property, separability statement or spatial reduction entered, which is (d).

Remarks

The factor $N$ and the normalization of $γ_{ρ}$ . The reduced preparation is normalized as a probability state on the one-particle space, while $A_{O}$ sums $N$ slot terms; the product-type case shows the two conventions consistently: a Slater determinant of $N$ orthonormal modes has reduced preparation the equal-weight mixture of those modes.
No separability claim. The mean law holds for arbitrary sector preparations, entangled or not; nothing in the argument factors $ρ$ , and the product-type clause is a special case.

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Direct and exchange contributions to a two-fermion mean energy and the repulsive-Coulomb ordering

Physical scope

The conditional two-fermion exchange computation of the declared N=2 bounded or Coulomb model: for a declared pair of orthonormal spatial orbitals and the two declared antisymmetric sector preparations obtained by combining them with the spin singlet and triplet, the mean pair energies are the sum of the one-body means plus the direct integral and minus or plus the exchange integral, and for a repulsive Coulomb pair interaction the spatially antisymmetric (spin-triplet) preparation has the lower mean energy. The comparison is a statement about the declared model's means; it derives no spin-statistics assignment, reports no measured level splitting and makes no claim about a real atom.

Statement

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 non-relativistic kinematics postulate P5 (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators) 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, Assigned sharp observables and the outcome law of the framework and Mean, second moment and variance of an assigned observable in a preparation, and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Take a declared spin- $\frac{1}{2}$ fermionic species with mass $m > 0$ , the spatial 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 with $N = 2$ , a bounded real external potential $V_{e x t}$ and one of the following two pair interactions: (bounded case) a bounded real even $W : R^{3} \to R$ ; (Coulomb case) $W (z) = κ / ∣ z ∣$ with $κ > 0$ and $V_{e x t}$ bounded. Let $u, v \in W^{2, 2} (R^{3}; C)$ be pairwise orthonormal spatial orbitals, put $ε_{u} : = ⟨ h u, u ⟩_{⋆}, ε_{v} : = ⟨ h v, v ⟩_{⋆}$ with $h = - ℏ^{2} Δ / (2 m) + V_{e x t}$ the declared one-particle operator and $⟨ \cdot, \cdot ⟩_{⋆}$ the $L^{2} (R^{3})$ pairing, and let $D$ and $X$ be the direct and exchange integrals $D = \int_{R^{6}} ∣ u (x) ∣^{2} ∣ v (y) ∣^{2} W (x - y) d x d y, X = \int_{R^{6}} \overline{u (x)} v (x) u (y) \overline{v (y)} W (x - y) d x d y$ of Slater direct and exchange calculation clauses (c),(d), as appropriate. Fix the standard orthonormal basis $e_{↑}, e_{↓}$ of the declared spin- $\frac{1}{2}$ factor $C^{2}$ , and define the unit spin vectors $χ_{-} = (e_{↑} \otimes e_{↓} - e_{↓} \otimes e_{↑}) / \sqrt{2}, χ_{+} = (e_{↑} \otimes e_{↓} + e_{↓} \otimes e_{↑}) / \sqrt{2},$ read as the singlet and symmetric $S_{z} = 0$ triplet of the declared basis. Let $Θ_{s p}$ be the scalar Euclidean product unitary and let $Θ_{2} : (L^{2} (R^{3}; C^{2}))^{\otimes 2} ⟶ L^{2} (R^{6}; C^{2} \otimes C^{2})$ be the full spin-product unitary of Finite spin components of an N-fold spatial tensor product clause (a). Put $φ_{\pm} = (u \otimes v \pm v \otimes u) / \sqrt{2}, f_{\pm} = Θ_{s p} φ_{\pm}, Φ_{\pm} = Θ_{2}^{- 1} (f_{\pm} χ_{\mp}), ρ_{\pm} = T_{Φ_{\pm}, Φ_{\pm}} .$ Thus $Φ_{\pm}$ are vectors in the declared one-particle tensor square, not untransported vectors in a regrouped spatial/spin product. They define the two declared preparations after the sector membership proved below. No reality assumption is imposed on the complex orbitals $u, v$ .

(a) declared preparations: $Φ_{\pm}$ are unit vectors of the antisymmetric sector $H_{2}^{-}$ of the model, and $ρ_{\pm}$ are declared sector preparations;

(b) mean energies: the means of the declared energy observable in the two preparations exist, are finite and equal $m_{ρ_{+}} (H_{2}^{-}) = ε_{u} + ε_{v} + D + X, m_{ρ_{-}} (H_{2}^{-}) = ε_{u} + ε_{v} + D - X;$ in particular the exchange splitting of the two preparations is $m_{ρ_{+}} - m_{ρ_{-}} = 2 X$ ;

(c) repulsive-Coulomb ordering: in the Coulomb case with $u, v \in W^{1, 2} (R^{3}; C)$ one has $X \geq 0$ and therefore $m_{ρ_{-}} (H_{2}^{-}) \leq m_{ρ_{+}} (H_{2}^{-})$ : the spatially antisymmetric, spin-triplet preparation has the lower mean pair energy of the two;

(d) scope: clauses (a)-(c) are conditional consequences of the declared model, the declared orbitals and the adopted spin-statistics sector of P6; they derive no spin-statistics theorem, assert no measured energy difference, compare no experimental spectrum, and make no claim that the declared orbitals approximate any atomic state.

Facts & Assumptions

Given: the declared spin-1/2 fermionic species, $N = 2$ , the external and pair data of the statement, the orthonormal orbitals $u, v \in W^{2, 2}$ , the spin vectors $χ_{\pm}$ and the vectors $Φ_{\pm}$ .

[F1]

Full spin-product transport. The unitary $Θ_{2}$ maps $(f e_{a}) \otimes (g e_{b})$ to $(x, y) \mapsto f (x) g (y) (e_{a} \otimes e_{b})$ . The transported exchange acts as $(U Ψ) (x, y) = R Ψ (y, x)$ , where $R (ζ \otimes η) = η \otimes ζ$ swaps the two spin factors; its antisymmetric projection is $P_{-} = (I - U) / 2 = Θ_{2} P_{-}^{(2)} Θ_{2}^{- 1}$ . Finite spin components preserve the $H^{2}$ domain, and a permutation-invariant scalar self-adjoint operator with that domain restricts self-adjointly to this combined sector (Finite spin components of an N-fold spatial tensor product clauses (a)–(c)). Tensor pairings and norms are those of Hilbert tensor construction: inner product, completion and product bases, and the tensor exchange is that of Permutation sector projections on the n-fold tensor power.

[F2]

Scalar spatial products. The scalar product unitary $Θ_{s p}$ has $Θ_{s p} (u \otimes v) (x, y) = u (x) v (y)$ (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})). Fubini applies to integrable products and the finite component pairings (Fubini's theorem for L^1 functions on a sigma-finite product).

