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Qm Math Fock And Equilibrium

1 · Prerequisites

2 · Summary

The Fock-space and equilibrium layer of the non-relativistic quantum mechanics mathematical prerequisites, entirely mathematical. The page begins with the completed-tensor-power Fock space and its sector projections, the creation and annihilation core operators with their contraction bounds, closed adjoint domains and the number operator; then single-mode oscillator coefficients, the fixed-mode number law and the single-mode thermal trace; then the countable-mode product criterion and the fermionic and dilute grand-canonical formulas with the finite Gibbs variational minimum and a divergent-entropy example; and finally the fixed-particle branch: bounded fixed- $N$ operators, the permutation-restricted and Coulomb self-adjointness of the fixed- $N$ Hamiltonian, the Slater direct and exchange calculation, the continuum transverse free Fock Hamiltonian, the cutoff dipole relative bound, the controlled Dyson series for vacuum emission and the golden-rule coefficient approximate identity.

Every item carries domain: mathematics and depends only on mathematical suppliers from qm-math-composites, qm-math-oscillator, qm-math-orbital-and-spin, qm-math-coulomb, qm-math-controlled-dynamics and published mathematics. Physical-sounding objects such as the transverse mode space, the dipole matrices, the coupling $η_{0}$ , the cutoff $Λ$ and the spectral transition density $J_{b}$ are defined as mathematical model data with declared hypotheses; no measurement postulate, statistical inference or empirical claim is made, and the scope paragraphs of the underlying items record where a physical identification would require a separate assumption. The companion page qm-math-fock-and-equilibrium-examples stays mathematics as well.

The central chain is Hilbert direct-sum construction, Fock sector projections, Creation contraction bounds and Closed adjoint sector domains for the algebraic structure, the product criterion Countable-mode product criterion for convergence of the equilibrium weights (with the fermionic formulas of Fermion and dilute grand-canonical formulas and the variational minimum of Finite Gibbs free-energy minimum), and the light-matter branch Cutoff dipole relative bound, Vacuum emission Dyson convergence, Golden-rule coefficient approximate identity for the finite-time self-adjoint model and its first-order coefficient limit.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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

Hilbert direct-sum construction

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space with inner product linear in the first argument (Hilbert space, Real and complex inner-product spaces and their induced length). Let $H^{\otimes 0} : = C$ , $H^{\otimes 1} : = H$ and $H^{\otimes (n + 1)} : = H^{\otimes n} \hat{\otimes} H$ be the completed tensor powers with elementary tensors $u_{1} \otimes \dots \otimes u_{n}$ (Hilbert tensor construction: inner product, completion and product bases); for $n \geq 1$ let $P_{+}^{(n)} \in B (H^{\otimes n})$ be the symmetric projection of Permutation sector projections on the n-fold tensor power and set $P_{+}^{(0)} : = {i d}_{C}$ . Write $ran P_{+}^{(n)}$ for its range and define

$F (H) : = {ψ = (ψ_{n})_{n \in N} : ψ_{n} \in ran P_{+}^{(n)} for all n, \sum_{n \in N} ∥ ψ_{n} ∥^{2} < \infty} .$

Then:

(a) for $ψ, φ \in F (H)$ the scalar series $\sum_{n \in N} ⟨ ψ_{n}, φ_{n} ⟩$ converges absolutely and defines a sesquilinear form, linear in the first argument, with respect to which $F (H)$ is a complex vector space with inner product $⟨ ψ, φ ⟩ : = \sum_{n \in N} ⟨ ψ_{n}, φ_{n} ⟩$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs, Series and absolute convergence in a normed space);

(b) $(F (H), ⟨ \cdot, \cdot ⟩)$ is a Hilbert space, i.e. the induced norm is complete (Hilbert space);

(c) the vacuum vector $Ω : = (1, 0, 0, \dots)$ satisfies $∥ Ω ∥ = 1$ , and the finite-particle space $F_{0} : = {ψ \in F (H) : ψ_{n} = 0 for all but finitely many n}$ is a dense linear subspace of $F (H)$ (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets);

(d) for every $n$ the map $ι_{n} : ran P_{+}^{(n)} \to F (H)$ , $ι_{n} (u) : = (0, \dots, 0, u, 0, \dots)$ (entry in coordinate $n$ ) is a linear isometry of $ran P_{+}^{(n)}$ onto the closed subspace of vectors supported in coordinate $n$ ;

(e) if $H$ admits a countable complete orthonormal family $(e_{i})_{i \in I}$ , $I$ at most countable (Orthonormal families, complete orthonormal systems and Hilbert bases), then $F (H)$ is separable (Separability: the existence of an at most countable dense subset);

(f) direct sums of self-adjoint operators: if for every $n \in N$ , including $n = 0$ , a self-adjoint operator $T_{n}$ on the Hilbert space $ran P_{+}^{(n)}$ with domain $D (T_{n}) \subseteq ran P_{+}^{(n)}$ is given (so $T_{0}$ is a self-adjoint operator on $C$ ), (Symmetric, self-adjoint and essentially self-adjoint operators, Unbounded linear operators: domain, graph and extension), and

$D (T) : = {ψ \in F (H) : ψ_{n} \in D (T_{n}) for all n, \sum_{n \in N} ∥ T_{n} ψ_{n} ∥^{2} < \infty}, (T ψ)_{n} : = T_{n} ψ_{n},$

then $T$ is self-adjoint and $D (T)$ is dense; if in addition every $T_{n}$ is bounded with $\sup_{n} ∥ T_{n} ∥ < \infty$ , then $T$ is bounded with $∥ T ∥ = \sup_{n} ∥ T_{n} ∥$ (A bounded linear operator between normed spaces).

No physical interpretation of $F (H)$ , of the tensor powers or of the direct-sum operators is asserted.

Facts & Assumptions

Given: the complex Hilbert space $H$ with inner product linear in the first argument, the completed tensor powers $H^{\otimes n}$ with symmetric projections $P_{+}^{(n)}$ , the set $F (H)$ , vectors $ψ, φ \in F (H)$ , and, in part (f), self-adjoint operators $T_{n}$ on $ran P_{+}^{(n)}$ .

[F1]

For every $n \in N$ , $P_{+}^{(n)}$ is an orthogonal projection (with $P_{+}^{(0)} = {i d}_{C}$ ), so $ran P_{+}^{(n)}$ is a closed linear subspace of the complete space $H^{\otimes n}$ ; for fixed $v \in H^{\otimes m}$ the map $u \mapsto u \otimes v$ is linear and bounded (Permutation sector projections on the n-fold tensor power, Hilbert tensor construction: inner product, completion and product bases).

[F2]

Cauchy-Schwarz holds in each sector by Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs. For square-summable scalar sequences, apply that theorem to the finite nonnegative real vectors $(∣ a_{0} ∣, \dots, ∣ a_{N} ∣)$ and $(∣ b_{0} ∣, \dots, ∣ b_{N} ∣)$ in $R^{N + 1}$ with inner product $⟨ a, b ⟩ : = \sum_{n = 0}^{N} a_{n} b_{n}$ , then let $N \to \infty$ in the nonnegative partial sums; this gives $\sum_{n} ∣ a_{n} b_{n} ∣ \leq (\sum_{n} ∣ a_{n} ∣^{2})^{1 / 2} (\sum_{n} ∣ b_{n} ∣^{2})^{1 / 2}$ .

[F3]

A self-adjoint operator is closed and symmetric; a densely defined symmetric operator $S$ satisfies $S \subseteq S^{*}$ (Symmetric, self-adjoint and essentially self-adjoint operators, Adjoint of a densely defined operator, Densely defined, closed and closable operators, and cores).

[F4]

A subset $A$ of a normed space is dense exactly when every vector is the norm limit of a sequence from $A$ (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets).

Proof

1.1F2algebra

[Absolutely convergent pairing] Let $ψ, φ \in F (H)$ . Coordinatewise addition preserves the defining square-summability because $∥ ψ_{n} + φ_{n} ∥^{2} \leq 2 ∥ ψ_{n} ∥^{2} + 2 ∥ φ_{n} ∥^{2}$ , and scalar multiplication does as well, so $F (H)$ is a complex vector space. By [F2] with $a_{n} : = ∥ ψ_{n} ∥$ , $b_{n} : = ∥ φ_{n} ∥$ , $\sum_{n} ∣ ⟨ ψ_{n}, φ_{n} ⟩ ∣ \leq \sum_{n} ∥ ψ_{n} ∥ ∥ φ_{n} ∥ \leq (\sum_{n} ∥ ψ_{n} ∥^{2})^{1 / 2} (\sum_{n} ∥ φ_{n} ∥^{2})^{1 / 2} = ∥ ψ ∥ ∥ φ ∥ < \infty$ , so the series defining $⟨ ψ, φ ⟩$ converges absolutely (Series and absolute convergence in a normed space); termwise addition and scalar multiplication give sesquilinearity and linearity in the first argument, inherited from the pairings on the $H^{\otimes n}$ . If $∥ ψ ∥^{2} = \sum_{n} ∥ ψ_{n} ∥^{2} = 0$ then $ψ_{n} = 0$ for every $n$ , so the displayed form is positive definite and is an inner product.

1.2F1algebra

[Completeness] Let $(ψ^{(k)})_{k}$ be a Cauchy sequence in $F (H)$ and let $ε > 0$ . For each $n$ , $∥ ψ_{n}^{(k)} - ψ_{n}^{(l)} ∥ \leq ∥ ψ^{(k)} - ψ^{(l)} ∥$ , so $(ψ_{n}^{(k)})_{k}$ is Cauchy in the complete space $H^{\otimes n}$ ([F1]) and converges to some $ψ_{n} \in H^{\otimes n}$ ; since $ran P_{+}^{(n)}$ is closed, $ψ_{n} \in ran P_{+}^{(n)}$ ([F1]). Choose $K$ with $∥ ψ^{(k)} - ψ^{(m)} ∥ \leq ε$ for $k, m \geq K$ . Fix $k \geq K$ . For every finite $N$ , $\sum_{n \leq N} ∥ ψ_{n} ∥^{2} \leq 2 \sum_{n \leq N} ∥ ψ_{n}^{(k)} ∥^{2} + 2 \sum_{n \leq N} ∥ ψ_{n} - ψ_{n}^{(k)} ∥^{2} \leq 2 ∥ ψ^{(k)} ∥^{2} + 2 ε^{2}$ , where the last term uses the limit as $m \to \infty$ in $\sum_{n \leq N} ∥ ψ_{n}^{(m)} - ψ_{n}^{(k)} ∥^{2} \leq ε^{2}$ . Taking the supremum over finite $N$ shows $ψ = (ψ_{n})_{n} \in F (H)$ . Also, for each $k \geq K$ and finite $N$ , $\sum_{n \leq N} ∥ ψ_{n}^{(k)} - ψ_{n} ∥^{2} \leq ε^{2}$ by letting $m \to \infty$ in $\sum_{n \leq N} ∥ ψ_{n}^{(k)} - ψ_{n}^{(m)} ∥^{2} \leq ε^{2}$ ; taking the supremum gives $∥ ψ^{(k)} - ψ ∥ \leq ε$ . Thus $ψ^{(k)} \to ψ$ and $F (H)$ is complete, proving (b).

1.3F1F4algebra

[Vacuum, density and embeddings] The vacuum has $∥ Ω ∥ = ∣ 1 ∣ = 1$ . For $ψ \in F (H)$ the truncations $ψ^{(N)} : = (ψ_{0}, \dots, ψ_{N}, 0, 0, \dots) \in F_{0}$ satisfy $∥ ψ - ψ^{(N)} ∥^{2} = \sum_{n > N} ∥ ψ_{n} ∥^{2} \to 0$ , so $F_{0}$ is dense by [F4]; $F_{0}$ is a linear subspace because each $ran P_{+}^{(n)}$ is one ([F1]). Each $ι_{n}$ is linear and $∥ ι_{n} (u) ∥^{2} = ∥ u ∥^{2}$ , and its range is the set of vectors supported in coordinate $n$ , a closed subspace identified isometrically with $ran P_{+}^{(n)}$ .

1.4F2F3algebra

[The direct sum is closed] Let $(T_{n})$ be as in (f) and let $(ψ^{(k)})_{k} \subseteq D (T)$ with $ψ^{(k)} \to ψ$ and $T ψ^{(k)} \to η$ in $F (H)$ . For each $n$ , $ψ_{n}^{(k)} \to ψ_{n}$ and $T_{n} ψ_{n}^{(k)} = (T ψ^{(k)})_{n} \to η_{n}$ ; since $T_{n}$ is self-adjoint, hence closed ([F3]), $ψ_{n} \in D (T_{n})$ and $T_{n} ψ_{n} = η_{n}$ . For every finite $N$ , continuity of the norm and the finite sum give $\sum_{n \leq N} ∥ T_{n} ψ_{n} ∥^{2} = \lim_{k} \sum_{n \leq N} ∥ T_{n} ψ_{n}^{(k)} ∥^{2} \leq \lim_{k} ∥ T ψ^{(k)} ∥^{2} = ∥ η ∥^{2}$ . Taking the supremum over finite partial sums shows $\sum_{n} ∥ T_{n} ψ_{n} ∥^{2} \leq ∥ η ∥^{2} < \infty$ , so $ψ \in D (T)$ and $T ψ = η$ ; thus $T$ is closed. It is symmetric because for $ψ, φ \in D (T)$ the middle series below converges absolutely by [F2] applied to the square-summable families $(∥ T_{n} ψ_{n} ∥)_{n}$ , $(∥ φ_{n} ∥)_{n}$ and $(∥ T_{n} φ_{n} ∥)_{n}$ : $⟨ T ψ, φ ⟩ = \sum_{n} ⟨ T_{n} ψ_{n}, φ_{n} ⟩ = \sum_{n} ⟨ ψ_{n}, T_{n} φ_{n} ⟩ = ⟨ ψ, T φ ⟩$ .

1.5F3algebra

[Density of the direct-sum domain] Let $ψ \in F (H)$ and $ε > 0$ . Choose $N$ so that $\sum_{n > N} ∥ ψ_{n} ∥^{2} < ε^{2} / 4$ . For each of the finitely many $n \leq N$ , the self-adjoint operator $T_{n}$ is densely defined, so choose $φ_{n} \in D (T_{n})$ with $∥ φ_{n} - ψ_{n} ∥ < ε / (2 \sqrt{N + 1})$ ; put $φ_{n} = 0$ for $n > N$ . The vector $φ$ belongs to $D (T)$ because it has finite support and every retained coordinate belongs to its operator domain. Moreover, $∥ φ - ψ ∥^{2} = \sum_{n \leq N} ∥ φ_{n} - ψ_{n} ∥^{2} + \sum_{n > N} ∥ ψ_{n} ∥^{2} < ε^{2} / 2 < ε^{2}$ . Thus $D (T)$ is dense in $F (H)$ .

2.1F2F3step 1.4step 1.5algebra

[The adjoint is the same diagonal sum] Let $φ \in D (T^{*})$ with $C : = ∥ T^{*} φ ∥$ , so $∣ ⟨ T ψ, φ ⟩ ∣ \leq C ∥ ψ ∥$ for all $ψ \in D (T)$ . Testing with $ψ$ supported in a single coordinate $n$ shows that the functional $ψ_{n} \mapsto ⟨ T_{n} ψ_{n}, φ_{n} ⟩$ on the dense domain $D (T_{n})$ is bounded by $C ∥ ψ_{n} ∥$ ; hence $φ_{n} \in D (T_{n}^{*}) = D (T_{n})$ and $T_{n} φ_{n} = (T^{*} φ)_{n}$ . Since $T^{*} φ \in F (H)$ , the coordinate identity gives $\sum_{n} ∥ T_{n} φ_{n} ∥^{2} = ∥ T^{*} φ ∥^{2} < \infty$ . Therefore $D (T^{*}) \subseteq D (T)$ and $T^{*} φ = T φ$ ; with $T \subseteq T^{*}$ from step 1.4, $T = T^{*}$ is self-adjoint, and $D (T)$ is dense by step 1.5. If every $T_{n}$ is bounded by $M : = \sup_{n} ∥ T_{n} ∥ < \infty$ , then $∥ T ψ ∥^{2} = \sum_{n} ∥ T_{n} ψ_{n} ∥^{2} \leq M^{2} ∥ ψ ∥^{2}$ for every $ψ \in F (H)$ and testing on unit vectors in individual nonzero sectors where $∥ T_{n} ∥$ approaches $M$ gives the reverse inequality.

3.1F1step 1.3algebra∎

[Separability] Assume $(e_{i})_{i \in I}$ is a complete orthonormal family of $H$ with $I$ at most countable. For $n \geq 1$ the elementary tensors $e_{i_{1}} \otimes \dots \otimes e_{i_{n}}$ , $(i_{1}, \dots, i_{n}) \in I^{n}$ , have dense linear span in $H^{\otimes n}$ by the product-bases statement of Hilbert tensor construction: inner product, completion and product bases; since $P_{+}^{(n)}$ is bounded, the image $P_{+}^{(n)}$ of a dense set is dense in $ran P_{+}^{(n)}$ (given $u = P_{+}^{(n)} u$ in the range and approximants $v_{j} \to u$ , continuity gives $P_{+}^{(n)} v_{j} \to u$ ). Hence the at most countable set $D : = {ι_{n} (P_{+}^{(n)} (e_{i_{1}} \otimes \dots \otimes e_{i_{n}})) : n \geq 1, i_{1}, \dots, i_{n} \in I} \cup {Ω}$ has dense span in $F (H)$ : any $ψ \in F (H)$ is the limit of its truncations, and each $ψ_{n}$ is approximated in $ran P_{+}^{(n)}$ by finite combinations from $D$ . A normed space with a countable set whose finite linear combinations are dense is separable.