[F3]

Direct and exchange integrals, including complex orbitals. For a real bounded even $W$ , the displayed $D, X$ converge absolutely, are real and satisfy $⟨ M_{W} f_{\pm}, f_{\pm} ⟩ = D \pm X$ for arbitrary complex orthonormal $u, v \in L^{2}$ . For $W (z) = κ / ∣ z ∣$ , $κ > 0$ , the same assertion holds for arbitrary complex orthonormal $u, v \in H^{1}$ , with $f_{\pm} \in D (M_{W})$ , $D \geq 0$ and $X \geq 0$ . These are Slater direct and exchange calculation clauses (c),(d); its proof 2.1 proves Coulomb positivity with only the $L^{1}$ overlap density, without an unclaimed $L^{2}$ or real-orbital hypothesis.

[F4]

The exact labelled realization with bounded external addition. Put $a = ℏ^{2} / (2 m)$ and $V_{b} (x, y) = V_{e x t} (x) + V_{e x t} (y)$ . In the bounded case the scalar labelled operator $T = - a Δ_{(x, y)} + V_{b} + W (x - y)$ is self-adjoint on $H^{2} (R^{6}; C)$ by CC realization of the bounded spatial operator on a finite-component H² domain, with the same finite-component realization on $H^{2} (R^{6}; C^{4})$ . In the Coulomb case apply Fixed-N Coulomb self-adjointness clauses (b),(c) with mathematical masses ${\tilde{m}}_{1} = {\tilde{m}}_{2} = m / (4 π^{2})$ , its parameter $ℏ$ unchanged, zero external singular coefficients and pair coefficient $κ$ . Its kinetic Fourier symbol is then $4 π^{2} a (∣ ξ ∣^{2} + ∣ η ∣^{2})$ , exactly the physical $- a Δ_{(x, y)}$ , so $T_{C} = - a Δ_{(x, y)} + κ / ∣ x - y ∣$ is self-adjoint on $H^{2}$ . The multiplier $M_{V_{b}}$ is bounded self-adjoint, with norm at most $2 ∥ V_{e x t} ∥_{\infty}$ and relative bound zero with respect to $T_{C}$ ; Kato-Rellich theorem therefore makes $T = T_{C} + M_{V_{b}}$ self-adjoint on the same exact domain and bounded below. In either case, $T$ preserves the spatial exchange symmetry because its Laplacian and real even pair kernel and identical external slots are exchange invariant. The full labelled operator $T_{4}$ acts componentwise on spin and its combined antisymmetric restriction, transported by $Θ_{2}^{- 1}$ , is the declared $H_{2}^{-}$ with exact domain $Θ_{2}^{- 1} (H^{2} \cap ran P_{-})$ by [F1]. No scalar-only application of a permutation theorem is used for the spin-four space. The one-particle $h = - a Δ + V_{e x t}$ is likewise self-adjoint on scalar $H^{2}$ by the CC bounded realization.

[F5]

Weak derivatives of spatial products. For $u, v \in H^{2} (R^{3}; C)$ the product $u (x) v (y)$ has weak derivatives $D_{x}^{α} D_{y}^{β} (u (x) v (y)) = (D^{α} u) (x) (D^{β} v) (y)$ for $∣ α ∣ + ∣ β ∣ \leq 2$ , obtained by iterated weak-derivative testing and Fubini. Their squared $L^{2} (R^{6})$ norms are $∥ D^{α} u ∥_{2}^{2} ∥ D^{β} v ∥_{2}^{2} < \infty$ , including mixed first derivatives. Thus the spatial products and $f_{\pm}$ lie in $H^{2} (R^{6})$ , without claiming an exact unproved Sobolev norm constant (Integer-order Sobolev spaces and their norms, Fubini's theorem for L^1 functions on a sigma-finite product). On these products the unbounded differential slot action is $(- a Δ_{x} - a Δ_{y} + V_{b}) (u (x) v (y)) = (h u) (x) v (y) + u (x) (h v) (y)$ ; every term lies in $L^{2}$ by its factor norms. This is an exact-domain calculation, not an application of bounded tensor-map rules to an unbounded $h$ .

[F6]

Finite means of domain preparations. For a unit vector $ψ$ in the domain of a self-adjoint assigned $A$ , the absolute first spectral moment is finite and its mean is $⟨ A ψ, ψ ⟩$ (Mean, second moment and variance of an assigned observable in a preparation clause (a), Assigned sharp observables and the outcome law of the framework clause (e)).

Proof

technique · direct; transport the spatial/spin products into the actual declared tensor square, verify their full-domain slot actions, and evaluate the real direct/exchange means

Given: The data of the statement and [F1]–[F6].

1.1givenF1F2algebra

[Unit preparations in the actual combined sector] Orthonormality of $u, v$ makes $u \otimes v$ and $v \otimes u$ orthogonal unit vectors, so $φ_{\pm}$ and $f_{\pm}$ are unit and satisfy $f_{\pm} (y, x) = \pm f_{\pm} (x, y)$ . The displayed spin definitions give $∥ χ_{\pm} ∥ = 1$ and $R χ_{\pm} = \pm χ_{\pm}$ . Hence $U (f_{\pm} χ_{\mp}) (x, y) = R f_{\pm} (y, x) χ_{\mp} = - f_{\pm} (x, y) χ_{\mp}$ . Their $L^{2} (R^{6}; C^{4})$ norm is one. By [F1], $Φ_{\pm} = Θ_{2}^{- 1} (f_{\pm} χ_{\mp})$ are unit vectors in the declared $H_{2}^{-}$ , so their pure projectors define the declared preparations. This proves (a).

2.1step 1.1F1F2F4F5algebra

[H² domain and exact one-body expectation] By [F5], each spatial product is in $H^{2} (R^{6})$ , hence so is $f_{\pm}$ . Multiplication by the fixed finite spin vector means a finite tuple of constant scalar multiples of $f_{\pm}$ , so $f_{\pm} χ_{\mp} \in H^{2} (R^{6}; C^{4})$ . Step 1.1 and [F4] then place $Φ_{\pm}$ in the exact domain of $H_{2}^{-}$ . Write $q (x, y) = u (x) v (y)$ , $r (x, y) = v (x) u (y)$ and $L = - a Δ_{x} - a Δ_{y} + V_{b}$ . By [F5], $L q = (h u) (x) v (y) + u (x) (h v) (y)$ and $L r = (h v) (x) u (y) + v (x) (h u) (y)$ . Fubini gives $⟨ L q, q ⟩ = ⟨ L r, r ⟩ = ε_{u} + ε_{v}$ , and $⟨ L q, r ⟩ = ⟨ h u, v ⟩ ⟨ v, u ⟩ + ⟨ u, v ⟩ ⟨ h v, u ⟩ = 0$ ; the reversed cross pairing vanishes in the same way. Thus $⟨ L f_{\pm}, f_{\pm} ⟩ = ε_{u} + ε_{v}$ . Spin independence and $∥ χ_{\mp} ∥ = 1$ give the same pairing in the full spin representation. The unbounded slot terms all had their stated domain inputs.