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

Fock sector projections

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space (Hilbert space) with completed tensor powers $H^{\otimes n}$ and symmetric projections $P_{+}^{(n)}$ as in Hilbert direct-sum construction, so that $F (H) = {ψ = (ψ_{n})_{n \in N} : ψ_{n} \in ran P_{+}^{(n)}, \sum_{n} ∥ ψ_{n} ∥^{2} < \infty}$ is a Hilbert space. For $n \in N$ define $Π_{n} : F (H) \to F (H), Π_{n} ψ : = (0, \dots, 0, ψ_{n}, 0, \dots)$ (the entry $ψ_{n}$ in coordinate $n$ and $0$ in all other coordinates). Then:

(a) each $Π_{n}$ is a bounded linear operator with $∥ Π_{n} ψ ∥ = ∥ ψ_{n} ∥$ and $∥ Π_{n} ∥ \leq 1$ ; it satisfies $Π_{n}^{2} = Π_{n}$ , $Π_{n}^{*} = Π_{n}$ (The Hilbert-space adjoint of a bounded operator), and $Π_{n}$ is the Hilbert orthogonal projection of $F (H)$ onto the closed subspace $ι_{n} (ran P_{+}^{(n)})$ of vectors supported in coordinate $n$ (The Hilbert orthogonal projection onto a closed subspace);

(b) $Π_{n} Π_{m} = 0$ for $n \neq m$ , so the ranges are pairwise orthogonal and $∥ Π_{n} ∥ = 1$ whenever $ran P_{+}^{(n)} \neq {0}$ (which holds for every $n$ when $H \neq {0}$ );

(c) for every $ψ \in F (H)$ the series $\sum_{n} Π_{n} ψ$ converges in norm to $ψ$ , the expansion is orthogonal, $ψ = \sum_{n \in N} Π_{n} ψ, ∥ ψ ∥^{2} = \sum_{n \in N} ∥ Π_{n} ψ ∥^{2},$ and the partial sums $\sum_{n \leq N} Π_{n}$ converge to the identity in the strong operator topology, i.e. $∥ ψ - \sum_{n \leq N} Π_{n} ψ ∥ \to 0$ for every $ψ \in F (H)$ (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets, A bounded linear operator between normed spaces);

(d) $Π_{n} = 0$ as an operator if and only if $ran P_{+}^{(n)} = {0}$ ; for $H \neq {0}$ no $Π_{n}$ vanishes.

No physical interpretation of the Fock space or of its sectors is asserted.

Facts & Assumptions

Given: the complex Hilbert space $H$ with completed tensor powers $H^{\otimes n}$ and symmetric projections $P_{+}^{(n)}$ ( $n \geq 1$ , $P_{+}^{(0)} = {i d}_{C}$ ), the Hilbert space $F (H)$ with inner product $⟨ ψ, φ ⟩ = \sum_{n} ⟨ ψ_{n}, φ_{n} ⟩$ , the vacuum $Ω = (1, 0, 0, \dots)$ , and vectors $ψ, φ \in F (H)$ .

[F1]

$F (H)$ is a Hilbert space; for $ψ \in F (H)$ one has $ψ_{n} \in ran P_{+}^{(n)}$ , $\sum_{n} ∥ ψ_{n} ∥^{2} = ∥ ψ ∥^{2} < \infty$ , and coordinatewise operations; the truncations $ψ^{(N)} = (ψ_{0}, \dots, ψ_{N}, 0, 0, \dots)$ satisfy $ψ^{(N)} \to ψ$ (Hilbert direct-sum construction).

[F2]

For each $n \geq 1$ , $P_{+}^{(n)}$ is an orthogonal projection on the Hilbert space $H^{\otimes n}$ , hence $ran P_{+}^{(n)}$ is a closed linear subspace (Permutation sector projections on the n-fold tensor power, Hilbert direct-sum construction).

[F3]

The Hilbert orthogonal projection onto a closed subspace $M$ is characterized by $P x \in M$ and $x - P x \in M^{⊥}$ for every $x$ ; it is self-adjoint, idempotent and contractive (The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F4]

For $ψ \in F (H)$ its truncation $ψ^{(N)}$ has squared norm $\sum_{n \leq N} ∥ ψ_{n} ∥^{2}$ and converges in norm to $ψ$ (Hilbert direct-sum construction).

Proof

1.1F1algebra

[Algebraic properties] For $ψ \in F (H)$ the tuple $Π_{n} ψ$ lies in $F (H)$ because its coordinate $n$ is $ψ_{n} \in ran P_{+}^{(n)}$ and all other coordinates vanish ([F1]); clearly $Π_{n} (ψ + c χ) = Π_{n} ψ + c Π_{n} χ$ , so $Π_{n}$ is linear, and $∥ Π_{n} ψ ∥^{2} = ∥ ψ_{n} ∥^{2} \leq ∥ ψ ∥^{2}$ gives $∥ Π_{n} ∥ \leq 1$ . Obviously $Π_{n}^{2} = Π_{n}$ . For the adjoint: $⟨ Π_{n} ψ, φ ⟩ = ⟨ ψ_{n}, φ_{n} ⟩ = ⟨ ψ, Π_{n} φ ⟩$ for all $ψ, φ \in F (H)$ , so $Π_{n}^{*} = Π_{n}$ by uniqueness of the Hilbert-space adjoint.

1.2F2F3algebra

[Orthogonality and ranges] For $n \neq m$ , $Π_{n} Π_{m} = 0$ because the tuple $Π_{m} ψ$ is supported in coordinate $m$ and $Π_{n}$ reads coordinate $n$ . The range of $Π_{n}$ is exactly the set of vectors of $F (H)$ supported in coordinate $n$ , a closed subspace that is identified with $ran P_{+}^{(n)}$ by the isometry $ι_{n}$ ; it is closed in $F (H)$ because if $Π_{n} ψ^{(k)} \to η$ , then $η_{m} = 0$ for $m \neq n$ and $η_{n} \in ran P_{+}^{(n)}$ by closedness ([F2]). For every $ψ$ , the difference $ψ - Π_{n} ψ$ has zero coordinate $n$ and is orthogonal to every vector supported there by the direct-sum inner product; hence [F3] identifies $Π_{n}$ with the orthogonal projection onto its range. This range identification also proves $Π_{n} = 0$ exactly when the sector is zero; when $ran P_{+}^{(n)} \neq {0}$ choose a unit $u$ there and note $∥ Π_{n} ι_{n} (u) ∥ = 1$ , so $∥ Π_{n} ∥ = 1$ . For $H \neq {0}$ and $n \geq 1$ one has $0 \neq e^{\otimes n} \in ran P_{+}^{(n)}$ for any $e \neq 0$ in $H$ , and $ran P_{+}^{(0)} = C$ , so no $Π_{n}$ vanishes. If $H = {0}$ , every positive tensor power is zero, $F (H)$ consists only of its scalar vacuum coordinate, $Π_{0} = I$ , and $Π_{n} = 0$ for $n \geq 1$ ; the same coordinate arguments apply. Thus (b), (d) hold.

2.1F1F4step 1.2algebra

[Orthogonal expansion] For $ψ \in F (H)$ put $ψ^{(N)} : = \sum_{n \leq N} Π_{n} ψ$ and $r^{(N)} : = ψ - ψ^{(N)}$ , whose coordinates are $ψ_{n}$ for $n > N$ and $0$ otherwise. Then $∥ r^{(N)} ∥^{2} = \sum_{n > N} ∥ ψ_{n} ∥^{2} \to 0$ by [F1], so $\sum_{n} Π_{n} ψ$ converges to $ψ$ in norm. The terms $Π_{n} ψ = ι_{n} (ψ_{n})$ are pairwise orthogonal by step 1.2, and [F4] gives $∥ ψ^{(N)} ∥^{2} = \sum_{n \leq N} ∥ ψ_{n} ∥^{2}$ ; passing to the limit and using continuity of the norm and of the inner product gives $∥ ψ ∥^{2} = \sum_{n} ∥ Π_{n} ψ ∥^{2}$ .

3.1step 1.1step 1.2step 2.1algebra∎

[Strong convergence and conclusion] Since $∥ ψ - \sum_{n \leq N} Π_{n} ψ ∥^{2} = \sum_{n > N} ∥ ψ_{n} ∥^{2} \to 0$ for every $ψ \in F (H)$ by step 2.1, the partial sums of $(Π_{n})$ converge to the identity in the strong operator topology; combined with steps 1.1, 1.2 and 2.1 this proves (a)-(d).

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

Creation contraction bounds

Statement

(a) Contraction map. For every $f \in H$ and $n \geq 0$ there is a unique bounded linear map $Ψ_{f}^{(n)} : H^{\otimes (n + 1)} \to H^{\otimes n}$ with $⟨ φ, Ψ_{f}^{(n)} χ ⟩ = ⟨ f \otimes φ, χ ⟩ (φ \in H^{\otimes n}, χ \in H^{\otimes (n + 1)})$ (Riesz representation for Hilbert spaces); it satisfies $∥ Ψ_{f}^{(n)} ∥ \leq ∥ f ∥$ , $Ψ_{f}^{(n)} (u_{1} \otimes \dots \otimes u_{n + 1}) = ⟨ u_{1}, f ⟩ u_{2} \otimes \dots \otimes u_{n + 1}$ on elementary tensors, $Ψ_{f}^{(n)} (ran P_{+}^{(n + 1)}) \subseteq ran P_{+}^{(n)}$ , and the map $f \mapsto Ψ_{f}^{(n)}$ is conjugate-linear.

(b) Creation and annihilation on the core. Define, for $f \in H$ and $ψ \in F_{0}$ , $(S_{+} (f) ψ)_{n} : = \sqrt{n} P_{+}^{(n)} (f \otimes ψ_{n - 1}) (n \geq 1), (S_{+} (f) ψ)_{0} : = 0,$ $(S_{-} (f) ψ)_{n} : = \sqrt{n + 1} Ψ_{f}^{(n)} (ψ_{n + 1}) (n \geq 0) .$ Then $S_{+} (f) ψ, S_{-} (f) ψ \in F_{0}$ ; $S_{+} (f)$ is linear in $f$ and $S_{-} (f)$ is conjugate-linear in $f$ ; and with $∥ ψ ∥_{1}^{2} : = \sum_{n} (n + 1) ∥ ψ_{n} ∥^{2} < \infty$ for $ψ \in F_{0}$ , $∥ S_{+} (f) ψ ∥ \leq ∥ f ∥ ∥ ψ ∥_{1}, ∥ S_{-} (f) ψ ∥ \leq ∥ f ∥ ∥ ψ ∥_{1} .$

(c) Adjoint relation. $⟨ S_{+} (f) φ, ψ ⟩ = ⟨ φ, S_{-} (f) ψ ⟩$ for all $f \in H$ and $φ, ψ \in F_{0}$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces).

(d) Canonical commutation relation. For all $f, g \in H$ and $ψ \in F_{0}$ , $S_{-} (f) S_{+} (g) ψ - S_{+} (g) S_{-} (f) ψ = ⟨ g, f ⟩ ψ,$ and consequently $∥ S_{+} (g) ψ ∥^{2} = ∥ S_{-} (g) ψ ∥^{2} + ∥ g ∥^{2} ∥ ψ ∥^{2} .$

Neither $S_{+} (f)$ nor $S_{-} (f)$ is asserted to be bounded on $F (H)$ , and no physical interpretation (particle creation, radiation or measurement) is asserted.

Facts & Assumptions

Given: the complex Hilbert space $H$ with inner product linear in the first argument, completed tensor powers $H^{\otimes n}$ with symmetric projections $P_{+}^{(n)}$ ( $n \geq 1$ ), the Fock space $F (H)$ and its dense finite-particle subspace $F_{0}$ , vectors $f, g, u, u_{1}, \dots, u_{n + 1} \in H$ , indices $n \geq 0$ , and $φ, ψ \in F_{0}$ .

[F1]

$F (H)$ is a Hilbert space with $⟨ ψ, φ ⟩ = \sum_{n} ⟨ ψ_{n}, φ_{n} ⟩$ ; $ψ \in F_{0}$ has finitely many nonzero coordinates; $F_{0}$ is dense (Hilbert direct-sum construction).

[F2]

$Π_{n}$ is the orthogonal projection onto the vectors supported in coordinate $n$ ; $∥ Π_{n} ∥ \leq 1$ , with equality when its sector is nonzero; the sectors are orthogonal (Fock sector projections).

[F3]

For $n \geq 1$ : $P_{+}^{(n)}$ is an orthogonal projection; every vector of $ran P_{+}^{(n)}$ is fixed by each permutation unitary $U_{π}$ ( $π \in S_{n}$ ), and conversely; for fixed $v$ the map $u \mapsto u \otimes v$ is linear and bounded with $∥ u \otimes v ∥ = ∥ u ∥ ∥ v ∥$ (Permutation sector projections on the n-fold tensor power, Hilbert tensor construction: inner product, completion and product bases).

[F4]

Cauchy-Schwarz in each Hilbert tensor power follows from Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs. For square-summable scalar sequences, apply that theorem to finite nonnegative real vectors $(∣ a_{0} ∣, \dots, ∣ a_{N} ∣)$ and $(∣ b_{0} ∣, \dots, ∣ b_{N} ∣)$ in $R^{N + 1}$ with inner product $⟨ a, b ⟩ : = \sum_{n = 0}^{N} a_{n} b_{n}$ , then take the limit of the nonnegative partial sums; this gives $\sum_{n} ∣ a_{n} b_{n} ∣ \leq (\sum_{n} ∣ a_{n} ∣^{2})^{1 / 2} (\sum_{n} ∣ b_{n} ∣^{2})^{1 / 2}$ .

[F5]

Riesz representation: every bounded linear functional on a Hilbert space is evaluation against a unique vector (Riesz representation for Hilbert spaces, A bounded linear operator between normed spaces).

Proof

1.1F3F4F5algebra

[The contraction map] Fix $f \in H$ , $n \geq 0$ and $χ \in H^{\otimes (n + 1)}$ . The map $φ \mapsto ⟨ f \otimes φ, χ ⟩$ is linear, and by [F4] and [F3] $∣ ⟨ f \otimes φ, χ ⟩ ∣ \leq ∥ f \otimes φ ∥ ∥ χ ∥ = ∥ f ∥ ∥ φ ∥ ∥ χ ∥$ , so it is bounded with norm at most $∥ f ∥ ∥ χ ∥$ ; by [F5] there is a unique $Ψ_{f}^{(n)} χ \in H^{\otimes n}$ representing it. The assignment $χ \mapsto Ψ_{f}^{(n)} χ$ is linear (the representing vectors of $χ \mapsto ⟨ f \otimes φ, χ ⟩$ depend linearly on $χ$ ), and $∥ Ψ_{f}^{(n)} χ ∥ = \sup_{∥ φ ∥ \leq 1} ∣ ⟨ f \otimes φ, χ ⟩ ∣ \leq ∥ f ∥ ∥ χ ∥$ gives $∥ Ψ_{f}^{(n)} ∥ \leq ∥ f ∥$ . For an elementary tensor, the tensor-product inner product and the first-linear convention give [F3, F4, F5, algebra] $⟨ f \otimes φ, u_{1} \otimes \dots \otimes u_{n + 1} ⟩ = ⟨ f, u_{1} ⟩ ⟨ φ, u_{2} \otimes \dots \otimes u_{n + 1} ⟩ = ⟨ φ, ⟨ u_{1}, f ⟩ u_{2} \otimes \dots \otimes u_{n + 1} ⟩ .$ This equality for every $φ$ and uniqueness of the representing vector imply $Ψ_{f}^{(n)} (u_{1} \otimes \dots \otimes u_{n + 1}) = ⟨ u_{1}, f ⟩ u_{2} \otimes \dots \otimes u_{n + 1}$ . Finally $Ψ_{c f}^{(n)} = \overline{c} Ψ_{f}^{(n)}$ for $c \in C$ since $⟨ u_{1}, c f ⟩ = \overset{ˉ}{c} ⟨ u_{1}, f ⟩$ on elementary tensors, so $f \mapsto Ψ_{f}^{(n)}$ is conjugate-linear.

2.1F3step 1.1algebra

[Symmetry preservation] Let $χ \in ran P_{+}^{(n + 1)}$ and let $σ \in S_{n}$ act on the last $n$ tensor factors by $U_{1 \otimes σ}$ ; by [F3] applied to the permutation $(1, σ) \in S_{n + 1}$ one has $U_{1 \otimes σ} χ = χ$ . Then for every $φ \in H^{\otimes n}$ , $⟨ φ, Ψ_{f}^{(n)} χ ⟩ = ⟨ f \otimes φ, χ ⟩ = ⟨ U_{1 \otimes σ}^{- 1} (f \otimes φ), χ ⟩ = ⟨ f \otimes U_{σ^{- 1}} φ, χ ⟩ = ⟨ U_{σ^{- 1}} φ, Ψ_{f}^{(n)} χ ⟩ = ⟨ φ, U_{σ} Ψ_{f}^{(n)} χ ⟩$ , the third equality because $U_{1 \otimes σ}^{- 1} = U_{1 \otimes σ^{- 1}}$ permutes only the last factors and fixes $f$ . Hence $U_{σ} Ψ_{f}^{(n)} χ = Ψ_{f}^{(n)} χ$ for every $σ \in S_{n}$ , so $Ψ_{f}^{(n)} χ \in ran P_{+}^{(n)}$ by the converse direction of [F3].