3.1step 2.1F1F2F3algebra

[Pair expectation and reality] In the bounded case, absolute values of $D$ and $X$ are bounded by $∥ W ∥_{\infty} ∥ u ∥_{2}^{2} ∥ v ∥_{2}^{2}$ ; the crossed term uses $\int ∣ u v ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ . In the Coulomb case $H^{2} \subset H^{1}$ , so [F3] gives absolute convergence and $f_{\pm} \in D (M_{W})$ . For either real even kernel, exchanging $x, y$ in the absolutely convergent $X$ integral gives $X = \overline{X}$ ; the two direct diagonal terms are equal to $D$ . Expanding $∣ f_{\pm} ∣^{2}$ therefore gives $⟨ M_{W} f_{\pm}, f_{\pm} ⟩ = D \pm X$ , as in [F3]. The full spin-independent pair multiplier has exactly this expectation in $f_{\pm} χ_{\mp}$ . No orbitals were required to be real.

4.1step 2.1step 3.1F1F4F6algebra

[Finite energy means and splitting] By step 2.1 the vectors lie in the self-adjoint sector energy domain of [F4], so [F6] guarantees finite energy means. The unitary $Θ_{2}$ preserves the energy pairings. Combining the one-body result of step 2.1 and pair result of step 3.1 gives $m_{ρ_{\pm}} (H_{2}^{-}) = ε_{u} + ε_{v} + D \pm X$ . Subtracting proves $m_{ρ_{+}} - m_{ρ_{-}} = 2 X$ . This proves (b) on the full stated bounded or Coulomb model, including the original bounded external potential.

5.1step 1.1step 4.1F3F4F6algebra∎

[Repulsive ordering and scope] In the repulsive Coulomb case [F3] gives $X \geq 0$ under the original complex $H^{1}$ hypothesis, which is already satisfied by the stated $H^{2}$ orbitals. Step 4.1 therefore gives $m_{ρ_{-}} (H_{2}^{-}) \leq m_{ρ_{+}} (H_{2}^{-})$ , proving (c). The $H^{1}$ interaction inequality does not alone assert an energy mean for vectors outside the energy domain. Step 1.1 supplies the actual adopted fermion sector, and every energy compared in step 4.1 is a finite mean of the declared preparation; no atomic-state approximation, measured splitting or derived spin-statistics rule was used. This proves (d) and completes the statement.

Remarks

What the exchange term is. $X$ is the integral of the overlap density $\overline{u} v$ against its translate; it is not an interaction between distinguishable particles, and its sign in the repulsive-Coulomb case is the content of clause (c). The theorem compares two declared preparations of the same model; it does not claim that a many-electron ground state is either of them.
Bounded versus Coulomb. The bounded case relies on the elementary expectation identity of the math supplier and requires only $u, v \in W^{2, 2}$ ; the Coulomb case additionally uses the Hardy-type bound of that supplier to make $D$ and $X$ finite and to obtain $X \geq 0$ .

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The canonical preparation as the unique free-energy minimiser of the declared finite-sector model

Physical scope

The conditional free-energy characterisation of the canonical preparation in the declared finite-sector many-body model at assigned temperature: the Gibbs operator is a declared preparation, its mean energy and entropy are the finite Gibbs sums, and it is the unique minimiser of the free-energy functional over all declared sector preparations, the minimum being the equilibrium free energy. The theorem is conditional on the adopted ensemble postulate and the declared model; it asserts no equilibration, no approach to equilibrium, no thermodynamic limit and no measured thermal quantity.

Statement

Adopt the canonical equilibrium-ensemble postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature together with the state, probability, dynamics and preparation postulates P1, P2, P4, P7 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions) in the readings of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model and Assigned sharp observables and the outcome law of the framework, and assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Work in the finite-dimensional sector class of the entropy definition: the declared fixed- $N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable has sector space $H_{N}^{σ}$ of finite dimension $D \geq 1$ with self-adjoint energy operator $H_{N}^{σ}$ , assigned temperature $T > 0$ , inverse temperature $β = 1 / (k_{B} T)$ and eigen-decomposition $H_{N}^{σ} = \sum_{j = 1}^{D} E_{j} T_{v_{j}, v_{j}}$ with orthonormal eigenbasis $(v_{1}, \dots, v_{D})$ and real eigenvalues $E_{1}, \dots, E_{D}$ . Write $q_{j} : = e^{- β E_{j}} / Z_{N}$ with $Z_{N} : = \sum_{j} e^{- β E_{j}}$ , $ρ_{β} : = \sum_{j} q_{j} T_{v_{j}, v_{j}}$ , and let $F (ρ) = tr (ρ H_{N}^{σ}) - k_{B} T s (ρ)$ and $F_{e q} : = - k_{B} T \log Z_{N}$ be the free-energy functional and equilibrium free energy of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model clauses (e). Then:

(a) the canonical preparation: $Z_{N} \in (0, \infty)$ is a finite sum, $q_{j} > 0$ , $\sum_{j} q_{j} = 1$ , and $ρ_{β}$ is a declared sector preparation of the model with $tr (ρ_{β} H_{N}^{σ}) = \frac{1}{Z_{N}} \sum_{j = 1}^{D} E_{j} e^{- β E_{j}}, s (ρ_{β}) = \log Z_{N} + β tr (ρ_{β} H_{N}^{σ});$

(b) free-energy minimum: for every declared sector preparation $ρ$ one has $F (ρ) \geq F_{e q} = - k_{B} T \log Z_{N}$ ;

(c) uniqueness: equality $F (ρ) = F_{e q}$ holds if and only if $ρ = ρ_{β}$ ; consequently the canonical preparation is the unique minimiser of the free-energy functional over the declared sector preparations, and $F (ρ_{β}) = F_{e q}$ ;

(d) degenerate case: if $H_{N}^{σ} = c i d$ for a real $c$ , then $Z_{N} = D e^{- β c}$ , $ρ_{β} = D^{- 1} i d$ , $s (ρ_{β}) = \log D$ and $F_{e q} = c - k_{B} T \log D$ ;

(e) scope: the statement is conditional on the adopted ensemble postulate and on the declared finite-dimensional model; it asserts no equilibration of any preparation towards $ρ_{β}$ , no rate of approach, no thermodynamic limit, no pressure or volume variable, and, for a laboratory sample, it inherits the tested identification of the preparation and temperature assignment of P7.

Facts & Assumptions

Given: the declared finite-dimensional sector model with energy operator $H_{N}^{σ}$ , eigen-decomposition and parameters $β$ , $T$ , $k_{B}$ ; the canonical preparation $ρ_{β}$ and the functional $F$ of the statement.

[F1]

Supplied finite Gibbs data. For the finite-dimensional self-adjoint $H_{N}^{σ}$ with eigenvalues $E_{1}, \dots, E_{D}$ and orthonormal eigenbasis $(v_{1}, \dots, v_{D})$ , and $β > 0$ : $Z_{N} = \sum_{j} e^{- β E_{j}} \in (0, \infty)$ , $ρ_{β} \in D (H_{N}^{σ})$ , $tr (ρ_{β} H_{N}^{σ}) = Z_{N}^{- 1} \sum_{j} e^{- β E_{j}} E_{j}$ , $tr (ρ_{β} H_{N}^{σ}) - β^{- 1} s (ρ_{β}) = - β^{- 1} \log Z_{N}$ , and for every density operator $ρ$ on $H_{N}^{σ}$ one has $tr (ρ H_{N}^{σ}) - β^{- 1} s (ρ) \geq - β^{- 1} \log Z_{N}$ with $s (ρ) \leq \log D$ and equality $s (ρ) = \log D$ exactly for $ρ = D^{- 1} i d$ (Finite Gibbs free-energy minimum clauses (a), (b), (c)).

[F2]

Spectral decomposition of a preparation. A density operator $ρ$ on the finite-dimensional space $H_{N}^{σ}$ has the form $ρ = \sum_{i = 1}^{D} w_{i} T_{u_{i}, u_{i}}$ with $w_{i} \geq 0$ , $\sum_{i} w_{i} = 1$ and $(u_{i})$ an orthonormal eigenbasis, by Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely and Rank-one operator calculus applied as in Finite Gibbs free-energy minimum clause (a); with $p_{i j} : = ∣ ⟨ u_{i}, v_{j} ⟩ ∣^{2}$ one has $\sum_{j} p_{i j} = 1$ and $\sum_{i} p_{i j} = 1$ , because $(u_{i})$ and $(v_{j})$ are orthonormal bases.

[F3]

Exact scalar logarithm suppliers. For $x > 0$ , $\log^{'} (x) = 1 / x$ and $\log x = \int_{1}^{x} d t / t$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t); $\log$ is continuous, $\log 1 = 0$ and $\log (x / y) = \log x - \log y$ for $x, y > 0$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm). For a continuous real function on $[a, b]$ differentiable on $(a, b)$ , $a < b$ , there is $ξ \in (a, b)$ with $f (b) - f (a) = f^{'} (ξ) (b - a)$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ). The strict tangent bound and its equality case are derived below from these statements, rather than assumed from a strict Jensen theorem.

Proof

technique · direct; express the free-energy difference as a sum of nonnegative scalar defects over the two orthonormal bases, with the zero-weight convention made explicit; the strict logarithm tangent bound determines the equality case, including degenerate Gibbs eigenspaces

Given: the data of the statement and [F1]-[F3].

1.1F1algebra

[The canonical preparation and its data] By [F1], $Z_{N} = \sum_{j} e^{- β E_{j}}$ is a finite sum with $0 < Z_{N} < \infty$ ; since $D \geq 1$ each term is positive, so $q_{j} = e^{- β E_{j}} / Z_{N} > 0$ with $\sum_{j} q_{j} = 1$ . By [F1] the operator $ρ_{β} = \sum_{j} q_{j} T_{v_{j}, v_{j}}$ is a density operator on $H_{N}^{σ}$ , hence a declared sector preparation of the model, and the two displayed identities are the clauses of [F1] together with the entropy definition. This proves (a).

1.2F3algebra

[Strict scalar defects, including zero weights] By [F3] and the mean value theorem applied to $\log$ between $1$ and $t > 0$ , $t \neq 1$ , one has $\log t = (t - 1) / ξ$ with $ξ$ strictly between $1$ and $t$ ; for $t > 1$ this is less than $t - 1$ , and for $0 < t < 1$ it is also less than $t - 1$ because the numerator is negative and $ξ < 1$ . Thus $\log t \leq t - 1$ , with equality exactly at $t = 1$ . For $w \geq 0$ , $r > 0$ define $b (w, r) : = w \log (w / r)$ when $w > 0$ and $b (0, r) : = 0$ , and $d (w, r) : = b (w, r) - w + r$ . If $w > 0$ , apply the bound to $t = r / w$ and the quotient law to get $d (w, r) \geq 0$ , with equality exactly at $w = r$ ; if $w = 0$ , $d (0, r) = r > 0$ . Likewise $h (q, r) : = \log r + (q - r) / r - \log q \geq 0$ for $q, r > 0$ , with equality exactly at $q = r$ , by applying the bound to $t = q / r$ .

1.3F1algebra

[The degenerate case] If $H_{N}^{σ} = c i d$ then $E_{j} = c$ for all $j$ and [F1] gives $Z_{N} = D e^{- β c}$ , $q_{j} = D^{- 1}$ , $ρ_{β} = D^{- 1} i d$ ; by [F1] $s (ρ_{β}) = \log D$ and $F_{e q} = - k_{B} T \log (D e^{- β c}) = c - k_{B} T \log D$ . This proves (d).

2.1F1F2F3step 1.2algebra

[The free-energy difference for every mixed preparation] Let $ρ = \sum_{i} w_{i} T_{u_{i}, u_{i}}$ be as in [F2] and put $p_{i j} = ∣ ⟨ u_{i}, v_{j} ⟩ ∣^{2}$ , $r_{i} = \sum_{j} p_{i j} q_{j}$ . Each $r_{i} > 0$ and $\sum_{i} r_{i} = \sum_{j} q_{j} \sum_{i} p_{i j} = 1$ . Use the entropy convention $η (w) = w \log w$ for $w > 0$ and $η (0) = 0$ (the notation $0 \log 0 = 0$ does not define $\log 0$ ). The trace pairing and the entropy definition give $tr (ρ H_{N}^{σ}) = \sum_{i, j} w_{i} p_{i j} E_{j}$ and $s (ρ) = - \sum_{i} η (w_{i})$ . Since $β E_{j} = - \log q_{j} - \log Z_{N}$ , $Δ : = β (F (ρ) - F_{e q}) = \sum_{i} (η (w_{i}) - w_{i} \sum_{j} p_{i j} \log q_{j})$ . Define $a_{i} : = \log r_{i} - \sum_{j} p_{i j} \log q_{j} = \sum_{j} p_{i j} h (q_{j}, r_{i})$ , the last equality using $\sum_{j} p_{i j} = 1$ and $\sum_{j} p_{i j} (q_{j} - r_{i}) = 0$ . Step 1.2 gives $a_{i} \geq 0$ , with equality exactly when $q_{j} = r_{i}$ for every $j$ with $p_{i j} > 0$ . By the quotient law for positive $w_{i}$ and by the explicit zero convention otherwise, $Δ = \sum_{i} b (w_{i}, r_{i}) + \sum_{i} w_{i} a_{i} = \sum_{i} d (w_{i}, r_{i}) + \sum_{i} w_{i} a_{i}$ , because $\sum_{i} (w_{i} - r_{i}) = 0$ . All summands are nonnegative, so $F (ρ) \geq F_{e q}$ , proving (b) for every density operator, including those with zero eigenvalues.