2.2F3step 1.1algebra

[Contraction bounds] For $ψ \in F_{0}$ , using $∥ P_{+}^{(n)} ∥ \leq 1$ ([F3]) and the tensor norm identity of [F3], $∥ S_{+} (f) ψ ∥^{2} = \sum_{n \geq 1} n ∥ P_{+}^{(n)} (f \otimes ψ_{n - 1}) ∥^{2} \leq \sum_{n \geq 1} n ∥ f \otimes ψ_{n - 1} ∥^{2} = ∥ f ∥^{2} \sum_{n \geq 0} (n + 1) ∥ ψ_{n} ∥^{2} = ∥ f ∥^{2} ∥ ψ ∥_{1}^{2}$ . Similarly $∥ S_{-} (f) ψ ∥^{2} = \sum_{n \geq 0} (n + 1) ∥ Ψ_{f}^{(n)} ψ_{n + 1} ∥^{2} \leq ∥ f ∥^{2} \sum_{n \geq 0} (n + 1) ∥ ψ_{n + 1} ∥^{2} \leq ∥ f ∥^{2} ∥ ψ ∥_{1}^{2}$ , where step 1.1 supplied $∥ Ψ_{f}^{(n)} ∥ \leq ∥ f ∥$ .

3.1F1F2step 1.1step 2.1algebra

[The operators on the core] For $ψ \in F_{0}$ only finitely many coordinates are nonzero, each in the appropriate range ([F1]); by [F2] and step 2.1 all displayed expressions $(S_{\pm} (f) ψ)_{n}$ lie in $ran P_{+}^{(n)}$ and are nonzero only for finitely many $n$ , so $S_{\pm} (f) ψ \in F_{0}$ . Linearity of $S_{+} (f)$ in $ψ$ and of $S_{+} (\cdot) ψ$ in $f$ , and conjugate-linearity of $S_{-} (f)$ in $f$ from step 1.1, are immediate from the formulas.

3.2F3step 1.1step 2.1algebra

[Per-sector commutator computation] For $n \geq 0$ define $A_{g}^{(n)} χ : = P_{+}^{(n + 1)} (g \otimes χ)$ from $ran P_{+}^{(n)}$ to $ran P_{+}^{(n + 1)}$ , and for $n \geq 1$ define $B_{f}^{(n)} χ : = Ψ_{f}^{(n - 1)} χ$ from $ran P_{+}^{(n)}$ to $ran P_{+}^{(n - 1)}$ . Let $χ = P_{+}^{(n)} (v_{1} \otimes \dots \otimes v_{n})$ . Since $P_{+}^{(n + 1)} (g \otimes χ) = P_{+}^{(n + 1)} (g \otimes v_{1} \otimes \dots \otimes v_{n})$ , averaging over the $(n + 1)!$ permutations and grouping by the first tensor factor gives [F3, step 1.1, step 2.1, algebra] $B_{f}^{(n + 1)} A_{g}^{(n)} χ = \frac{1}{n + 1} (⟨ g, f ⟩ χ + \sum_{j = 1}^{n} ⟨ v_{j}, f ⟩ P_{+}^{(n)} (g \otimes v_{1} \otimes \dots \otimes \hat{v_{j}} \otimes \dots \otimes v_{n})) .$ The same grouping over the $n!$ permutations of $χ$ gives $B_{f}^{(n)} χ = \frac{1}{n} \sum_{j = 1}^{n} ⟨ v_{j}, f ⟩ P_{+}^{(n - 1)} (v_{1} \otimes \dots \otimes \hat{v_{j}} \otimes \dots \otimes v_{n}),$ so applying $A_{g}^{(n - 1)}$ and multiplying by $n$ yields exactly the sum over $j$ in the preceding display. Therefore $(n + 1) B_{f}^{(n + 1)} A_{g}^{(n)} χ - n A_{g}^{(n - 1)} B_{f}^{(n)} χ = ⟨ g, f ⟩ χ$ . The maps in this identity are bounded by step 1.1 and contractivity of $P_{+}^{(m)}$ ([F3]); the displayed $χ$ have dense span in $ran P_{+}^{(n)}$ ([F3]), so the identity extends to every vector in that sector. For the vacuum scalar $c \in C$ , $Ψ_{f}^{(0)} P_{+}^{(1)} (g \otimes c) = ⟨ g, f ⟩ c$ and the annihilation-then-creation term vanishes, so the commutator identity also holds at $n = 0$ .

4.1F1step 1.1step 3.1algebra

[Adjoint relation] Let $φ, ψ \in F_{0}$ . Using the definition of $S_{+}$ and the self-adjointness of $P_{+}^{(n + 1)}$ , and the fact that $ψ_{n + 1} \in ran P_{+}^{(n + 1)}$ , $⟨ S_{+} (f) φ, ψ ⟩ = \sum_{n \geq 0} \sqrt{n + 1} ⟨ P_{+}^{(n + 1)} (f \otimes φ_{n}), ψ_{n + 1} ⟩ = \sum_{n \geq 0} \sqrt{n + 1} ⟨ f \otimes φ_{n}, ψ_{n + 1} ⟩$ . By the defining property of $Ψ_{f}^{(n)}$ in step 1.1 this equals $\sum_{n \geq 0} \sqrt{n + 1} ⟨ φ_{n}, Ψ_{f}^{(n)} ψ_{n + 1} ⟩ = ⟨ φ, S_{-} (f) ψ ⟩$ , both sums being finite.

4.2step 3.1step 3.2algebra

[Canonical commutation relation] Let $f, g \in H$ and $ψ \in F_{0}$ . For $n \geq 1$ , the definitions of $S_{\pm}$ and step 3.2 give $(S_{-} (f) S_{+} (g) ψ)_{n} = (n + 1) B_{f}^{(n + 1)} A_{g}^{(n)} ψ_{n}$ and $(S_{+} (g) S_{-} (f) ψ)_{n} = n A_{g}^{(n - 1)} B_{f}^{(n)} ψ_{n}$ . For $n = 0$ , the first composition is $Ψ_{f}^{(0)} P_{+}^{(1)} (g \otimes ψ_{0}) = ⟨ g, f ⟩ ψ_{0}$ and the second is zero. Thus each coordinate of the difference is $⟨ g, f ⟩ ψ_{n}$ ; all coordinates outside a finite set vanish, so the two sides agree.

5.1step 4.1step 4.2algebra∎

[The norm identity] Let $g \in H$ and $ψ \in F_{0}$ . Taking the inner product of the commutation relation of step 4.2 (with $f = g$ ) with $ψ$ gives $⟨ S_{-} (g) S_{+} (g) ψ, ψ ⟩ - ⟨ S_{+} (g) S_{-} (g) ψ, ψ ⟩ = ∥ g ∥^{2} ∥ ψ ∥^{2}$ . By the adjoint relation of step 4.1, $⟨ S_{-} (g) S_{+} (g) ψ, ψ ⟩ = ⟨ S_{+} (g) ψ, S_{+} (g) ψ ⟩ = ∥ S_{+} (g) ψ ∥^{2}$ and $⟨ S_{+} (g) S_{-} (g) ψ, ψ ⟩ = ⟨ S_{-} (g) ψ, S_{-} (g) ψ ⟩ = ∥ S_{-} (g) ψ ∥^{2}$ ; hence $∥ S_{+} (g) ψ ∥^{2} = ∥ S_{-} (g) ψ ∥^{2} + ∥ g ∥^{2} ∥ ψ ∥^{2}$ . This completes the proof of (a)-(d).

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

Closed adjoint sector domains

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space (Hilbert space) with completed tensor powers $H^{\otimes n}$ , symmetric projections $P_{+}^{(n)}$ , contraction maps $Ψ_{f}^{(n)}$ , Fock space $F (H)$ and core operators $S_{\pm} (f)$ on the finite-particle space $F_{0}$ as in Creation contraction bounds. For $f \in H$ define $D (A_{+} (f)) : = {ψ \in F (H) : \sum_{n \in N} (n + 1) ∥ P_{+}^{(n + 1)} (f \otimes ψ_{n}) ∥^{2} < \infty}, (A_{+} (f) ψ)_{n} : = \sqrt{n} P_{+}^{(n)} (f \otimes ψ_{n - 1}) (n \geq 1), (A_{+} (f) ψ)_{0} : = 0,$ $D (A_{-} (f)) : = {ψ \in F (H) : \sum_{n \in N} (n + 1) ∥ Ψ_{f}^{(n)} (ψ_{n + 1}) ∥^{2} < \infty}, (A_{-} (f) ψ)_{n} : = \sqrt{n + 1} Ψ_{f}^{(n)} (ψ_{n + 1}) .$ Put $D_{1} : = {ψ \in F (H) : \sum_{n} (n + 1) ∥ ψ_{n} ∥^{2} < \infty}$ . Then:

(a) $A_{+} (f)$ and $A_{-} (f)$ are well-defined linear operators on $F (H)$ with the displayed domains, and they are the closure and the adjoint of the core operator: $A_{+} (f) = \overline{S_{+} (f)} = S_{-} (f)^{*}, A_{-} (f) = S_{+} (f)^{*} = \overline{S_{-} (f)}$ (Adjoint of a densely defined operator, Densely defined, closed and closable operators, and cores, Symmetric, self-adjoint and essentially self-adjoint operators);

(b) both $A_{\pm} (f)$ are closed, densely defined and satisfy $A_{+} (f)^{*} = A_{-} (f)$ , $A_{-} (f)^{*} = A_{+} (f)$ ; in particular $F_{0} \subseteq D_{1} \subseteq D (A_{\pm} (f))$ ;

(c) for every $ψ \in D_{1}$ , $∥ A_{+} (f) ψ ∥ \leq ∥ f ∥ ∥ ψ ∥_{1}$ and $∥ A_{-} (f) ψ ∥ \leq ∥ f ∥ ∥ ψ ∥_{1}$ , where $∥ ψ ∥_{1}^{2} = \sum_{n} (n + 1) ∥ ψ_{n} ∥^{2}$ ;

(d) for every $g \in H$ and $ψ \in D_{1}$ the identity $∥ A_{+} (g) ψ ∥^{2} = ∥ A_{-} (g) ψ ∥^{2} + ∥ g ∥^{2} ∥ ψ ∥^{2}$ holds.

No boundedness of $A_{\pm} (f)$ on $F (H)$ , no self-adjointness of $A_{\pm} (f)$ , and no physical interpretation is asserted.

Facts & Assumptions

Given: the complex Hilbert space $H$ , the Fock space $F (H)$ with dense finite-particle subspace $F_{0}$ , the core operators $S_{\pm} (f) : F_{0} \to F_{0}$ and contraction maps $Ψ_{f}^{(n)}$ , vectors $f, g \in H$ , and vectors $φ, ψ \in F (H)$ .

[F1]

$F (H)$ is a Hilbert space, $F_{0}$ is dense, and $∥ ψ ∥^{2} = \sum_{n} ∥ ψ_{n} ∥^{2}$ (Hilbert direct-sum construction).

[F2]

$S_{+} (f)$ and $S_{-} (f)$ are densely defined linear (resp. conjugate-linear in $f$ ) operators on $F_{0}$ with the sector formulas $(S_{+} (f) ψ)_{n} = \sqrt{n} P_{+}^{(n)} (f \otimes ψ_{n - 1})$ , $(S_{-} (f) ψ)_{n} = \sqrt{n + 1} Ψ_{f}^{(n)} (ψ_{n + 1})$ , and $⟨ S_{+} (f) φ, ψ ⟩ = ⟨ φ, S_{-} (f) ψ ⟩$ for $φ, ψ \in F_{0}$ ; $Ψ_{f}^{(n)} (ran P_{+}^{(n + 1)}) \subseteq ran P_{+}^{(n)}$ (Creation contraction bounds).

[F3]

If $T$ is a densely defined operator with $S \subseteq T$ for a densely defined $S$ , then $T^{*} \subseteq S^{*}$ ; if $T$ is closed and $\overline{S} = T$ then $T = S^{* *}$ ; adjoints are closed operators (Adjoint of a densely defined operator, Densely defined, closed and closable operators, and cores, Symmetric, self-adjoint and essentially self-adjoint operators).

[F4]

Cauchy-Schwarz in the Hilbert space $F (H)$ : $∣ ⟨ η, θ ⟩ ∣ \leq ∥ η ∥ ∥ θ ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

Proof

1.1F1F2F3algebra

[The raising operator and its closure] For every $n$ the map $ψ \mapsto P_{+}^{(n + 1)} (f \otimes ψ_{n})$ is bounded with norm at most $∥ f ∥$ by [F2], so $D (A_{+} (f))$ is a linear subspace containing $F_{0}$ and $A_{+} (f)$ is a linear operator extending $S_{+} (f)$ with $(A_{+} (f) ψ)_{n} \in ran P_{+}^{(n)}$ for all $ψ \in D (A_{+} (f))$ and $\sum_{n} ∥ (A_{+} (f) ψ)_{n} ∥^{2} < \infty$ , so $A_{+} (f) ψ \in F (H)$ . If $ψ^{(k)} \in D (A_{+} (f))$ with $ψ^{(k)} \to ψ$ and $A_{+} (f) ψ^{(k)} \to η$ , then for each $m \geq 1$ continuity of $ψ \mapsto P_{+}^{(m)} (f \otimes ψ_{m - 1})$ gives $η_{m} = \sqrt{m} P_{+}^{(m)} (f \otimes ψ_{m - 1})$ , and $η_{0} = 0$ . For every finite $N$ , continuity of the norm and the finite sum gives $\sum_{n = 0}^{N} (n + 1) ∥ P_{+}^{(n + 1)} (f \otimes ψ_{n}) ∥^{2} = \lim_{k} \sum_{n = 0}^{N} (n + 1) ∥ P_{+}^{(n + 1)} (f \otimes ψ_{n}^{(k)}) ∥^{2} \leq \lim_{k} ∥ A_{+} (f) ψ^{(k)} ∥^{2} = ∥ η ∥^{2} .$ Taking the supremum over $N$ shows $ψ \in D (A_{+} (f))$ ; the coordinate formulas then give $A_{+} (f) ψ = η$ , so $A_{+} (f)$ is closed. For $ψ \in D (A_{+} (f))$ the truncations $ψ^{(N)}$ lie in $F_{0}$ , satisfy $ψ^{(N)} \to ψ$ , and $A_{+} (f) ψ^{(N)} \to A_{+} (f) ψ$ because the defining series of $ψ$ converges; hence $F_{0}$ is a core and $A_{+} (f) = \overline{S_{+} (f)}$ .

1.2F1F2F3F4algebra

[The lowering operator is the core adjoint] For $ψ \in F (H)$ set $η_{n} : = \sqrt{n + 1} Ψ_{f}^{(n)} ψ_{n + 1}$ . For $φ \in F_{0}$ , [F2] gives $⟨ S_{+} (f) φ, ψ ⟩ = \sum_{n \geq 0} ⟨ φ_{n}, η_{n} ⟩$ , a finite sum because $φ$ has finite support. If $\sum_{n} ∥ η_{n} ∥^{2} < \infty$ , Cauchy--Schwarz shows this functional is bounded on $F_{0}$ and its representing vector is $(η_{n})_{n} = A_{-} (f) ψ$ . Conversely, if the functional is bounded with norm $C$ , let $φ^{(N)}$ have coordinates $η_{n}$ for $n \leq N$ and zero thereafter. Then $φ^{(N)} \in F_{0}$ and $S_{N} : = \sum_{n \leq N} ∥ η_{n} ∥^{2} = ∣ ⟨ S_{+} (f) φ^{(N)}, ψ ⟩ ∣ \leq C ∥ φ^{(N)} ∥ = C \sqrt{S_{N}}$ , so $S_{N} \leq C^{2}$ for every $N$ . Hence $(η_{n})_{n} \in F (H)$ and the functional is represented by $A_{-} (f) ψ$ . Thus $ψ \in D (S_{+} (f)^{*})$ exactly when $\sum_{n} (n + 1) ∥ Ψ_{f}^{(n)} ψ_{n + 1} ∥^{2} < \infty$ , and $S_{+} (f)^{*} = A_{-} (f)$ . As an adjoint $A_{-} (f)$ is closed, and it is densely defined because it contains $F_{0}$ .

1.3F1F2algebra

[Weighted domain and bounds] For $ψ \in D_{1}$ , $\sum_{n} (n + 1) ∥ P_{+}^{(n + 1)} (f \otimes ψ_{n}) ∥^{2} \leq ∥ f ∥^{2} \sum_{n} (n + 1) ∥ ψ_{n} ∥^{2} < \infty$ and likewise $\sum_{n} (n + 1) ∥ Ψ_{f}^{(n)} (ψ_{n + 1}) ∥^{2} \leq ∥ f ∥^{2} \sum_{n} (n + 1) ∥ ψ_{n + 1} ∥^{2} < \infty$ , so $D_{1} \subseteq D (A_{\pm} (f))$ and the bounds of (c) follow from the same estimates; since $F_{0} \subseteq D_{1}$ , this also gives the density in (b).

2.1F2F3step 1.1step 1.2algebra

[Mutual adjoints] Since $A_{+} (f) = \overline{S_{+} (f)} = S_{+} (f)^{* *}$ by step 1.1, [F3] gives $A_{+} (f)^{*} = S_{+} (f)^{* * *} = S_{+} (f)^{*} = A_{-} (f)$ by step 1.2. Applying the same step 1.2 with the roles of $S_{+}$ and $S_{-}$ exchanged (the adjoint relation of [F2] is symmetric in the two core operators) gives $S_{-} (f)^{*} = A_{+} (f)$ , hence $A_{-} (f)^{*} = S_{+} (f)^{* *} = A_{+} (f)$ . Consequently both operators are closed with the stated adjoint pairs, proving (a) and (b) apart from the domain inclusion.

3.1F1F2step 1.1step 1.3algebra∎

[The weighted norm identity] Let $ψ \in D_{1}$ and let $ψ^{(N)}$ be its truncations; then $ψ^{(N)} \to ψ$ and, by the sector formulas and the convergence of the defining sums of $ψ$ , $A_{\pm} (g) ψ^{(N)} \to A_{\pm} (g) ψ$ . The identity of Creation contraction bounds (d) applies to $ψ^{(N)} \in F_{0}$ ; passing to the limit in $∥ A_{+} (g) ψ^{(N)} ∥^{2} = ∥ A_{-} (g) ψ^{(N)} ∥^{2} + ∥ g ∥^{2} ∥ ψ^{(N)} ∥^{2}$ using continuity of the norm and of addition in $F (H)$ gives the displayed identity for $ψ$ . This proves (d) and completes the proof.

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

Number operator domain

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space with completed tensor powers $H^{\otimes n}$ , symmetric projections $P_{+}^{(n)}$ and Fock space $F (H)$ as in Hilbert direct-sum construction. Define the number operator by $D (N) : = {ψ \in F (H) : \sum_{n \in N} n^{2} ∥ ψ_{n} ∥^{2} < \infty}, (N ψ)_{n} : = n ψ_{n} .$ Then:

(a) $N$ is a densely defined self-adjoint operator on $F (H)$ with the displayed domain; $N \geq 0$ in the sense that $⟨ N ψ, ψ ⟩ \geq 0$ for $ψ \in D (N)$ , $N Ω = 0$ , and $F_{0}$ is a core for $N$ (Symmetric, self-adjoint and essentially self-adjoint operators);

(b) for every $n$ , every nonzero $u \in ran P_{+}^{(n)}$ satisfies $N ι_{n} (u) = n ι_{n} (u)$ , and for $H \neq {0}$ the spectrum of $N$ is $σ (N) = N$ ;

(c) for $f \in H$ and $ψ \in F_{0}$ the core operators of Creation contraction bounds satisfy $N S_{+} (f) ψ - S_{+} (f) N ψ = S_{+} (f) ψ, N S_{-} (f) ψ - S_{-} (f) N ψ = - S_{-} (f) ψ;$

(d) $D (N) \subseteq D_{1} : = {ψ : \sum_{n} (n + 1) ∥ ψ_{n} ∥^{2} < \infty}$ , and for every $ψ \in D (N)$ , $∥ N ψ ∥^{2} = \sum_{n \in N} n^{2} ∥ ψ_{n} ∥^{2}, ⟨ N ψ, ψ ⟩ = \sum_{n \in N} n ∥ ψ_{n} ∥^{2} .$

No physical interpretation of $N$ , and no assertion about particle number measurement, is made.

Facts & Assumptions

Given: the complex Hilbert space $H$ , the Fock space $F (H)$ with sectors $ran P_{+}^{(n)}$ and coordinate embeddings $ι_{n}$ , the dense finite-particle subspace $F_{0}$ , the vacuum $Ω$ , the number operator $N$ with domain $D (N)$ , vectors $f \in H$ and $ψ \in F (H)$ , and $n \in N$ .

[F1]

$F (H)$ is a Hilbert space with inner product $⟨ ψ, φ ⟩ = \sum_{n} ⟨ ψ_{n}, φ_{n} ⟩$ , $F_{0}$ is dense, and each $ran P_{+}^{(n)}$ is a closed subspace of the Hilbert space $H^{\otimes n}$ (Hilbert direct-sum construction, Permutation sector projections on the n-fold tensor power).

[F2]

For a densely defined operator $A$ , a vector $y$ belongs to $D (A^{*})$ exactly when there is $z$ with $⟨ A x, y ⟩ = ⟨ x, z ⟩$ for every $x \in D (A)$ ; then $A^{*} y = z$ . A densely defined symmetric operator satisfies $A \subseteq A^{*}$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators). Cauchy-Schwarz holds in each sector by Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs; for square-summable coordinate norms apply that theorem to finite nonnegative real vectors $(a_{0}, \dots, a_{N})$ and $(b_{0}, \dots, b_{N})$ in $R^{N + 1}$ with inner product $⟨ a, b ⟩ : = \sum_{n = 0}^{N} a_{n} b_{n}$ , then take increasing partial sums. These definitions require only the declared Countable Choice; self-adjointness of $N$ will be proved directly rather than invoking the stronger choice hypotheses of the general direct-sum theorem.

[F3]

The core operators satisfy $(S_{+} (f) ψ)_{n} = \sqrt{n} P_{+}^{(n)} (f \otimes ψ_{n - 1})$ , $(S_{-} (f) ψ)_{n} = \sqrt{n + 1} Ψ_{f}^{(n)} (ψ_{n + 1})$ on $F_{0}$ (Creation contraction bounds).

Proof

1.1F1F2algebra

[Coordinate adjoint, vacuum and positivity] The coordinate Hilbert-space construction used in [F1] is valid under the present hypotheses: the sector spaces are complete closed tensor subspaces, the pairing converges by Cauchy-Schwarz, and a Cauchy sequence has coordinate limits. For each finite coordinate sum, the Cauchy norm bound passes to the limit; taking the supremum over those finite sums proves that the limit is square summable and that convergence holds in the full sum. Finite-coordinate truncations give density of $F_{0}$ . Thus these Hilbert and density facts do not require invoking the Axiom of Choice premise of the general operator-sum theorem. [F1, F2, algebra] Every vector of $F_{0}$ lies in $D (N)$ , so $N$ is densely defined. For $ψ, φ \in D (N)$ , the sequence form of Cauchy-Schwarz in [F2] makes the coordinate pairing series absolutely convergent and gives $⟨ N ψ, φ ⟩ = \sum_{n} n ⟨ ψ_{n}, φ_{n} ⟩ = ⟨ ψ, N φ ⟩$ ; hence $N \subseteq N^{*}$ . If $φ \in D (N^{*})$ and $η = N^{*} φ$ , test the adjoint identity with every vector $ι_{n} (u)$ in a single sector. It gives $⟨ n u, φ_{n} ⟩ = ⟨ u, η_{n} ⟩$ for all $u$ , so $η_{n} = n φ_{n}$ for every $n$ , including $η_{0} = 0$ . Consequently $\sum_{n} n^{2} ∥ φ_{n} ∥^{2} = ∥ η ∥^{2} < \infty$ , proving $φ \in D (N)$ and $N^{*} φ = N φ$ . Thus $N = N^{*}$ on exactly the displayed domain. For $ψ \in D (N)$ , $⟨ N ψ, ψ ⟩ = \sum_{n} n ∥ ψ_{n} ∥^{2} \geq 0$ , and the vacuum satisfies $N Ω = 0$ . The finite-coordinate truncations $ψ^{(M)}$ lie in $F_{0}$ and satisfy $ψ^{(M)} \to ψ$ and $N ψ^{(M)} \to N ψ$ by convergence of $\sum_{n} n^{2} ∥ ψ_{n} ∥^{2}$ , so $F_{0}$ is a graph core. This retains all conclusions of (a) under its original Countable Choice hypothesis.

1.2F3algebra

[Commutation on the core] Let $f \in H$ and $ψ \in F_{0}$ ; all vectors below lie in $F_{0}$ by [F3]. Using the sector formulas of [F3], for $n \geq 1$ the $n$ -th coordinate of $S_{+} (f) N ψ$ is $\sqrt{n} P_{+}^{(n)} (f \otimes (n - 1) ψ_{n - 1}) = (n - 1) (S_{+} (f) ψ)_{n}$ , while the $n$ -th coordinate of $N S_{+} (f) ψ$ is $n (S_{+} (f) ψ)_{n}$ ; both coordinates vanish for $n = 0$ . Hence $N S_{+} (f) ψ - S_{+} (f) N ψ = S_{+} (f) ψ$ . Similarly, the $n$ -th coordinate of $S_{-} (f) N ψ$ is $\sqrt{n + 1} Ψ_{f}^{(n)} ((n + 1) ψ_{n + 1}) = (n + 1) (S_{-} (f) ψ)_{n}$ , and that of $N S_{-} (f) ψ$ is $n (S_{-} (f) ψ)_{n}$ ; their difference is $- (S_{-} (f) ψ)_{n}$ for every $n$ , proving the second identity.

2.1F1F2step 1.1algebra

[Eigenvalues and spectrum] Let $n \in N$ and $u \in ran P_{+}^{(n)}$ , $u \neq 0$ . The vector $ι_{n} (u)$ lies in $D (N)$ with $(N ι_{n} (u))_{m} = m (ι_{n} (u))_{m}$ , which is $n ι_{n} (u)$ ; thus every $n \in N$ is an eigenvalue when $H \neq {0}$ , since then each sector contains $e^{\otimes n} \neq 0$ (and the vacuum sector is $C$ ). Now fix $λ \in C ∖ N$ and put $c_{n} : = (n - λ)^{- 1}$ . If $n > 2 ∣ λ ∣$ , then $∣ n - λ ∣ \geq n - ∣ λ ∣ > n / 2$ , so $∣ c_{n} ∣ < 2 / n \leq 2$ and $n ∣ c_{n} ∣ < 2$ ; among the finitely many remaining indices each denominator is nonzero because $λ \notin N$ . Hence $\sup_{n} ∣ c_{n} ∣ < \infty$ and $\sup_{n} n ∣ c_{n} ∣ < \infty$ . Define $R_{λ} ψ : = (c_{n} ψ_{n})_{n}$ . Then $∥ R_{λ} ψ ∥^{2} = \sum_{n} ∣ c_{n} ∣^{2} ∥ ψ_{n} ∥^{2} \leq (\sup_{n} ∣ c_{n} ∣)^{2} ∥ ψ ∥^{2}$ and $\sum_{n} n^{2} ∥ c_{n} ψ_{n} ∥^{2} \leq (\sup_{n} n ∣ c_{n} ∣)^{2} ∥ ψ ∥^{2}$ , so $R_{λ}$ is bounded from $F (H)$ to $D (N)$ with its graph norm. Coordinatewise, $(N - λ I) R_{λ} = I$ on $F (H)$ and $R_{λ} (N - λ I) = I$ on $D (N)$ ; hence $R_{λ}$ is a bounded inverse of $N - λ I$ . Therefore $σ (N) \subseteq N$ , and the exhibited eigenvalues give $σ (N) = N$ for $H \neq {0}$ .

3.1F1step 1.1algebra∎

[Domain inclusion and moments] If $ψ \in D (N)$ then $\sum_{n} (n + 1) ∥ ψ_{n} ∥^{2} = \sum_{n} n ∥ ψ_{n} ∥^{2} + \sum_{n} ∥ ψ_{n} ∥^{2} \leq \sum_{n} n^{2} ∥ ψ_{n} ∥^{2} + ∥ ψ ∥^{2} < \infty$ (using $n \leq n^{2}$ ), so $D (N) \subseteq D_{1}$ . The coordinate formula $(N ψ)_{n} = n ψ_{n}$ and orthogonality of the sectors give $∥ N ψ ∥^{2} = \sum_{n} ∥ n ψ_{n} ∥^{2} = \sum_{n} n^{2} ∥ ψ_{n} ∥^{2}$ and $⟨ N ψ, ψ ⟩ = \sum_{n} n ∥ ψ_{n} ∥^{2}$ , both series converging absolutely; this proves (d) and completes the proof of (a)-(d).

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Fixed-mode number law

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H : = C$ with its standard inner product (linear in the first argument), let $F (C)$ be its Fock space as in Hilbert direct-sum construction, let $e_{n} \in F (C)$ be the unit vector supported in coordinate $n$ (so $(e_{n})_{n \in N}$ is a complete orthonormal family of $F (C)$ ), let $N$ be the number operator and let $A_{\pm} (1)$ be the maximal creation and annihilation operators of Closed adjoint sector domains for the vector $1 \in C$ . Then:

(a) oscillator coefficients: for every $n \in N$ , $A_{+} (1) e_{n} = \sqrt{n + 1} e_{n + 1}, A_{-} (1) e_{n} = \sqrt{n} e_{n - 1} (n \geq 1), A_{-} (1) e_{0} = 0;$

(b) coherent vectors: for every $z \in C$ the series $ψ_{z} : = e^{- ∣ z ∣^{2} / 2} \sum_{n \in N} \frac{z^{n}}{\sqrt{n!}} e_{n}$ converges absolutely in $F (C)$ (The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument), $∥ ψ_{z} ∥ = 1$ , $ψ_{z} \in D (A_{-} (1))$ with $A_{-} (1) ψ_{z} = z ψ_{z},$ and for all $z, w \in C$ , $⟨ ψ_{z}, ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2 + z \overline{w}};$

(c) number law: for every $n \in N$ , $∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = e^{- ∣ z ∣^{2}} \frac{∣ z ∣^{2 n}}{n!},$ and $\sum_{n \in N} ∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = 1$ .

The numbers in (c) are squared moduli of overlaps of unit vectors; identifying them with outcome probabilities of a number measurement requires a separately adopted measurement postulate, and no such physical claim is made here.

Facts & Assumptions

Given: the one-dimensional Hilbert space $C$ with its standard inner product, the Fock space $F (C)$ with complete orthonormal family $(e_{n})_{n \in N}$ , the maximal operators $A_{\pm} (1)$ and the number operator $N$ on $F (C)$ , and $z, w \in C$ , $n \in N$ .

[F1]

$F (C)$ is a Hilbert space with $⟨ e_{m}, e_{n} ⟩ = δ_{m n}$ and $∥ ψ ∥^{2} = \sum_{n} ∣ ⟨ e_{n}, ψ ⟩ ∣^{2}$ , and it is complete so that every absolutely convergent series converges (Hilbert direct-sum construction).

[F2]

For $H = C$ one has $C^{\otimes m} = C$ , $P_{+}^{(m)} = i d$ and $ran P_{+}^{(m)} = C$ for every $m$ (Hilbert tensor construction: inner product, completion and product bases, Permutation sector projections on the n-fold tensor power).

[F3]

$A_{\pm} (1)$ are closed operators. For a finite-sector vector $ψ$ , their coordinate formulas are $(A_{+} (1) ψ)_{0} = 0$ , $(A_{+} (1) ψ)_{n} = \sqrt{n} ψ_{n - 1}$ for $n \geq 1$ , and $(A_{-} (1) ψ)_{n} = \sqrt{n + 1} ψ_{n + 1}$ for $n \geq 0$ , under the identifications of [F2] (Closed adjoint sector domains).

[F5]

The complex exponential satisfies $\exp (z + w) = \exp (z) \exp (w)$ for all complex $z, w$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential).

[F4]

For every $z \in C$ the series $\sum_{n} z^{n} / n!$ converges absolutely, so its value $\exp z$ is defined and the partial sums of $\sum_{n} ∣ z ∣^{2 n} / n!$ converge to $\exp (∣ z ∣^{2})$ (The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument).

Proof

1.1F2F3algebra

[Oscillator coefficients] For $e_{m}$ supported in coordinate $m$ , the raising formula of [F3] has its only nonzero coordinate at $m + 1$ , with value $\sqrt{m + 1}$ , so $A_{+} (1) e_{m} = \sqrt{m + 1} e_{m + 1}$ . For $m \geq 1$ the lowering formula has its only nonzero coordinate at $m - 1$ , with value $\sqrt{m}$ , so $A_{-} (1) e_{m} = \sqrt{m} e_{m - 1}$ ; for $m = 0$ every lowering coordinate is zero. This proves (a).

1.2F1F4F5algebra

[Convergence and normalization] Let $z \in C$ and $S_{N} : = e^{- ∣ z ∣^{2} / 2} \sum_{n \leq N} z^{n} / \sqrt{n!} e_{n}$ . Orthogonality of the $e_{n}$ gives $∥ S_{N} - S_{M} ∥^{2} = e^{- ∣ z ∣^{2}} \sum_{M < n \leq N} ∣ z ∣^{2 n} / n!$ , and $\sum_{n} ∣ z ∣^{2 n} / n! = \exp (∣ z ∣^{2})$ by [F4], so $(S_{N})$ is Cauchy and converges in the complete space $F (C)$ to a vector $ψ_{z}$ ; moreover $∥ ψ_{z} ∥^{2} = \lim_{N} ∥ S_{N} ∥^{2} = e^{- ∣ z ∣^{2}} \exp (∣ z ∣^{2}) = 1$ , so $∥ ψ_{z} ∥ = 1$ . For $z = 0$ only the zeroth term survives, so convergence is absolute. For $z \neq 0$ , convergence is absolute because the ratio of consecutive moduli of $\sum_{n} ∣ z ∣^{n} / \sqrt{n!}$ is $∣ z ∣ / \sqrt{n + 1} \to 0$ , so that series converges by the ratio test (Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence); and $\sum_{n} ∣ z ∣^{2 n} / n! = \exp (∣ z ∣^{2})$ is the exponential series of the real number $∣ z ∣^{2} \geq 0$ ([F4]).