3.1F1F2step 1.2step 2.1algebra

[Equality, strict zero-weight exclusion and degeneracies] If $F (ρ) = F_{e q}$ , the finite sum of nonnegative defects in step 2.1 vanishes. Hence $d (w_{i}, r_{i}) = 0$ for every $i$ ; step 1.2 gives $w_{i} = r_{i} > 0$ , since $d (0, r_{i}) = r_{i} > 0$ excludes every zero weight. Thus $w_{i} a_{i} = 0$ forces $a_{i} = 0$ for every row, and step 2.1 gives $q_{j} = r_{i} = w_{i}$ whenever $p_{i j} > 0$ . In the expansion of $u_{i}$ in the orthonormal basis $(v_{j})$ , the coefficients at all $q_{j} \neq w_{i}$ therefore vanish, so $ρ_{β} u_{i} = w_{i} u_{i}$ . Since also $ρ u_{i} = w_{i} u_{i}$ and $(u_{i})$ is a basis, the two operators agree: $ρ = ρ_{β}$ . This argument allows arbitrary changes of basis within any repeated-eigenvalue Gibbs eigenspace. Conversely $ρ = ρ_{β}$ gives equality by [F1]. Together with step 2.1, this proves (c) with no full-rank assumption on the competing preparation.

4.1step 1.1step 1.2step 2.1step 1.3step 3.1∎

[Conclusion and scope] Step 1.1 proves (a), step 2.1 proves (b), step 3.1 proves (c), and step 1.3 proves (d). The argument is an optimisation inside the declared finite-dimensional sector at the adopted ensemble parameters; no equilibration, rate, thermodynamic limit or measured thermal quantity entered, which is (e).

Remarks

Why the minimiser is unique. The equality analysis shows that a minimiser must be diagonal in the same eigenbasis with the same weights; it excludes the block-diagonal counterexample in which $ρ$ is diagonal in a different basis, because equal weights on a degenerate eigenspace reassemble the same projector.
The adopted part. The minimum property is a theorem about the functional; that a physical sample is described by $ρ_{β}$ remains the adopted assumption of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature, not a consequence of the optimisation.

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

Physical scope

The declared free-mode thermal models of the pair: a single bosonic mode with energy operator epsilon N on the one-mode Fock space, a single fermionic orbital with two-dimensional occupation space and grand-canonical shift, and countably many independent modes with positive energies, each equipped with the adopted canonical preparation at assigned temperature. The theorem records the model's thermal occupation means, the partition-product convergence criterion and the dilute remainders; it asserts no thermodynamic limit, no continuum density of states, no Bose-Einstein condensation, no measured occupation and no thermalisation of any sample.

Statement

Adopt the canonical equilibrium-ensemble postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature for the mode models declared below, the state and probability postulates P1, P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes) in the readings of Assigned sharp observables and the outcome law of the framework and Mean, second moment and variance of an assigned observable in a preparation, and the identical-particle and spin postulate P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations); assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Temperatures are assigned as in Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, so that $β = 1 / (k_{B} T)$ in $J^{- 1}$ . Declare the occupation readouts as assigned sharp observables $(N, 1)$ and $(N_{-}, 1)$ , with unit particle-count, which is dimensionless. The total particle-number operator $N$ in the countable-mode reading has the same unit; every occupation mean and higher number moment below is read in the corresponding dimensionless count unit. Then:

(a) single bosonic mode: let $F (C)$ be the one-mode Fock space with number operator $N$ and orthonormal number basis $(e_{n})$ (Fixed-mode number law, Number operator domain), let $ε > 0$ and let the model declare the energy operator $H : = ε N$ on $F (C)$ ; the model's canonical preparation at temperature $T$ is $ρ_{β} = e^{- β H} / Z$ with $Z = (1 - e^{- β ε})^{- 1}$ , the number readout $N$ is an assigned sharp observable with finite thermal mean $⟨ N ⟩_{β} = \frac{1}{e^{β ε} - 1} = \frac{1}{e^{ε / (k_{B} T)} - 1},$ and all moments $⟨ N^{k} ⟩_{β}$ , $k \geq 1$ , are finite;

(b) single fermionic orbital and grand-canonical shift: on $h = C^{2}$ with occupation operator $N_{-}$ and $ε_{1}, μ \in R$ , the model's canonical (grand-canonical) preparation $F = Z^{- 1} \sum_{k = 0}^{1} e^{- β (ε_{1} - μ) k} T_{u_{k}, u_{k}}$ has $Z = 1 + e^{- β (ε_{1} - μ)}$ , $tr F = 1$ , and occupation mean $⟨ N_{-} ⟩ = \frac{1}{e^{β (ε_{1} - μ)} + 1};$ the same formula is the model's thermal mean when $μ = 0$ and $ε_{1} > 0$ ;

(c) countably many independent modes: let $(ε_{j})_{j \geq 1} \subseteq (0, \infty)$ and $μ \in R$ , and let the model declare the free-mode energies $ε_{j}$ (one mode case of clause (a)) with the product reading of Countable-mode product criterion; then the bosonic product partition is finite exactly when $\sum_{j} e^{- β ε_{j}} < \infty$ and $μ < \inf_{j} ε_{j}$ , and the fermionic product partition is finite exactly when $\sum_{j} e^{- β ε_{j}} < \infty$ ; in the finite cases the products equal $\prod_{j} (1 - e^{- β (ε_{j} - μ)})^{- 1}$ and $\prod_{j} (1 + e^{- β (ε_{j} - μ)})$ respectively;

(d) dilute remainders: for $q = e^{- β (ε - μ)}$ the model's occupation means satisfy the exact identities $\frac{q}{1 - q} = q + \frac{q^{2}}{1 - q} (bosonic, 0 < q < 1), \frac{q}{1 + q} = q - \frac{q^{2}}{1 + q} (fermionic, q > 0),$ so both differ from $q$ by at most $q^{2} / (1 - q)$ and $q^{2} / (1 + q)$ respectively;

(e) scope: clauses (a)-(d) are conditional statements about the declared mode models together with the adopted ensemble assignment; they assert no thermodynamic limit, no continuum of radiation modes, no Bose-Einstein condensation, no measured occupation number, and no thermalisation of a sample. All numerical equalities are exact identities of the declared finite-mode (or product) models at the assigned temperature.