2.1F3step 1.1step 1.2algebra

[Annihilation eigenvalue] Every finite combination $S_{N} \in F_{0}$ lies in the domain of the closed operator $A_{-} (1)$ , and $A_{-} (1) S_{N} = e^{- ∣ z ∣^{2} / 2} \sum_{1 \leq n \leq N} z^{n} / \sqrt{n!} \sqrt{n} e_{n - 1} = z e^{- ∣ z ∣^{2} / 2} \sum_{0 \leq m \leq N - 1} z^{m} / \sqrt{m!} e_{m} \to z ψ_{z}$ by step 1.2. Since $S_{N} \to ψ_{z}$ and $A_{-} (1)$ is closed, $ψ_{z} \in D (A_{-} (1))$ and $A_{-} (1) ψ_{z} = z ψ_{z}$ .

2.2F1F4F5step 1.2algebra

[Overlaps and the number law] For $z, w \in C$ , the series $ψ_{z}$ and $ψ_{w}$ have absolutely convergent coordinate expansions, so the inner product may be computed termwise: $⟨ ψ_{z}, ψ_{w} ⟩ = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2} \sum_{n} z^{n} \overline{w^{n}} / n! = e^{- (∣ z ∣^{2} + ∣ w ∣^{2}) / 2 + z \overline{w}}$ , the last equality by the exponential series [F4] and its addition law [F5]. In particular $∥ ψ_{z} ∥^{2} = ⟨ ψ_{z}, ψ_{z} ⟩ = e^{- ∣ z ∣^{2} + ∣ z ∣^{2}} = 1$ . Since $∣ ⟨ e_{n}, ψ_{z} ⟩ ∣^{2} = ∣ e^{- ∣ z ∣^{2} / 2} {\overline{z}}^{n} / \sqrt{n!} ∣^{2} = e^{- ∣ z ∣^{2}} ∣ z ∣^{2 n} / n!$ and $\sum_{n} ∣ z ∣^{2 n} / n! = \exp (∣ z ∣^{2})$ , the weights in (c) sum to $1$ .

3.1step 1.1step 1.2step 2.1step 2.2algebra∎

[Conclusion and scope] Steps 1.1-2.2 prove (a), (b) and (c); the square roots $\sqrt{n!}$ and $\sqrt{n}$ are the positive real roots. The final statement records that no measurement postulate, and hence no probability interpretation of the displayed weights, is asserted; the Fock vectors and the eigenvalue relation are purely mathematical.

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Single-mode thermal trace

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $ω > 0$ and $β > 0$ be real parameters, let $F (C)$ be the one-mode Fock space with its complete orthonormal family $(e_{n})_{n \in N}$ and number operator $N$ as in Fixed-mode number law and Number operator domain, and define the operator $G_{β} \in B (F (C))$ by $G_{β} e_{n} : = e^{- β ω n} e_{n} (n \in N)$ together with bounded linear extension. Then:

(a) $G_{β}$ is a bounded self-adjoint positive operator with $∥ G_{β} ∥ = 1$ , and $G_{β} = ⨁_{n \in N} e^{- β ω n} i d$ on the Fock sectors (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum);

(b) the series of weights converge, $Z (β) : = \sum_{n \in N} e^{- β ω n} = \frac{1}{1 - e^{- β ω}}, \sum_{n \in N} n e^{- β ω n} = \frac{e^{- β ω}}{(1 - e^{- β ω})^{2}};$

(c) $G_{β}$ is trace class with $∥ G_{β} ∥_{1} = tr G_{β} = Z (β)$ (Trace class operator, Trace of a trace class operator);

(d) $ρ_{β} : = G_{β} / Z (β)$ is a density operator on $F (C)$ (Density operators and scalar spectral probability), and its number expectation, defined by the absolutely convergent spectral sum $⟨ N ⟩_{β} : = \sum_{n} n e^{- β ω n} / Z (β)$ , equals $⟨ N ⟩_{β} = \frac{e^{- β ω}}{1 - e^{- β ω}} = \frac{1}{e^{β ω} - 1};$ more generally $\sum_{n} n^{k} e^{- β ω n} < \infty$ for every $k \geq 1$ .

The parameter $β$ is a positive real parameter of the model; its identification with an inverse temperature, and the interpretation of $ρ_{β}$ as a physical thermal state, are additional physical assumptions not asserted here.

Facts & Assumptions

Given: the real parameters $ω > 0$ , $β > 0$ , the one-mode Fock space $F (C)$ with complete orthonormal family $(e_{n})_{n \in N}$ and number operator $N$ , and the bounded operator $G_{β}$ defined on the orthonormal family by $G_{β} e_{n} = e^{- β ω n} e_{n}$ .

[F1]

$(e_{n})_{n \in N}$ is a complete orthonormal family of the Hilbert space $F (C)$ ; $N$ is self-adjoint with $N e_{n} = n e_{n}$ (Fixed-mode number law, Number operator domain).

[F2]

For real scalars $c_{n}$ on every sector $ran P_{+}^{(n)}$ , including $c_{0}$ on $C$ , the direct-sum theorem makes $⨁_{n} c_{n} i d$ bounded and self-adjoint when $\sup_{n} ∣ c_{n} ∣ < \infty$ , with norm $\sup_{n} ∣ c_{n} ∣$ (Hilbert direct-sum construction (f)); when $c_{n} \geq 0$ , positivity follows from $⟨ (⨁_{n} c_{n} i d) ψ, ψ ⟩ = \sum_{n} c_{n} ∥ ψ_{n} ∥^{2} \geq 0$ . A bounded operator agreeing with another bounded operator on a complete orthonormal family agrees everywhere (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F3]

Geometric series: $\sum_{n \geq 0} q^{n} = 1 / (1 - q)$ for $∣ q ∣ < 1$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges); Tonelli for nonnegative double series permits the interchange of the summation order, even for the extended value $+ \infty$ (Tonelli's theorem for double series of nonnegative extended real numbers).

[F4]

The Fourier-coefficient map $Φ : F (C) \to ℓ^{2} (N, C)$ , $Φ (ψ) = (⟨ ψ, e_{n} ⟩)_{n}$ , is a linear bijection with $∥ Φ (ψ) ∥_{2} = ∥ ψ ∥$ (A Hilbert space with a given orthonormal basis is $ℓ^{2}$ of the index set).

[F5]

Bounded finite-rank operators are compact, norm limits of compact operators into a Hilbert space are compact, and a compact operator with a nuclear representation of summable rank-one norms is trace class with trace norm no greater than that sum (Bounded finite rank operators are compact, Norm limit of compact operators is compact, Nuclear series characterizes trace norm).

[F6]

If $(u_{n})$ is a countable orthonormal family and $w_{n} \geq 0$ with $\sum_{n} w_{n} = 1$ , then $ρ : = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ converges in trace norm to a density operator with $tr ρ = 1$ (Density operators and scalar spectral probability (a)).

[F7]

The complex exponential satisfies $\exp (z + w) = \exp (z) \exp (w)$ for all complex $z, w$ and agrees with the real exponential on real inputs ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential).

[F8]

For a trace-class operator $T$ and a supplied Hilbert basis $(e_{n})$ , $tr T = \sum_{n} ⟨ T e_{n}, e_{n} ⟩$ and $∣ tr T ∣ \leq ∥ T ∥_{1}$ (Trace is absolutely convergent and basis independent).

Proof

1.1F1F2algebra

[The diagonal weights] For every $n$ the operator $c_{n} i d$ on the nonzero complex Hilbert space $ran P_{+}^{(n)} = C$ has the single real eigenvalue $e^{- β ω n} \in (0, 1]$ , so the direct sum $⨁_{n} e^{- β ω n} i d$ is bounded and self-adjoint with norm $\sup_{n} e^{- β ω n} = 1$ by [F2]; its quadratic form is $\sum_{n} e^{- β ω n} ∥ ψ_{n} ∥^{2} \geq 0$ , so it is positive. By [F1] this direct sum agrees with $G_{β}$ on the complete orthonormal family $(e_{n})$ and therefore equals $G_{β}$ : a bounded operator vanishing on all $e_{n}$ is zero. This proves (a).

1.2F3F7algebra

[The weight sums] Put $q : = e^{- β ω} \in (0, 1)$ . The geometric series [F3] gives $Z (β) = \sum_{n \geq 0} q^{n} = 1 / (1 - q)$ . For the first moment, $\sum_{n \geq 1} n q^{n} = \sum_{n \geq 1} \sum_{m = 1}^{n} q^{n} = \sum_{m \geq 1} \sum_{n \geq m} q^{n}$ , where the interchange of the nonnegative double series is licensed by [F3]; the inner sum is $\sum_{n \geq m} q^{n} = q^{m} / (1 - q)$ by the geometric series, so the total is $\frac{1}{1 - q} \sum_{m \geq 1} q^{m} = \frac{q}{(1 - q)^{2}} < \infty$ . The identity $q / (1 - q) = 1 / (e^{β ω} - 1)$ follows from $e^{β ω} e^{- β ω} = e^{0} = 1$ by [F7]. This proves (b).

2.1F4F5F8step 1.2algebra

[Trace class and its trace] Put $q_{n} : = e^{- β ω n}$ and $R_{N} : = \sum_{n = 0}^{N} q_{n} T_{e_{n}, e_{n}}$ . Each $R_{N}$ is finite rank and compact by [F5]. Parseval [F4] gives $∥ (G_{β} - R_{N}) ψ ∥^{2} = \sum_{n > N} q_{n}^{2} ∣ ⟨ ψ, e_{n} ⟩ ∣^{2} \leq q_{N + 1}^{2} ∥ ψ ∥^{2}$ , so $R_{N} \to G_{β}$ in operator norm and [F5] makes $G_{β}$ compact. The expansion $G_{β} ψ = \sum_{n \geq 0} ⟨ ψ, e_{n} ⟩ (q_{n} e_{n})$ is a nuclear representation with sum $\sum_{n} ∥ e_{n} ∥ ∥ q_{n} e_{n} ∥ = \sum_{n} q_{n} = Z (β)$ ; [F5] therefore gives $∥ G_{β} ∥_{1} \leq Z (β)$ and trace class. By [F8], $tr G_{β} = \sum_{n} ⟨ G_{β} e_{n}, e_{n} ⟩ = \sum_{n} q_{n} = Z (β)$ and $Z (β) = ∣ tr G_{β} ∣ \leq ∥ G_{β} ∥_{1}$ . Thus $∥ G_{β} ∥_{1} = tr G_{β} = Z (β)$ , proving (c).

2.2F3F6step 1.1step 1.2algebra

[The density operator and its number expectation] With $w_{n} : = e^{- β ω n} / Z (β) \geq 0$ and $\sum_{n} w_{n} = 1$ by step 1.2, [F6] gives that $ρ_{β} = \sum_{n} w_{n} T_{e_{n}, e_{n}}$ is a density operator; since $T_{e_{n}, e_{n}}$ is the rank-one projection onto $C e_{n}$ and $G_{β} = \sum_{n} e^{- β ω n} T_{e_{n}, e_{n}}$ (agreement on the complete orthonormal family, step 1.1), one has $ρ_{β} = G_{β} / Z (β)$ . The display defines $⟨ N ⟩_{β}$ as the absolutely convergent sum $\sum_{n} n w_{n}$ ; by step 1.2, $⟨ N ⟩_{β} = \frac{e^{- β ω}}{(1 - e^{- β ω})^{2}} / \frac{1}{1 - e^{- β ω}} = \frac{e^{- β ω}}{1 - e^{- β ω}} = \frac{1}{e^{β ω} - 1}$ . For each integer $k \geq 1$ , put $a : = β ω / 2 > 0$ and $b_{n} : = n^{k} e^{- a n}$ for $n \geq 1$ . Then $b_{n + 1} / b_{n} = (1 + 1 / n)^{k} e^{- a} \to e^{- a} < 1$ , so for some $N$ and $r < 1$ this ratio is at most $r$ for all $n \geq N$ ; hence $(b_{n})$ is bounded. Thus $n^{k} e^{- β ω n} = b_{n} e^{- a n} \leq C_{k} e^{- a n}$ for a finite $C_{k}$ , and the geometric series with ratio $e^{- a} < 1$ gives $\sum_{n} n^{k} e^{- β ω n} < \infty$ .

3.1step 1.1step 1.2step 2.1step 2.2algebra∎

[Conclusion and scope] Steps 1.1-2.2 prove (a)-(d); the displayed series are ordinary absolutely convergent real series and the trace statements use only the supplied spectral data. The closing paragraph of the statement records that the physical reading of $β$ and $ρ_{β}$ requires additional assumptions, so no physical interpretation is asserted here.

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Countable-mode product criterion

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $H$ be a complex Hilbert space with a countable complete orthonormal family $(e_{j})_{j \geq 1}$ (Orthonormal families, complete orthonormal systems and Hilbert bases), let $(ε_{j})_{j \geq 1} \subseteq (0, \infty)$ , let $β > 0$ , and let $F (H)$ be the Fock space of Hilbert direct-sum construction. Let $M$ be the set of finite nondecreasing sequences $I = (j_{1} \leq \dots \leq j_{n})$ with $n \geq 0$ (the empty sequence included); for $I = (j_{1} \leq \dots \leq j_{n})$ put $v_{I} : = P_{+}^{(n)} (e_{j_{1}} \otimes \dots \otimes e_{j_{n}}) \in F (H) (n \geq 1), v_{\emptyset} : = Ω, λ_{I} : = ε_{j_{1}} + \dots + ε_{j_{n}} (empty sum 0),$ and $u_{I} : = v_{I} / ∥ v_{I} ∥$ . Then:

(a) $(u_{I})_{I \in M}$ is a complete orthonormal family of $F (H)$ (Hilbert space);

(b) the diagonal operator $D (D_{β}) : = {ψ \in F (H) : \sum_{I \in M} e^{- 2 β λ_{I}} ∣ ⟨ ψ, u_{I} ⟩ ∣^{2} < \infty}, D_{β} ψ : = \sum_{I} e^{- β λ_{I}} ⟨ ψ, u_{I} ⟩ u_{I}$ is bounded, self-adjoint and positive with $∥ D_{β} ∥ \leq 1$ ; it is trace class exactly when $\sum_{I} e^{- β λ_{I}} < \infty$ , and then $tr D_{β} = \sum_{I} e^{- β λ_{I}}$ (Trace class operator, Trace of a trace class operator);

(c) for every $N \geq 1$ , $\sum_{\begin{array}{c} I \in M \\ I \subseteq {1, \dots, N} \end{array}} e^{- β λ_{I}} = \prod_{j = 1}^{N} \frac{1}{1 - e^{- β ε_{j}}}$ , and $\sum_{I \in M} e^{- β λ_{I}} = \lim_{N \to \infty} \prod_{j = 1}^{N} \frac{1}{1 - e^{- β ε_{j}}} \in (0, \infty];$

(d) product criterion: $\sum_{I} e^{- β λ_{I}} < \infty$ if and only if $\sum_{j \geq 1} e^{- β ε_{j}} < \infty$ , and in that case $\sum_{I \in M} e^{- β λ_{I}} = \prod_{j = 1}^{\infty} \frac{1}{1 - e^{- β ε_{j}}} < \infty;$

(e) consequently $D_{β}$ is trace class if and only if $\sum_{j} e^{- β ε_{j}} < \infty$ , with the trace given by the infinite product in (d).

The operator $D_{β}$ is a purely mathematical diagonal operator on $F (H)$ ; no thermal, thermodynamic or measurement interpretation is asserted.

Facts & Assumptions

Given: a complex Hilbert space $H$ with countable complete orthonormal family $(e_{j})_{j \geq 1}$ , positive numbers $ε_{j}$ and $β$ , the Fock space $F (H)$ with tensor powers $H^{\otimes n}$ and symmetric projections $P_{+}^{(n)}$ , the family $(u_{I})_{I \in M}$ and the numbers $λ_{I}$ , and the diagonal operator $D_{β}$ .

[F1]

$F (H)$ is a Hilbert space; $ψ_{n} \in ran P_{+}^{(n)}$ ; $∥ ψ ∥^{2} = \sum_{n} ∥ ψ_{n} ∥^{2}$ ; the sectors are orthogonal (Hilbert direct-sum construction, Fock sector projections).

[F2]

$P_{+}^{(n)}$ is the orthogonal projection $(1 / n!) \sum_{π \in S_{n}} U_{π}$ onto the vectors fixed by all permutation unitaries; for fixed $m$ the tensor maps are isometric: $∥ u_{1} \otimes \dots \otimes u_{m} ∥ = ∥ u_{1} ∥ \dots ∥ u_{m} ∥$ ; the elementary tensors $e_{i_{1}} \otimes \dots \otimes e_{i_{n}}$ over all $n$ -tuples have dense linear span in $H^{\otimes n}$ (Permutation sector projections on the n-fold tensor power, Hilbert tensor construction: inner product, completion and product bases).

[F3]

The diagonal lemma: for a complete orthonormal family $(f_{n})_{n \in N}$ , with $N = {0, 1, 2, \dots}$ , of a Hilbert space and real numbers $μ_{n}$ , the operator with domain ${ψ : \sum_{n} μ_{n}^{2} ∣ ⟨ ψ, f_{n} ⟩ ∣^{2} < \infty}$ and $ψ \mapsto \sum_{n} μ_{n} ⟨ ψ, f_{n} ⟩ f_{n}$ is self-adjoint (Diagonal adjoint and graph-core lemma).