Facts & Assumptions

Given: the mode models of the statement with their energies, the assigned temperature $T$ and inverse temperature $β = 1 / (k_{B} T)$ , and the adopted canonical preparations.

[F1]

One-mode bosonic calculation. For $ω > 0$ and $β > 0$ , on the one-mode Fock space with number operator $N$ and orthonormal basis $(e_{n})$ , the bounded operator $G_{β} e_{n} = e^{- β ω n} e_{n}$ is positive with $Z (β) = tr G_{β} = (1 - e^{- β ω})^{- 1}$ , $ρ_{β} = G_{β} / Z (β)$ is a density operator, $⟨ N ⟩_{β} = 1 / (e^{β ω} - 1)$ , and $\sum_{n} n^{k} e^{- β ω n} < \infty$ for every $k \geq 1$ (Single-mode thermal trace clauses (a)-(d)).

[F2]

Fermionic and grand-canonical formulas. For $β > 0$ , $(ε_{j}) \subseteq (0, \infty)$ and $μ \in R$ , with $q_{j} = e^{- β (ε_{j} - μ)}$ : the two-dimensional occupation operator $N_{-}$ on $C^{2}$ has $Z = 1 + e^{- β (ε_{1} - μ)}$ and mean occupancy $1 / (e^{β (ε_{1} - μ)} + 1)$ ; the products $Z_{B} = \sup_{N} \prod_{j \in N} (1 - q_{j})^{- 1}$ and $Z_{F} = \sup_{N} \prod_{j \in N} (1 + q_{j})$ are finite exactly under the stated summability and positivity conditions; and the single-mode dilute identities and remainder bounds hold, all as statements about diagonal weights (Fermion and dilute grand-canonical formulas clauses (a)-(c)).

[F3]

Countable-mode product criterion. For a separable $H$ with countable complete orthonormal family $(e_{j})$ and weights $e^{- β λ_{I}}$ over finite multisets, the diagonal operator $D_{β}$ is trace class exactly when $\sum_{j} e^{- β ε_{j}} < \infty$ , with trace $\prod_{j} (1 - e^{- β ε_{j}})^{- 1}$ in the finite case (Countable-mode product criterion clauses (b), (d), (e)).

[F4]

Observable reading and moments. An assigned sharp observable has outcome law $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , and its mean in a pure or mixed preparation is the corresponding spectral sum when finite; the number operator $N$ is self-adjoint on its domain with $N e_{n} = n e_{n}$ (Assigned sharp observables and the outcome law of the framework, Mean, second moment and variance of an assigned observable in a preparation, Number operator domain, Density operators and scalar spectral probability).

[F5]

The finite-particle bosonic and fermionic occupation bases for a supplied countable mode basis are constructed in Fixed-particle-number occupation bases and the countable-mode partition criterion (a); the vacuum is the scalar degree-zero sector of Hilbert direct-sum construction. On a supplied countable orthonormal basis, arbitrary real diagonal weights define a self-adjoint operator on its exact squared-weight domain, with finite coordinate sums as a graph core (Diagonal adjoint and graph-core lemma (a)-(e)).

Proof

technique · direct; instantiate the supplied one-mode, fermionic and product calculations at the declared energies and read the occupation operators as assigned observables

Given: the data of the statement and [F1]-[F5].

1.1F1F4algebra

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

1.2F2algebra

[The fermionic orbital and the shift] For the stated arbitrary real $ε_{1}, μ$ , 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 $q = e^{- β (ε_{1} - μ)} > 0$ and $tr G = Z = 1 + q$ . Thus $F = G / Z$ has trace one and occupation mean $q / (1 + q) = 1 / (e^{β (ε_{1} - μ)} + 1)$ with $q = e^{- β (ε_{1} - μ)}$ ; for $μ = 0$ , $ε_{1} > 0$ this is the model's thermal mean at the declared energy $ε_{1}$ . This proves (b) and the identities of (d).

1.3F3F5constructalgebra

[The number-variable space and complete basis] Let $H_{0}$ be the one-particle Hilbert space with supplied countable orthonormal basis $(e_{j})_{j \geq 1}$ of [F3]. For either sign define $H_{0}^{σ} = C$ and $F_{σ} : = {(ψ_{N})_{N \geq 0} : ψ_{N} \in H_{N}^{σ}, \sum_{N} ∥ ψ_{N} ∥^{2} < \infty}$ with inner product $\sum_{N} ⟨ ψ_{N}, φ_{N} ⟩$ , 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 $F_{-}$ , 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 $(b_{n}^{σ})$ by $C^{σ} : = {n \in N^{N_{> 0}} : n has finite support, n_{j} \leq 1 if σ = -}$ , with $∣ n ∣ = \sum_{j} n_{j}$ ; for minus use the Slater vector of the occupied subset. Enumerate these labels by finite integer sequences and apply the diagonal lemma of [F5].

2.1F2F4F5step 1.3algebra

[Exact generator and trace] On this basis define the self-adjoint diagonal $H$ , $N$ and $G$ with respective eigenvalues $E_{n} = \sum_{j} ε_{j} n_{j}$ , $∣ n ∣$ and $g_{n} = \sum_{j} (ε_{j} - μ) n_{j}$ , each on its exact domain ${\sum_{n} a_{n} b_{n}^{σ} : \sum_{n} ∣ w_{n} ∣^{2} ∣ a_{n} ∣^{2} < \infty}$ for the corresponding weight $w_{n}$ , by [F5]. The self-adjoint $G$ is the closure of the common finite-occupation action $H - μ N$ , 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 $g_{n} \geq 0$ . In the finite fermionic-product case summability gives $ε_{j} \to \infty$ , so $g_{n} \geq \sum_{j : ε_{j} < μ} (ε_{j} - μ) > - \infty$ , the sum having finitely many terms. The coordinate spectral PVM is $E_{G} (B) \sum_{n} a_{n} b_{n}^{σ} = \sum_{n} 1_{B} (g_{n}) a_{n} b_{n}^{σ}$ , and spectral integration gives the bounded positive $e^{- β G}$ with weights $\prod_{j} q_{j}^{n_{j}}$ , where $q_{j} = e^{- β (ε_{j} - μ)}$ . 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 $C^{σ}$ , so the full sum is the increasing limit of those products; if a bosonic $q_{j} \geq 1$ , 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 $q_{j}^{k}$ , $k \geq 0$ 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.