[F4]

Once step 2.1 establishes that $(u_{I})_{I \in M}$ is a complete orthonormal family, its Fourier coefficient map has squared coefficient sum $\sum_{I} ∣ ⟨ ψ, u_{I} ⟩ ∣^{2} = ∥ ψ ∥^{2}$ (A Hilbert space with a given orthonormal basis is $ℓ^{2}$ of the index set). Bounded finite-rank operators are compact, norm limits of compact operators into a Hilbert space are compact, and a compact operator is trace class exactly when it has a nuclear representation with summable rank-one norms (Bounded finite rank operators are compact, Norm limit of compact operators is compact, Nuclear series characterizes trace norm).

[F5]

Geometric series and nonnegative double series: $\sum_{k \geq 0} q^{k} = 1 / (1 - q)$ for $0 \leq q < 1$ ; the order of summation of a nonnegative double series may be interchanged (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, Tonelli's theorem for double series of nonnegative extended real numbers).

[F6]

For $x \geq 1$ , $\log x = \int_{1}^{x} d t / t$ and $0 \leq \log x \leq x - 1$ , $\log x \geq 1 - 1 / x$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , The natural logarithm as the inverse of the exponential function); consequently $q \leq - \log (1 - q) \leq q / (1 - q)$ for $0 < q < 1$ (substitute $x = 1 / (1 - q) \geq 1$ ).

[F7]

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

[F8]

Every nondecreasing real sequence bounded above converges to the supremum of its range (A nondecreasing sequence bounded above converges to the supremum of its range, and a nonincreasing sequence bounded below to the infimum).

[F9]

Once step 2.1 supplies the Hilbert basis $(u_{I})$ , the basis sum for any trace-class $T$ is absolutely convergent and equals $tr T$ (Trace is absolutely convergent and basis independent).

Proof

1.1F1F2algebra

[Orthonormality] Let $I = (j_{1} \leq \dots \leq j_{n})$ , $I^{'} = (j_{1}^{'} \leq \dots \leq j_{m}^{'})$ and first suppose $n = m \geq 1$ . Using $P_{+}^{(n)} = P_{+}^{(n) *}$ and $P_{+}^{(n)} χ = χ$ for $χ \in ran P_{+}^{(n)}$ , [F2] gives $⟨ v_{I}, v_{I^{'}} ⟩ = \frac{1}{n!} \sum_{π \in S_{n}} ⟨ e_{j_{1}} \otimes \dots \otimes e_{j_{n}}, U_{π} (e_{j_{1}^{'}} \otimes \dots \otimes e_{j_{n}^{'}}) ⟩$ . Each summand is $1$ exactly when $π$ maps the tuple $(j_{1}^{'}, \dots, j_{n}^{'})$ to $(j_{1}, \dots, j_{n})$ and $0$ otherwise, by orthonormality of the $e_{j}$ ; since both tuples are nondecreasing, such $π$ exists only if $I = I^{'}$ , and then the number of such $π$ is the product of the factorials of the multiplicities of $I$ . Hence $⟨ v_{I}, v_{I^{'}} ⟩ = 0$ for $I \neq I^{'}$ and $∥ v_{I} ∥^{2} \geq 1 / n! > 0$ , so $(u_{I})$ is an orthonormal family; tensors from different $n$ are orthogonal by [F1].

1.2F5algebra

[Factoring the finite sums] Fix $N \geq 1$ . The map $I \mapsto (n_{1}, \dots, n_{N})$ assigning to a multiset $I \subseteq {1, \dots, N}$ the number $n_{j}$ of occurrences of $j$ is a bijection onto $N^{N}$ , and $λ_{I} = \sum_{j} n_{j} ε_{j}$ ; by [F5] and the geometric series with $q = e^{- β ε_{j}} \in (0, 1)$ , $\sum_{I \subseteq {1, \dots, N}} e^{- β λ_{I}} = \sum_{(n_{1}, \dots, n_{N}) \in N^{N}} \prod_{j = 1}^{N} e^{- β n_{j} ε_{j}} = \prod_{j = 1}^{N} \sum_{n \geq 0} e^{- β n ε_{j}} = \prod_{j = 1}^{N} \frac{1}{1 - e^{- β ε_{j}}}$ , the interchange of the finite product with the sums following by induction from [F5] (Tonelli for nonnegative series).

2.1F1F2step 1.1algebra

[Completeness] The ordered elementary tensors $e_{i_{1}} \otimes \dots \otimes e_{i_{n}}$ have dense span in $H^{\otimes n}$ ([F2]); applying the bounded projection $P_{+}^{(n)}$ gives a dense subset of $ran P_{+}^{(n)}$ , and the vectors $P_{+}^{(n)} (e_{i_{1}} \otimes \dots \otimes e_{i_{n}})$ over ordered tuples coincide with those over nondecreasing tuples because $P_{+}^{(n)} U_{π} = P_{+}^{(n)}$ ([F2]). Hence the span of ${u_{I} : ∣ I ∣ = n}$ is dense in $ran P_{+}^{(n)}$ ; combining the sectors and using that every $ψ \in F (H)$ is the limit of its truncations ([F1]) shows that the span of all $u_{I}$ is dense in $F (H)$ . With step 1.1 this proves (a).

2.2F5F6F7F8step 1.2algebra

[The infinite product and the criterion] The finite sums of step 1.2 increase with $N$ because all terms are positive, and every finite multiset $I$ lies in an initial segment, so $\sum_{I \in M} e^{- β λ_{I}} = \lim_{N} \sum_{I \subseteq {1, \dots, N}} e^{- β λ_{I}} = \lim_{N} \prod_{j \leq N} (1 - e^{- β ε_{j}})^{- 1} \in (0, \infty]$ . Put $q_{j} = e^{- β ε_{j}} \in (0, 1)$ . For every $j$ , [F6] gives $q_{j} \leq - \log (1 - q_{j}) \leq q_{j} / (1 - q_{j})$ . If infinitely many $q_{j} > 1 / 2$ , both $\sum_{j} q_{j}$ and $\sum_{j} - \log (1 - q_{j})$ diverge, the latter already bounded below by the former termwise. Otherwise only finitely many such terms occur, all finite, and on the remaining tail $q_{j} \leq - \log (1 - q_{j}) \leq 2 q_{j}$ ; therefore the logarithmic sum is finite exactly when $\sum_{j} q_{j}$ is finite. By the product and reciprocal laws in [F7], $\log \prod_{j \leq N} (1 - q_{j})^{- 1} = \sum_{j \leq N} - \log (1 - q_{j})$ . If these logarithmic partial sums are bounded, the increasing reciprocal products are bounded as well, since $\log$ is increasing and $\exp$ is its increasing inverse; hence [F8] gives a finite limit. Conversely, if the reciprocal products have a finite limit, they are bounded and [F7] makes their logarithmic partial sums bounded. In the finite case, if $P_{N}$ denotes the reciprocal partial product and $P_{N} \to P$ , continuity of $\log$ gives $\log P = \sum_{j} - \log (1 - q_{j})$ , so $P = \exp (\sum_{j} - \log (1 - q_{j}))$ . This proves (c) and (d) in both large-factor branches without assuming summability in advance.

3.1F3F4F9step 1.1step 2.1algebra

[The diagonal operator and the $M$ -indexed trace criterion] The set $M$ is countably infinite: group its finite nondecreasing sequences by the integer $n + \sum_{r = 1}^{n} j_{r}$ , list each finite group lexicographically (its sequences have length and entries bounded by the group integer), skip empty groups, and concatenate the groups (the singleton sequences ensure infinitely many entries). Fix the resulting enumeration $I_{1}, I_{2}, \dots$ and put $μ_{m} : = e^{- β λ_{I_{m}}}$ . Put $μ_{I} : = e^{- β λ_{I}}$ for every $I$ . By step 2.1, $(f_{n})_{n \geq 0} : = (u_{I_{n + 1}})_{n \geq 0}$ is a complete orthonormal family, and [F3] applies with coefficients $μ_{I_{n + 1}}$ . Reindexing its sums by this bijection shows that the diagonal operator $D_{β}$ is self-adjoint; Parseval from [F4] gives $∥ D_{β} ψ ∥^{2} = \sum_{I} μ_{I}^{2} ∣ ⟨ ψ, u_{I} ⟩ ∣^{2} \leq ∥ ψ ∥^{2}$ , so its domain is all of $F (H)$ and it is bounded with norm at most one. Its coefficients are positive, so $⟨ D_{β} ψ, ψ ⟩ = \sum_{I} μ_{I} ∣ ⟨ ψ, u_{I} ⟩ ∣^{2} \geq 0$ ; hence it is positive. If $\sum_{I} μ_{I} < \infty$ , define $R_{N} ψ : = \sum_{m = 1}^{N} μ_{m} ⟨ ψ, u_{I_{m}} ⟩ u_{I_{m}}$ . Each $R_{N}$ is finite rank and compact by [F4], while Parseval gives $∥ D_{β} - R_{N} ∥ \leq \sup_{m > N} μ_{m} \leq \sum_{m > N} μ_{m} \to 0$ . Thus $D_{β}$ is compact by [F4]. Its coefficient expansion is the nuclear representation $D_{β} ψ = \sum_{m \geq 1} ⟨ ψ, u_{I_{m}} ⟩ (μ_{m} u_{I_{m}})$ , whose nuclear sum is $\sum_{m} μ_{m} < \infty$ ; [F4] therefore proves that $D_{β}$ is trace class. Conversely, if $D_{β}$ is trace class, [F9] applied to the supplied basis $(u_{I})$ gives $tr D_{β} = \sum_{I} ⟨ D_{β} u_{I}, u_{I} ⟩ = \sum_{I} μ_{I} < \infty$ . Hence $D_{β}$ is trace class exactly when the $M$ -indexed sum is finite, and then its trace is that sum.

4.1step 3.1step 2.2algebra∎

[Conclusion] Combining step 3.1 with step 2.2 gives (e): $D_{β}$ is trace class if and only if $\sum_{j} e^{- β ε_{j}} < \infty$ , and then $tr D_{β} = \sum_{I} e^{- β λ_{I}} = \prod_{j \geq 1} (1 - e^{- β ε_{j}})^{- 1}$ . Parts (a)-(e) are proved, and no thermal or measurement interpretation is claimed.

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Fermion and dilute grand-canonical formulas

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $β > 0$ and let $(ε_{j})_{j \geq 1} \subseteq (0, \infty)$ and $μ \in R$ be real data; write $q_{j} : = e^{- β (ε_{j} - μ)}$ .

(a) Single fermionic orbital. On $h : = C^{2}$ with orthonormal basis $(u_{0}, u_{1})$ define the bounded self-adjoint operator $N_{-}$ by $N_{-} u_{0} = 0$ , $N_{-} u_{1} = u_{1}$ (A bounded linear operator between normed spaces). Then the diagonal operator $F : = \sum_{k = 0}^{1} e^{- β (ε_{1} - μ) k} T_{u_{k}, u_{k}}$ is positive, self-adjoint, finite rank, and $tr F = 1 + e^{- β (ε_{1} - μ)}, \frac{\sum_{k = 0}^{1} k e^{- β (ε_{1} - μ) k}}{tr F} = \frac{1}{e^{β (ε_{1} - μ)} + 1}$ (Trace class operator, Trace of a trace class operator, Trace is absolutely convergent and basis independent).

(b) Countable grand-canonical factor products. Define $N : = {N \subseteq {1, 2, \dots} : N finite}$ and set $Z_{B} : = \sup_{N \in N} \prod_{j \in N} \frac{1}{1 - q_{j}} \in (0, \infty], Z_{F} : = \sup_{N \in N} \prod_{j \in N} (1 + q_{j}) \in [1, \infty],$ with the convention $Z_{B} : = + \infty$ if some $j$ has $ε_{j} - μ \leq 0$ . Then: $Z_{B} < \infty ⟺ \sum_{j} e^{- β ε_{j}} < \infty and μ < \inf_{j} ε_{j}, Z_{F} < \infty ⟺ \sum_{j} e^{- β ε_{j}} < \infty,$ and in these finite cases $Z_{B} = \prod_{j \geq 1} (1 - q_{j})^{- 1}$ and $Z_{F} = \prod_{j \geq 1} (1 + q_{j})$ .

(c) Mode occupancies and dilute remainders. For a single mode with $0 < q < 1$ (boson) and $q > 0$ (fermion), $\frac{\sum_{n \geq 0} n q^{n}}{\sum_{n \geq 0} q^{n}} = \frac{q}{1 - q} = q + \frac{q^{2}}{1 - q}, \frac{\sum_{n = 0}^{1} n q^{n}}{\sum_{n = 0}^{1} q^{n}} = \frac{q}{1 + q} = q - \frac{q^{2}}{1 + q};$ consequently the deviations from $q$ are bounded by $q^{2} / (1 - q)$ and $q^{2} / (1 + q)$ respectively, and both are at most $2 q^{2}$ when $q \leq 1 / 2$ .

All statements are mathematical statements about diagonal weights and products; no identification of $β, μ, ε_{j}$ with physical temperatures, chemical potentials or energies, and no thermal or thermodynamic conclusion, is asserted.

Facts & Assumptions

Given: the real data $β > 0$ , $(ε_{j})_{j \geq 1} \subseteq (0, \infty)$ , $μ \in R$ , the numbers $q_{j} = e^{- β (ε_{j} - μ)}$ , the two-dimensional Hilbert space $h = C^{2}$ with orthonormal basis $(u_{0}, u_{1})$ , the finite-rank operators $T_{u_{k}, u_{k}}$ , and $0 < q < 1$ .

[F1]

For $0 \leq q < 1$ , the geometric series gives $\sum_{n \geq 0} q^{n} = (1 - q)^{- 1}$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges). Tonelli for nonnegative double series (Tonelli's theorem for double series of nonnegative extended real numbers) gives $\sum_{n \geq 1} n q^{n} = \sum_{m \geq 1} \sum_{n \geq m} q^{n} = (1 - q)^{- 1} \sum_{m \geq 1} q^{m} = q / (1 - q)^{2}$ . This scalar derivation uses no Axiom-of-Choice operator supplier.

[F2]

A finite-rank operator is trace class; the trace is independent of the chosen Hilbert basis and equals the sum of the diagonal entries in any supplied Hilbert basis (Trace class operator, Trace of a trace class operator, Trace is absolutely convergent and basis independent).

[F3]

For $x \geq 1$ , $\log x = \int_{1}^{x} d t / t$ and $1 - 1 / x \leq \log x \leq x - 1$ ; hence for $0 \leq q \leq 1 / 2$ , $\log (1 + q) \leq q$ and $\log (1 + q) \geq q - q^{2}$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , The natural logarithm as the inverse of the exponential function).

[F4]

The scalar bosonic logarithmic comparison is $q \leq - \log (1 - q) \leq q / (1 - q)$ for $0 < q < 1$ : apply the integral bounds $1 - 1 / x \leq \log x \leq x - 1$ of [F3] at $x = (1 - q)^{- 1}$ . This uses only The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ and The natural logarithm as the inverse of the exponential function, not a Hilbert-space or spectral product theorem.

[F5]

The natural logarithm has the product, quotient and reciprocal laws and is continuous and strictly increasing on $(0, \infty)$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

[F6]

The complex exponential satisfies $\exp (z + w) = \exp (z) \exp (w)$ for all complex $z, w$ and extends the real exponential ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential).

[F7]

A nondecreasing real sequence bounded above converges to the supremum of its range (A nondecreasing sequence bounded above converges to the supremum of its range, and a nonincreasing sequence bounded below to the infimum).

Proof

1.1F1F2algebra

[Single orbital] The vector $u_{k}$ is an eigenvector of $\sum_{m = 0}^{1} m T_{u_{m}, u_{m}}$ with eigenvalue $k$ , so $e^{- β (ε_{1} - μ) N_{-}} = \sum_{k = 0}^{1} e^{- β (ε_{1} - μ) k} T_{u_{k}, u_{k}} = F$ on a complete orthonormal family, hence everywhere; $F$ is a positive diagonal finite-rank operator with eigenvalues $1$ and $e^{- β (ε_{1} - μ)} > 0$ . By [F2] its trace is $1 + e^{- β (ε_{1} - μ)}$ and the weighted average of the eigenvalues $0, 1$ is $\frac{0 \cdot 1 + 1 \cdot e^{- β (ε_{1} - μ)}}{1 + e^{- β (ε_{1} - μ)}} = \frac{1}{e^{β (ε_{1} - μ)} + 1}$ . This proves (a).

1.2F1algebra

[Dilute remainders] For $0 < q < 1$ , the bosonic signed deviation is $\frac{q}{1 - q} - q = \frac{q^{2}}{1 - q} > 0$ . For every $q > 0$ , the fermionic signed deviation is $\frac{q}{1 + q} - q = - \frac{q^{2}}{1 + q} < 0$ . Their absolute values are $q^{2} / (1 - q)$ and $q^{2} / (1 + q)$ on these respective domains; when $0 < q \leq 1 / 2$ , each is bounded by $2 q^{2}$ because $1 - q \geq 1 / 2$ and $1 + q \geq 1$ . The bosonic geometric sums are [F1], and the fermionic finite sums are $1 + q$ and $q$ . This proves (c).

1.3F3F5F6F7algebra

[Fermionic products] If infinitely many $j$ have $q_{j} > 1 / 2$ , the product over any $k$ of these indices is at least $(3 / 2)^{k}$ , so $Z_{F} = + \infty$ and $\sum_{j} q_{j} = + \infty$ . Otherwise their finitely many factors are finite positive constants, and [F3] gives $q_{j} / 2 \leq q_{j} - q_{j}^{2} \leq \log (1 + q_{j}) \leq q_{j}$ on the remaining tail. Thus the nonnegative logarithmic partial sums are bounded exactly when $\sum_{j} q_{j} < \infty$ . Since $\prod_{j \leq n} (1 + q_{j}) = \exp (\sum_{j \leq n} \log (1 + q_{j}))$ , the increasing partial products have a finite limit exactly in that case. Every finite subset is contained in an initial segment and every factor is at least one, so their supremum equals that limit. Finally $\sum_{j} q_{j} = e^{β μ} \sum_{j} e^{- β ε_{j}}$ , giving precisely the fermionic equivalence in (b).

1.4F3F4F5F6F7algebra

[Bosonic products] If some $ε_{j} \leq μ$ , then $q_{j} \geq 1$ and $Z_{B} = + \infty$ by the stated convention; also $μ < \inf_{j} ε_{j}$ fails. Suppose all $q_{j} < 1$ . By [F4], $q_{j} \leq - \log (1 - q_{j})$ for every index. If infinitely many $q_{j} > 1 / 2$ , both corresponding sums diverge; otherwise discard only that finite set and use $q_{j} \leq - \log (1 - q_{j}) \leq 2 q_{j}$ on its complement. Consequently $\sum_{j} - \log (1 - q_{j}) < \infty$ exactly when $\sum_{j} q_{j} < \infty$ , and the reciprocal partial products, whose logarithms are these partial sums, have a finite limit exactly then. Their limit equals the finite-subset supremum because the factors are at least one. If $\sum_{j} q_{j} < \infty$ , then $q_{j} \to 0$ , so $ε_{j} \to + \infty$ since $q_{j} = e^{- β (ε_{j} - μ)}$ and $β > 0$ . In particular, all but finitely many $ε_{j}$ exceed $ε_{1}$ ; hence the infimum is attained in a nonempty finite initial set. Since every $ε_{j} > μ$ , this gives $μ < \inf_{j} ε_{j}$ . Conversely, $μ < \inf_{j} ε_{j}$ and $\sum_{j} e^{- β ε_{j}} < \infty$ imply all $q_{j} < 1$ and $\sum_{j} q_{j} = e^{β μ} \sum_{j} e^{- β ε_{j}} < \infty$ , so the scalar criterion just proved gives finite $Z_{B}$ . This proves the full bosonic equivalence without assuming attainment in the hypotheses and without invoking an Axiom-of-Choice spectral theorem.

2.1step 1.1step 1.2step 1.3step 1.4algebra∎

[Conclusion] Steps 1.1, 1.2, 1.3 and 1.4 establish (a), (c), the fermionic half of (b) and the bosonic half of (b) respectively; in the finite cases the displayed products are the limits, and the mode occupancies in (c) are the normalized geometric sums of [F1]. The closing paragraph of the statement records that no physical identification of the parameters or of the products with thermodynamic quantities is asserted.

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

Finite Gibbs free-energy minimum

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a finite-dimensional complex Hilbert space of dimension $d \geq 1$ with inner product linear in the first argument (Hilbert space), let $H : H \to H$ be a self-adjoint linear operator with real eigenvalues $E_{1}, \dots, E_{d}$ (repeated according to multiplicity) and orthonormal eigenbasis $(v_{1}, \dots, v_{d})$ , and let $β > 0$ . Write $Z : = \sum_{j = 1}^{d} e^{- β E_{j}} \in (0, \infty), ρ_{β} : = \frac{1}{Z} \sum_{j = 1}^{d} e^{- β E_{j}} T_{v_{j}, v_{j}},$ where $T_{u, v} x = ⟨ x, v ⟩ u$ (Density operators and scalar spectral probability). Let $D$ be the set of density operators on $H$ , i.e. positive self-adjoint trace-class operators $ρ$ with $tr ρ = 1$ (Trace class operator, Trace of a trace class operator), and for $ρ \in D$ with eigenvalues $w_{1}, \dots, w_{d} \geq 0$ (counted with multiplicity and in any order) define the entropy $s (ρ) : = - \sum_{i : w_{i} > 0} w_{i} \log w_{i} \in [0, \log d]$ (The natural logarithm as the inverse of the exponential function). Then:

(a) $ρ_{β} \in D$ , $tr (ρ_{β} H) = \frac{1}{Z} \sum_{j} e^{- β E_{j}} E_{j}$ , and $tr (ρ_{β} H) - β^{- 1} s (ρ_{β}) = - β^{- 1} \log Z;$

(b) for every $ρ \in D$ , $tr (ρ H) - β^{- 1} s (ρ) \geq - β^{- 1} \log Z;$

(c) for every $ρ \in D$ , $s (ρ) \leq \log d$ , with $s (ρ) = \log d$ if and only if $ρ = d^{- 1} i d$ ; when $H = 0$ one has $Z = d$ , $ρ_{β} = d^{- 1} i d$ and the bound in (b) is attained there.

The combination $tr (ρ H) - β^{- 1} s (ρ)$ is a purely mathematical functional of $(ρ, H, β)$ ; identifying $β$ with an inverse physical temperature and $ρ_{β}$ with a thermal equilibrium state are additional physical assumptions not asserted here.

Facts & Assumptions

Given: the finite-dimensional complex Hilbert space $H$ of dimension $d \geq 1$ , the self-adjoint operator $H$ with eigenvalues $E_{j}$ and orthonormal eigenbasis $(v_{j})$ , the number $β > 0$ , the partition sum $Z$ , the Gibbs weight operator $ρ_{β}$ , and an arbitrary density operator $ρ \in D$ with eigenvalues $w_{i}$ and orthonormal eigenbasis $(u_{i})$ .

[F1]

Finite-dimensional spectral theorem: every self-adjoint (indeed normal) endomorphism of $H$ has an orthonormal basis of eigenvectors; the eigenvalues are real for $H$ , and $ρ$ has an orthonormal eigenbasis with $w_{i} \geq 0$ , $\sum_{i} w_{i} = tr ρ = 1$ (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, Density operators and scalar spectral probability).

[F2]

On a finite-dimensional space every operator is trace class, the trace is basis-independent, and for a diagonal operator the trace is the sum of its eigenvalues (Trace class operator, Trace of a trace class operator, Trace is absolutely convergent and basis independent).

[F3]

For $x > 0$ , $\log^{'} (x) = 1 / x$ and $\log x = \int_{1}^{x} d t / t$ ; for $x \geq 1$ , $1 - 1 / x \leq \log x \leq x - 1$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, The natural logarithm as the inverse of the exponential function, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ). The inequality $\log x \leq x - 1$ for every $x > 0$ , with equality only at $x = 1$ , is derived in step 1.3.

[F4]

For positive $x, y$ , $\log (x y) = \log x + \log y$ , $\log (x / y) = \log x - \log y$ and $\log (1 / x) = - \log x$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

Proof

1.1F1F2F4algebra

[The Gibbs state and its entropy] The weights $q_{j} : = e^{- β E_{j}} / Z$ are positive with $\sum_{j} q_{j} = 1$ , so by [F1] and [F2] $ρ_{β} = \sum_{j} q_{j} T_{v_{j}, v_{j}}$ is a positive self-adjoint operator with orthonormal eigenbasis $(v_{j})$ , eigenvalues $q_{j}$ and trace $1$ ; hence $ρ_{β} \in D$ . Its entropy is $s (ρ_{β}) = - \sum_{j} q_{j} \log q_{j} = \sum_{j} q_{j} (β E_{j} + \log Z) = β tr (ρ_{β} H) + \log Z$ by [F2], where $tr (ρ_{β} H) = \sum_{j} q_{j} E_{j}$ . Rearranging gives $tr (ρ_{β} H) - β^{- 1} s (ρ_{β}) = - β^{- 1} \log Z$ , which is (a).

1.2F1algebra

[The doubly stochastic matrix] Let $p_{i j} : = ∣ ⟨ u_{i}, v_{j} ⟩ ∣^{2}$ . Since $(u_{i})$ and $(v_{j})$ are orthonormal bases, [F1] gives $p_{i j} \geq 0$ , $\sum_{j} p_{i j} = ∥ u_{i} ∥^{2} = 1$ and $\sum_{i} p_{i j} = ∥ v_{j} ∥^{2} = 1$ : the matrix $(p_{i j})$ is doubly stochastic. For $r_{i} : = \sum_{j} p_{i j} q_{j}$ one has $r_{i} > 0$ because all $q_{j} > 0$ and $\sum_{j} p_{i j} = 1$ , and $\sum_{i} r_{i} = \sum_{j} q_{j} \sum_{i} p_{i j} = 1$ .

1.3F1F3algebra

[The entropy bound and equality case] For $x > 1$ , the integral formula in [F3] gives $\log x = \int_{1}^{x} d t / t < x - 1$ , because $1 / t < 1$ for $t > 1$ ; for $0 < x < 1$ , put $y = 1 / x > 1$ . On $[1, y)$ one has $1 / t > 1 / y = x$ , so $\log y = \int_{1}^{y} d t / t > x (y - 1) = 1 - x$ , and the reciprocal law gives $\log x = - \log y < x - 1$ . Thus $\log x \leq x - 1$ for every $x > 0$ , with equality exactly at $x = 1$ . Let $S : = {i : w_{i} > 0}$ and $k : = ∣ S ∣$ , so $1 \leq k \leq d$ . Then Since $0 < w_{i} \leq 1$ , every term $- w_{i} \log w_{i} = w_{i} \log (1 / w_{i})$ is nonnegative. Also $s (ρ) - \log d = \sum_{i \in S} w_{i} \log \frac{1}{d w_{i}} \leq \sum_{i \in S} w_{i} (\frac{1}{d w_{i}} - 1) = \frac{k}{d} - 1 \leq 0.$ If $s (ρ) = \log d$ , then $k = d$ and equality holds in every positive-weight logarithm inequality, since each individual difference $w_{i} [(1 / (d w_{i}) - 1) - \log (1 / (d w_{i}))]$ is nonnegative and their finite sum is zero. The equality characterization above forces $1 / (d w_{i}) = 1$ for all $i$ , so $w_{i} = 1 / d$ and $ρ = \sum_{i} (1 / d) T_{u_{i}, u_{i}} = d^{- 1} i d$ . Conversely $ρ = d^{- 1} i d$ has entropy $\log d$ . When $H = 0$ all $E_{j} = 0$ , so $Z = d$ and $ρ_{β} = d^{- 1} i d$ , and the bound of (b) is attained there.

2.1F2F3F4step 1.1step 1.2step 1.3algebra

[Relative entropy is nonnegative] With $w_{i}$ the eigenvalues of $ρ$ in the orthonormal eigenbasis $(u_{i})$ of [F1], $\log ρ = \sum_{i : w_{i} > 0} (\log w_{i}) T_{u_{i}, u_{i}}$ and $\log ρ_{β} = \sum_{j} (\log q_{j}) T_{v_{j}, v_{j}}$ (finite-dimensional spectral calculus). For the product with $ρ$ , the symbol $\log ρ$ here denotes the logarithm on the positive spectral support extended by zero on $\ker ρ$ ; it does not assign a value to $\log 0$ . Thus by [F2] $tr (ρ \log ρ) = \sum_{i : w_{i} > 0} w_{i} \log w_{i}$ and $tr (ρ \log ρ_{β}) = \sum_{i : w_{i} > 0} w_{i} \sum_{j} p_{i j} \log q_{j}$ . For each $i$ with $w_{i} > 0$ , $r_{i} = \sum_{j} p_{i j} q_{j} > 0$ and $\sum_{j} p_{i j} = 1$ by step 1.2. Applying $\log x \leq x - 1$ from step 1.3 to $x = q_{j} / r_{i}$ and summing with weights $p_{i j}$ gives $\sum_{j} p_{i j} \log (q_{j} / r_{i}) \leq \sum_{j} p_{i j} (q_{j} / r_{i} - 1) = 0$ , hence $\sum_{j} p_{i j} \log q_{j} \leq \log r_{i}$ . Therefore $tr (ρ \log ρ) - tr (ρ \log ρ_{β}) \geq \sum_{i : w_{i} > 0} w_{i} (\log w_{i} - \log r_{i}) = \sum_{i : w_{i} > 0} w_{i} \log (w_{i} / r_{i})$ . Writing $w_{i} \log (w_{i} / r_{i}) = - w_{i} \log (r_{i} / w_{i})$ and again applying step 1.3 to $x = r_{i} / w_{i} > 0$ gives $w_{i} \log (w_{i} / r_{i}) \geq w_{i} - r_{i}$ ; summing, $\sum_{i : w_{i} > 0} (w_{i} - r_{i}) = 1 - \sum_{i : w_{i} > 0} r_{i} = \sum_{i : w_{i} = 0} r_{i} \geq 0$ , since $\sum_{i} r_{i} = 1$ by step 1.2. Therefore $tr (ρ \log ρ) \geq tr (ρ \log ρ_{β})$ .

3.1F2F4step 1.1step 2.1algebra

[The lower bound] By the spectral description of $ρ_{β}$ in step 1.1, $tr (ρ \log ρ_{β}) = \sum_{j} (\log q_{j}) tr (ρ T_{v_{j}, v_{j}}) = \sum_{j} (- β E_{j} - \log Z) ⟨ ρ v_{j}, v_{j} ⟩ = - β tr (ρ H) - \log Z$ , since $\sum_{j} ⟨ ρ v_{j}, v_{j} ⟩ = tr ρ = 1$ and $tr (ρ H) = \sum_{j} E_{j} ⟨ ρ v_{j}, v_{j} ⟩$ (because $H = \sum_{j} E_{j} T_{v_{j}, v_{j}}$ ). Combining with step 2.1 gives $tr (ρ \log ρ) \geq - β tr (ρ H) - \log Z$ ; multiplying by $β^{- 1}$ and rearranging, using $s (ρ) = - tr (ρ \log ρ)$ , gives the bound (b).

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

[Conclusion and scope] Steps 1.1-3.1 prove (a), (b) and (c), including the asserted equality in (a) and the attainment in (c). The final paragraph of the statement records that the thermodynamic reading of the functional and of $ρ_{β}$ requires additional physical assumptions, which are not part of this mathematical statement.

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Entropy-divergence example

Example

Let $w_{n} : = {\begin{cases} 0, & n \in {0, 1}, \\ \frac{1}{C n (\log n)^{2}}, & n \geq 2, \end{cases} C : = \sum_{n \geq 2} \frac{1}{n (\log n)^{2}},$ where $\log$ is the natural logarithm. This example verifies concretely that:

(a) the normalizing constant is finite and positive, $C \in (0, \infty)$ , so that $\sum_{n \geq 0} w_{n} = 1$ (Series and absolute convergence in a normed space);

(b) the entropy of the weight family diverges, $s (w) : = - \sum_{n \geq 0} w_{n} \log w_{n} = \sum_{n \geq 0} w_{n} \log \frac{1}{w_{n}} = + \infty,$ where the sum is the nonnegative extended sum of the nonnegative terms $w_{n} \log (1 / w_{n})$ ; for a zero weight, both expressions $- w_{n} \log w_{n}$ and $w_{n} \log (1 / w_{n})$ are defined to have value $0$ . Thus the two zero weights at $n = 0, 1$ contribute $0$ rather than requiring a logarithm or reciprocal at zero;

(c) Consequently, a probability weight on a countable set can have infinite entropy, while every finite probability vector $p = (p_{1}, \dots, p_{d})$ with $p_{i} \geq 0$ and $\sum_{i} p_{i} = 1$ has entropy $s_{d} (p) : = - \sum_{i : p_{i} > 0} p_{i} \log p_{i} \leq \log d$ (zero entries contribute $0$ ). The finite bound is verified below.

Verification

Given: the weight family $(w_{n})_{n \in N}$ and the constant $C$ defined above.

[F1] Cauchy condensation: for a nonnegative nonincreasing family $(a_{k})_{k \geq 1}$ , $\sum_{k \geq 1} a_{k}$ converges if and only if $\sum_{j \geq 0} 2^{j} a_{2^{j}}$ converges (For a nonincreasing nonnegative sequence, $\sum a_{k}$ converges iff $\sum 2^{k} a_{2^{k}}$ converges).

[F2] $p$ -series: $\sum_{k \geq 1} k^{- p}$ converges if and only if $p > 1$ ; in particular $\sum_{k \geq 1} k^{- 2} < \infty$ and $\sum_{k \geq 1} k^{- 1} = \infty$ (The p-series for a real exponent p converges exactly when p is greater than one).

[F3] For $x \geq 1$ , $\log x = \int_{1}^{x} d t / t$ with $0 \leq \log x \leq x - 1$ and $1 - 1 / x \leq \log x$ (The natural logarithm as the inverse of the exponential function, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[F4] The natural logarithm is continuous, strictly increasing and onto $R$ , satisfies $\log 1 = 0$ , the product and reciprocal laws, and is inverse to the real exponential (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The natural logarithm as the inverse of the exponential function).

1.1F1F2F3algebra

[Finiteness of $C$ ] The family $a_{k} : = 1 / (k (\log k)^{2})$ , $k \geq 2$ , is nonnegative and nonincreasing for $k \geq 3$ (the denominator $k (\log k)^{2}$ is increasing by [F3] applied to $\log$ ), so condensation [F1] applied to the tail from $k \geq 2$ reduces $\sum_{k \geq 2} a_{k}$ to $\sum_{j \geq 1} 2^{j} a_{2^{j}} = \sum_{j \geq 1} \frac{2^{j}}{2^{j} (\log 2^{j})^{2}} = \frac{1}{(\log 2)^{2}} \sum_{j \geq 1} \frac{1}{j^{2}}$ , which converges by [F2]; the finitely many terms with $k = 2$ do not affect convergence. Since every term is positive, $C \in (0, \infty)$ and therefore $\sum_{n} w_{n} = C / C = 1$ .