3.1F4step 1.1step 1.2step 1.3step 2.1algebra

[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).

4.1step 1.1step 1.2step 1.3step 2.1step 3.1∎

[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).

Remarks

No continuum claim. The product criterion covers the declared countable discrete occupation bases only. These suppliers establish no continuum partition-trace result; such a model requires its own one-particle space, measure, self-adjoint energy realization and trace analysis.
The chemical potential. $μ$ is an assigned model parameter in joules with its uncertainty; the bosonic condition $ε_{j} - μ > 0$ is required for each geometric factor, while the fermionic factors are finite for every real $μ$ .

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Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one

Physical scope

The conditional conservation statement of the declared finite-sector model: under the unitary flow of the model's self-adjoint energy operator, every declared sector preparation keeps its entropy, the adopted canonical preparation is stationary, and a preparation different from it is never mapped to it at any time, so the model's unitary dynamics alone does not produce the canonical preparation. The theorem is about the exact closed-system model and asserts nothing about open systems, about equilibration of individual readouts, about time averages, or about any laboratory sample.

Statement

Adopt the autonomous-dynamics postulate P4 (Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution, the state postulate P1 and the observable postulate P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes) in the readings of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable and Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, and adopt the canonical equilibrium-ensemble postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature for the canonical preparation $ρ_{β}$ ; assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Work in the finite-sector class of the entropy definition: the declared model has finite-dimensional sector space $H_{N}^{σ}$ , self-adjoint energy operator $H_{N}^{σ}$ with flow $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ , and every declared sector preparation evolves as $ρ (t) = U (t) ρ (0) U (t)^{*}$ under P4 clause (iii). Then:

(a) entropy is conserved: for every declared sector preparation $ρ (0)$ of the model and every $t \in R$ , $s (ρ (t)) = s (ρ (0));$

(b) the canonical preparation is stationary: $U (t) ρ_{β} U (t)^{*} = ρ_{β}$ for every $t$ , and every thermal mean $tr (ρ_{β} A)$ of an assigned observable is time independent;

(c) no unitary production of the canonical preparation: if $ρ (0) \neq ρ_{β}$ then $ρ (t) \neq ρ_{β}$ for every $t \in R$ ; equivalently, no evolution of this isolated model maps a non-canonical declared preparation to the canonical one at any time;

(d) scope: clauses (a)-(c) concern the model's exact unitary flow in the finite-sector class and the adopted canonical preparation. They assert nothing about open systems or environment coupling, about the time behaviour of individual readouts or their time averages, about an approach of any readout to a thermal value in a limit, about environments or measurement models, and about any laboratory sample; in particular clause (c) is not a claim that no physical system ever approaches thermal equilibrium, but the statement that the model's P4 dynamics alone does not do so.

Facts & Assumptions

Given: the declared finite-sector model with self-adjoint $H_{N}^{σ}$ , flow $U (t)$ , canonical preparation $ρ_{β}$ of the ensemble postulate, and a declared sector preparation $ρ (0)$ with $ρ (t) = U (t) ρ (0) U (t)^{*}$ .

[F1]

Unitary flow and density evolution. $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ is a strongly continuous unitary group with $U (t)^{*} = U (t)^{- 1}$ and $U (t + s) = U (t) U (s)$ , and $ρ (t) = U (t) ρ (0) U (t)^{*}$ is a density operator for every $t$ (Assigned energy observable, unitary flow and the two pictures of autonomous evolution clauses (b), (c), Density operators and scalar spectral probability).

[F2]

Spectral decomposition and entropy. Every density operator $ρ$ on the finite-dimensional $H_{N}^{σ}$ has an orthonormal eigenbasis $(u_{i})$ with eigenvalues $w_{i} \geq 0$ , and $s (ρ) = - \sum_{i : w_{i} > 0} w_{i} \log w_{i}$ depends only on the eigenvalue multiset (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, Rank-one operator calculus, Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model clause (b), The natural logarithm as the inverse of the exponential function).

[F3]

Canonical preparation. $ρ_{β} = Z_{N}^{- 1} e^{- β H_{N}^{σ}}$ with $Z_{N} = \sum_{j} e^{- β E_{j}}$ is diagonal in an orthonormal eigenbasis $(v_{j})$ of $H_{N}^{σ}$ with strictly positive weights $q_{j} = Z_{N}^{- 1} e^{- β E_{j}}$ , and it is a declared sector preparation (Finite Gibbs free-energy minimum clause (a), The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature clause (b)).

Proof

technique · direct; compare eigenvalue multisets under unitary conjugation and use that $\rho_\beta$ is a spectral function of the generator

Given: the data of the statement and [F1]-[F3].

1.1F1F2algebra

[Entropy conservation] Let $ρ (0) = \sum_{i} w_{i} T_{u_{i}, u_{i}}$ be the spectral decomposition of [F2]. By [F1], $U (t)$ is unitary, so $(U (t) u_{i})$ is an orthonormal basis and $ρ (t) = \sum_{i} w_{i} T_{U (t) u_{i}, U (t) u_{i}}$ : the eigenvalues of $ρ (t)$ are the same $w_{i}$ with the same multiplicities. By [F2] the entropy depends only on that multiset, so $s (ρ (t)) = s (ρ (0))$ for every $t$ . This proves (a).

1.2F1F3algebra

[Stationarity of the canonical preparation] By [F3] $ρ_{β}$ is diagonal in the eigenbasis $(v_{j})$ of $H_{N}^{σ}$ ; the same basis diagonalizes $U (t)$ with eigenvalues $e^{- i t E_{j} / ℏ}$ , so $U (t) T_{v_{j}, v_{j}} U (t)^{*} = T_{v_{j}, v_{j}}$ for every $j$ and hence $U (t) ρ_{β} U (t)^{*} = ρ_{β}$ . Every thermal mean $tr (ρ_{β} A)$ is therefore time independent by [F1]. This proves (b).

2.1F1step 1.2algebra

[No unitary production of the canonical preparation] Suppose $ρ (t) = ρ_{β}$ for some $t$ . Applying the unitary $U (t)^{*}$ on both sides and using $U (t)^{*} = U (t)^{- 1}$ from [F1], $ρ (0) = U (t)^{*} ρ_{β} U (t) = ρ_{β}$ by step 1.2 applied to $- t$ . Hence $ρ (0) \neq ρ_{β}$ implies $ρ (t) \neq ρ_{β}$ for every $t$ , which is (c).

3.1step 1.1step 1.2step 2.1∎

[Conclusion and scope] Steps 1.1-2.1 prove (a)-(c). The argument used only the unitary flow, the eigenvalue multiset of a preparation and the spectral form of $ρ_{β}$ ; no environment, measurement model, readout equilibration or laboratory sample entered, which is (d).

Remarks

Entropy versus readout equilibration. Clause (a) constrains the entropy of the full sector preparation; it does not prevent the outcome law of a chosen readout from approaching a stationary average under the flow. Clause (c) is the sharp statement about the preparation itself.
Open systems. The theorem does not apply to a reduced description obtained by tracing out an environment; such a description requires its own model assumption, as P4 clause (v) records.

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Scope and empirical status of the many-body and equilibrium models

Physical scope

The recorded scope and empirical status of the many-body and equilibrium page: which statements are model definitions, adopted ensemble assumptions or conditional results, which quantities are declared empirical parameters, what is explicitly not derived (thermalisation, thermodynamic limit, infinite-dimensional entropy, spin-statistics), and the absence of experiment items or measured values on this page.

Recorded scope

The page declares and works with the finite-mode and spatial realizations of a declared fixed- $N$ many-body model (The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable): a species sector of P6, a one-particle energy operator, a pair interaction and the self-adjoint restriction of the labelled Hamiltonian to the adopted symmetric or antisymmetric sector. The page also treats separately declared number-variable free-mode spaces, including the one-mode and countable-mode occupation models of Thermal occupation means of the declared free-mode models: Bose and Fermi factors, product convergence and dilute remainders, under reading (a)(ii) of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature; these are additional model declarations, not realizations supplied by the fixed- $N$ definition. Every result is conditional on its own stated model and parameters:

the non-interacting finite-mode model is diagonal in the occupation basis with the stated occupation eigenvalues, partition sum and ground energy (Diagonalisation of the non-interacting finite-mode sector: occupation eigenvalues, partition sum and ground energy);
the one-body readout of a bounded self-adjoint one-particle quantity has mean equal to $N$ times the pairing of the one-body reduced preparation with that quantity (The one-body readout law: sector means of summed bounded one-particle quantities and their reduced forms);
for two orthonormal orbitals satisfying the stated regularity and a repulsive Coulomb pair interaction, the spatially antisymmetric (spin-triplet) sector preparation has mean pair energy $D - X$ , no greater than the spatially symmetric (spin-singlet) preparation's $D + X$ , where $D$ and $X \geq 0$ are the direct and exchange integrals of the cited theorem; the splitting is $2 X$ , and the triplet's mean pair energy is strictly lower exactly when $X > 0$ (Direct and exchange contributions to a two-fermion mean energy and the repulsive-Coulomb ordering);
in the finite-sector class the canonical preparation is the unique minimiser of the free-energy functional at the assigned $β = 1 / (k_{B} T)$ (The canonical preparation as the unique free-energy minimiser of the declared finite-sector model);
in the separately declared number-variable mode models, the free-mode thermal occupation means are the Bose and Fermi factors with the stated product convergence criterion and dilute remainders (Thermal occupation means of the declared free-mode models: Bose and Fermi factors, product convergence and dilute remainders);
in the finite-sector class, the model's unitary flow preserves the entropy of every declared sector preparation and never maps a preparation different from the canonical one to it (Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one).

Adopted assumptions and parameters

The canonical ensemble is the adopted assumption of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature; the maximum-entropy and free-energy arguments are motivations inside the model, and the free-energy minimum theorem is a statement about the functional, not a derivation of the ensemble from P1-P4. The Boltzmann constant is the exact SI defining constant, while the temperature, chemical potential, masses, couplings and any potential parameters are assigned model parameters with uncertainty (Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, Dimensions, SI unit conventions and quantity types of the framework). The assignment of a concrete sample, mode family or detector to these model objects is a tested identification under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

Explicitly not asserted

The page does not assert: thermalization or equilibration of any preparation (the opposite conservation statement is proved for the closed model); a thermodynamic limit, pressure, volume, phase transitions or condensation; entropy or free energy for infinite-dimensional sectors, for which $ρ \log ρ$ need not be trace class; any spin-statistics theorem, which remains the adopted sector assignment of P6; any experimental value, count or measured level splitting. The mode models are declared idealizations: a finitely many-mode or product-of-modes reading of a free system, not a derivation from a spatial continuum model. The exchange computation of the page compares two declared preparations of one model; it does not claim that any atom's spectrum equals the difference computed.

Source limits

The physical reading of the Gibbs distribution and of the relations among $U$ , $S$ and $F$ follows the treatments of Binney-Skinner and Burmistrov-Lunkin at the indicated locators; both are textbook and lecture-note presentations whose thermodynamic identifications are heuristic model assumptions rather than mathematical derivations, and the present page therefore records them as adopted assumptions. Burmistrov-Lunkin work with $β$ as a formal inverse-energy parameter and set $k_{B} = 1$ implicitly; the SI reading $β = 1 / (k_{B} T)$ with the exact BIPM value of $k_{B}$ used here is our own convention, not a claim made at their locators. No source supplies measured temperature-dependent data for the declared mode models on this page, and no experiment item is created from these sources.

5 · Examples, experiments and counterexamples

None yet.

Qm Many Body And Equilibrium Models

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The one-body reduction trace law of the finite-mode permutation sectors

Statement

Facts & Assumptions

Proof

Remarks

Closed formulas for the two-level canonical weights, mean energy, entropy and free energy

Statement

Facts & Assumptions

Proof

Remarks

CC realization of the bounded spatial operator on a finite-component H² domain

Statement

Facts & Assumptions

Proof

Finite spin components of an N-fold spatial tensor product

Statement

Facts & Assumptions

Proof

Fixed-particle-number occupation bases and the countable-mode partition criterion

Statement

Facts & Assumptions

Proof

Spectral first moments and trace products with a density operator

Statement

Facts & Assumptions

Proof

The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable

Definition

Remarks

The declared one-body reduced operator and the one-body readout of a sector preparation

Definition

Remarks

Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model

Definition

Remarks

The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature

Postulate

Remarks

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

Statement

Facts & Assumptions

Proof

Remarks

The one-body readout law: sector means of summed bounded one-particle quantities and their reduced forms

Statement

Facts & Assumptions

Proof

Remarks

Direct and exchange contributions to a two-fermion mean energy and the repulsive-Coulomb ordering

Statement

Facts & Assumptions

Proof

Remarks

The canonical preparation as the unique free-energy minimiser of the declared finite-sector model

Statement

Facts & Assumptions

Proof

Remarks

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

Statement

Facts & Assumptions

Proof

Remarks

Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one

Statement

Facts & Assumptions

Proof

Remarks

Scope and empirical status of the many-body and equilibrium models

Recorded scope

Adopted assumptions and parameters

Explicitly not asserted

Source limits

5 · Examples, experiments and counterexamples

Sources