1.2F1F2algebra

[Divergence of the comparison series] Again by [F1], $\sum_{k \geq 3} \frac{1}{k \log k}$ converges if and only if $\sum_{j \geq 2} \frac{2^{j}}{2^{j} \log (2^{j})} = \frac{1}{\log 2} \sum_{j \geq 2} \frac{1}{j}$ converges, which fails by [F2] with $p = 1$ ; hence $\sum_{k \geq 3} 1 / (k \log k) = + \infty$ .

1.3F3F4algebra

[The entropy lower bound] Put $R : = \max (0, - \frac{1}{2} \log C)$ . Since $\log$ is onto, choose $x_{0} > 0$ with $\log x_{0} = e^{R}$ ; for every integer $n \geq \max {2, x_{0}}$ , monotonicity gives $\log n \geq e^{R}$ and then $\log \log n \geq R$ . Thus $\log C + 2 \log \log n \geq 0$ . By the product and reciprocal laws [F4], $\log \frac{1}{w_{n}} = \log C + \log n + 2 \log \log n \geq \log n > 0$ for all such $n$ , and $w_{n} \log \frac{1}{w_{n}} \geq \frac{1}{C n (\log n)^{2}} \log n = \frac{1}{C n \log n}$ .

2.1step 1.1step 1.2step 1.3algebra

[Divergence of the entropy] Since all terms $w_{n} \log (1 / w_{n})$ are nonnegative, the extended sum $s (w) = \sum_{n \geq 0} w_{n} \log (1 / w_{n})$ dominates any finite partial sum; by step 1.3 it dominates $\frac{1}{C} \sum_{N \leq n \leq M} \frac{1}{n \log n}$ for all large $N \leq M$ , and step 1.2 shows that these partial sums are unbounded as $M \to \infty$ . Hence $s (w) = + \infty$ , proving (b) and, with step 1.1, the countable-set claims (a)-(b).

3.1F3F4step 1.1step 2.1algebra∎

[The finite-dimensional entropy bound] Let $S : = {i : p_{i} > 0}$ and $k : = ∣ S ∣$ , so $1 \leq k \leq d$ . For every $x \geq 1$ , [F3] gives $\log x \leq x - 1$ ; for $0 < x < 1$ , put $y = 1 / x > 1$ and use [F3] with [F4] to get $\log x = - \log y \leq - (1 - 1 / y) = x - 1$ . Therefore, using the logarithm laws [F4] and $\sum_{i \in S} p_{i} = 1$ , $s_{d} (p) - \log d = \sum_{i \in S} p_{i} \log \frac{1}{d p_{i}} \leq \sum_{i \in S} p_{i} (\frac{1}{d p_{i}} - 1) = \frac{k}{d} - 1 \leq 0.$ This proves the finite-dimensional bound in (c); together with steps 1.1-2.1 it establishes (a)-(c).

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Variational ground gap overlap

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a self-adjoint operator on a nonzero complex Hilbert space $H$ (Symmetric, self-adjoint and essentially self-adjoint operators, Hilbert space) which is bounded below, let $E_{0} : = \inf σ (H) \in R$ , let $E$ be the projection-valued measure of $H$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), put $P : = E ({E_{0}})$ , and assume that $g > 0$ is a spectral gap: $σ (H) \cap (E_{0}, E_{0} + g) = \emptyset .$ For $u \in D (∣ H ∣^{1 / 2})$ write $E_{u} (B) : = ⟨ E (B) u, u ⟩$ for Borel sets $B \subseteq R$ (Scalar and complex measures from a pvm) and put $q_{H} [u] : = \int_{R} λ d E_{u} (λ)$ ; by Density operators and scalar spectral probability (c) this integral exists for $u \in D (∣ H ∣^{1 / 2})$ , and for $u \in D (H)$ it equals $⟨ H u, u ⟩$ . Then for every $u \in D (∣ H ∣^{1 / 2})$ with $∥ u ∥ = 1$ :

(a) $q_{H} [u] - E_{0} = \int_{R} (λ - E_{0}) d E_{u} (λ) \geq g ∥ (I - P) u ∥^{2}$ ;

(b) if $q_{H} [u] \leq E_{0} + ε$ for some $ε \geq 0$ , then $∥ (I - P) u ∥^{2} \leq ε / g$ ; in particular, if $ε < g$ then $P u \neq 0$ .

No claim is made about the existence of a ground eigenvector, about $E_{0}$ being an eigenvalue, or about any physical interpretation of the trial vector.

Facts & Assumptions

Given: the self-adjoint operator $H$ bounded below with bottom $E_{0} = \inf σ (H)$ , its projection-valued measure $E$ , the ground projection $P = E ({E_{0}})$ , a gap $g > 0$ , and a vector $u \in D (∣ H ∣^{1 / 2})$ with $∥ u ∥ = 1$ .

[F1]

For a self-adjoint $H$ and unit $u \in D (∣ H ∣^{1 / 2})$ , the scalar measure $E_{u} (B) = ⟨ E (B) u, u ⟩$ is a probability measure and $\int ∣ λ ∣ d E_{u} < \infty$ , so its signed first moment $\int λ d E_{u}$ is finite; if $u \in D (H)$ , that moment equals $⟨ H u, u ⟩$ (Density operators and scalar spectral probability (c), Scalar and complex measures from a pvm, Spectral theorem for unbounded self-adjoint operators (PVM form)). No square-root expectation of $H$ is asserted when $H$ may have negative spectrum.

[F2]

$E ({E_{0}}) = P$ is an orthogonal projection; for every Borel set $B$ , $E_{u} (B) = ∥ E (B) u ∥^{2}$ , and $E (B) = 0$ implies $E_{u} (B) = 0$ (Projection valued measure).

[F3]

$E_{0} = \inf σ (H)$ means $σ (H) \subseteq [E_{0}, \infty)$ , so $E ((- \infty, E_{0})) = 0$ ; the gap hypothesis means $E ((E_{0}, E_{0} + g)) = 0$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, Symmetric, self-adjoint and essentially self-adjoint operators).

Proof

1.1F1F2F3algebra

[Support of the scalar measure] By [F2] and [F3], $E_{u} ((- \infty, E_{0})) = ∥ E ((- \infty, E_{0})) u ∥^{2} = 0$ and $E_{u} ((E_{0}, E_{0} + g)) = ∥ E ((E_{0}, E_{0} + g)) u ∥^{2} = 0$ ; hence $E_{u}$ is concentrated on ${E_{0}} \cup [E_{0} + g, \infty)$ and, since $E_{u}$ is a probability measure ([F1]), $E_{u} (R) = ∥ u ∥^{2} = 1$ , $E_{u} ({E_{0}}) = ∥ P u ∥^{2}$ and $E_{u} ([E_{0} + g, \infty)) = ∥ (I - P) u ∥^{2}$ .

2.1F1step 1.1algebra

[The first-moment inequality] Write $(λ - E_{0})^{+} : = (λ - E_{0}) 1_{[E_{0}, \infty)} (λ)$ , a nonnegative Borel function equal to $λ - E_{0}$ on the support of $E_{u}$ by step 1.1. Since $\int ∣ λ ∣ d E_{u} < \infty$ by [F1] and $E_{u} (R) = 1$ , $\int (λ - E_{0}) d E_{u} = \int λ d E_{u} - E_{0} = q_{H} [u] - E_{0}$ . On the support, $(λ - E_{0})^{+} \geq g 1_{[E_{0} + g, \infty)} (λ)$ , so $q_{H} [u] - E_{0} \geq g E_{u} ([E_{0} + g, \infty)) = g ∥ (I - P) u ∥^{2}$ , which is (a); if in addition $q_{H} [u] \leq E_{0} + ε$ , then $g ∥ (I - P) u ∥^{2} \leq ε$ .

3.1step 2.1algebra∎

[Failure of the ground component] If $ε < g$ , then $∥ (I - P) u ∥^{2} \leq ε / g < 1 = ∥ u ∥^{2}$ , so $∥ P u ∥^{2} = ∥ u ∥^{2} - ∥ (I - P) u ∥^{2} > 0$ and $P u \neq 0$ ; this completes the proof of (b) and of the statement.

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

Bounded fixed-N operator

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $N \geq 1$ , let $m_{1}, \dots, m_{N} > 0$ and $ℏ > 0$ , write points of $R^{3 N}$ as $X = (x_{1}, \dots, x_{N})$ with $x_{i} \in R^{3}$ , and let $F_{2}$ be the Plancherel transform of Plancherel theorem with the convention $\hat{f} (ξ) = \int f (X) e^{- 2 π i X \cdot ξ} d X$ on $L^{1}$ (Fourier transform on complex L1 classes). Define the quadratic weight and the free kinetic operator $κ (ξ_{1}, \dots, ξ_{N}) : = \sum_{i = 1}^{N} \frac{ℏ^{2}}{2 m_{i}} ∣ ξ_{i} ∣^{2}, K : = F_{2}^{*} M_{κ} F_{2}, D (K) : = {u \in L^{2} (R^{3 N}; C) : κ F_{2} u \in L^{2}}$ (Real multiplication operators and free plus bounded-potential Schrödinger operators). Let $V : R^{3 N} \to R$ be measurable and bounded, with $∥ V ∥_{\infty} : = ess sup ∣ V ∣ < \infty$ , and let $M_{V}$ be multiplication by $V$ (Integer-order Sobolev spaces and their norms, Hilbert space). Then:

(a) $K$ is a self-adjoint positive operator with domain $D (K)$ , and $D (K)$ is the Sobolev space $W^{2, 2} (R^{3 N}; C)$ as a set, with $∥ \cdot ∥_{W^{2, 2}}$ equivalent to the graph norm $u \mapsto (∥ u ∥^{2} + ∥ K u ∥^{2})^{1 / 2}$ (Integer-order W^{k,2} and H^k agree with equivalent norms, Adjoint of a densely defined operator);

(b) $H : = K + M_{V}$ with $D (H) : = D (K)$ is self-adjoint and bounded below by $- ∥ V ∥_{\infty}$ ; the space $C_{c}^{\infty} (R^{3 N}; C)$ is a core for both $K$ and $H$ (Symmetric, self-adjoint and essentially self-adjoint operators, Compactly supported smooth functions are dense in W^{k,p}(R^n), Relative boundedness with respect to an operator);

(c) for every $r \geq 1$ the operator $K \otimes I_{r} + M_{V} \otimes I_{r}$ on $L^{2} (R^{3 N}; C) \otimes C^{r}$ , with domain $D (K) \otimes C^{r}$ , is self-adjoint and bounded below by $- ∥ V ∥_{\infty}$ .

In particular, if $V_{i} : R^{3} \to R$ and $W_{i j} : R^{3} \to R$ ( $1 \leq i < j \leq N$ ) are measurable and bounded and $V (X) : = \sum_{i} V_{i} (x_{i}) + \sum_{i < j} W_{i j} (x_{i} - x_{j})$ , then $∥ V ∥_{\infty} \leq \sum_{i} ∥ V_{i} ∥_{\infty} + \sum_{i < j} ∥ W_{i j} ∥_{\infty} < \infty$ and (a)-(c) apply. No Hunziker-van Winter-Zhislin threshold statement, no stability of matter, no thermodynamic-limit and no physical interpretation is asserted.

Facts & Assumptions

Given: the integers $N, r \geq 1$ , masses $m_{i} > 0$ , the constant $ℏ > 0$ , the weight $κ$ and free operator $K$ , a real bounded measurable $V$ , and the perturbation $M_{V}$ .

[F1]

For a real measurable finite-valued function $a$ on $R^{d}$ , the multiplier $M_{a} u = a u$ on $D (M_{a}) = {u : a u \in L^{2}}$ is self-adjoint; the Plancherel transform is unitary with $∥ F_{2} u ∥ = ∥ u ∥$ (Real multiplication operators and free plus bounded-potential Schrödinger operators (a), Plancherel theorem, Fourier transform on complex L1 classes).

[F2]

Unitary equivalence preserves self-adjointness: if $U$ is unitary and $A$ is self-adjoint with domain $D (A)$ , then $U^{*} A U$ with domain $U^{- 1} D (A)$ is self-adjoint, and $K \geq 0$ is equivalent to $⟨ K u, u ⟩ = ∥ M_{κ}^{1 / 2} F_{2} u ∥^{2} \geq 0$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators, Hilbert space).

[F3]

Fourier characterization of Sobolev spaces: $W^{2, 2} (R^{d}; C) = {u : (1 + ∣ ξ ∣^{2}) F_{2} u \in L^{2}}$ with equivalent norm, and $C_{c}^{\infty} (R^{d}; C)$ is dense in $W^{k, p}$ (Integer-order W^{k,2} and H^k agree with equivalent norms, Integer-order Sobolev spaces and their norms, Compactly supported smooth functions are dense in W^{k,p}(R^n)).

[F4]

Kato-Rellich: if $A$ is self-adjoint and $B$ is symmetric with $D (A) \subseteq D (B)$ and $∥ B u ∥ \leq a ∥ A u ∥ + b ∥ u ∥$ on $D (A)$ for some $a < 1$ , then $A + B$ on $D (A)$ is self-adjoint, and if moreover $A \geq γ$ then $A + B \geq γ - \max {a ∣ γ ∣ + b, b / (1 - a)}$ in the quadratic-form sense (Kato-Rellich theorem, Relative boundedness with respect to an operator).

[F5]

If $S \subseteq T$ are operators on the same Hilbert space with $S$ symmetric and $D (S)$ a core for the self-adjoint $T$ , then $S$ is essentially self-adjoint with $\overline{S} = T$ (Symmetric, self-adjoint and essentially self-adjoint operators, Densely defined, closed and closable operators, and cores).

Proof

1.1F1F2algebra

[Self-adjointness of $K$ ] The function $κ$ is real, measurable and finite-valued, so $M_{κ}$ is self-adjoint on $D (M_{κ}) = {v : κ v \in L^{2}}$ by [F1]; conjugating by the unitary $F_{2}$ ([F1]) gives that $K = F_{2}^{*} M_{κ} F_{2}$ is self-adjoint with $D (K) = F_{2}^{- 1} D (M_{κ}) = {u : κ F_{2} u \in L^{2}}$ ([F2]). For $u \in D (K)$ , $⟨ K u, u ⟩ = ⟨ M_{κ} F_{2} u, F_{2} u ⟩ = \int κ ∣ F_{2} u ∣^{2} \geq 0$ , so $K \geq 0$ .

1.2F4algebra

[Finitely many components] Let $K_{1}, \dots, K_{r}$ be Hilbert spaces and let $T_{a}$ be self-adjoint on $K_{a}$ for $a = 1, \dots, r$ . On the direct sum $⨁_{a} K_{a}$ with domain ${(u_{a}) : u_{a} \in D (T_{a})}$ define $T ((u_{a})) : = (T_{a} u_{a})$ . Then $T$ is symmetric (componentwise), and if $(u^{(k)})$ is in the domain with $u^{(k)} \to u$ and $T u^{(k)} \to η$ , then componentwise $u_{a}^{(k)} \to u_{a}$ and $T_{a} u_{a}^{(k)} \to η_{a}$ ; closedness of $T_{a}$ gives $u_{a} \in D (T_{a})$ , $T_{a} u_{a} = η_{a}$ and $\sum_{a} ∥ T_{a} u_{a} ∥^{2} = ∥ η ∥^{2} < \infty$ , so $T$ is closed. Testing the adjoint componentwise shows $D (T^{*}) = D (T)$ and $T^{*} = T$ ; a symmetric densely defined operator whose adjoint has the same domain is self-adjoint. If all $T_{a} = T_{0}$ and $K_{a} = K_{0}$ , then $T \geq - ∥ V ∥_{\infty}$ whenever $T_{0} \geq - ∥ V ∥_{\infty}$ , because the quadratic form is the sum of the components.

2.1F3F5step 1.1algebra

[Identification with $W^{2, 2}$ and the core] For $ξ \in R^{3 N}$ one has $c_{\min} (1 + ∣ ξ ∣^{2}) \leq 1 + κ (ξ) \leq c_{\max} (1 + ∣ ξ ∣^{2})$ for suitable positive constants $c_{\min}, c_{\max}$ depending only on $N, m_{i}, ℏ$ ; hence $κ F_{2} u \in L^{2}$ if and only if $(1 + ∣ ξ ∣^{2}) F_{2} u \in L^{2}$ , i.e. $u \in W^{2, 2} (R^{3 N}; C)$ , with equivalent norms by [F3]. Since $C_{c}^{\infty}$ is dense in $W^{2, 2}$ ([F3]) and the graph norm of $K$ is equivalent to the $W^{2, 2}$ norm, $C_{c}^{\infty}$ is a core for $K$ , and $K$ is the unique self-adjoint extension of $K ∣_{C_{c}^{\infty}}$ ([F5]).

3.1F4step 1.1step 2.1algebra

[Bounded perturbation] Since $∥ V ∥_{\infty} < \infty$ , the operator $M_{V}$ is bounded with $∥ M_{V} ∥ \leq ∥ V ∥_{\infty}$ ; it is symmetric as multiplication by a real function, and on $D (K)$ the bound $∥ M_{V} u ∥ \leq 0 \cdot ∥ K u ∥ + ∥ V ∥_{\infty} ∥ u ∥$ exhibits the admissible pair $(a, b) = (0, ∥ V ∥_{\infty})$ . By [F4] with $A = K$ and $γ = 0$ , $H = K + M_{V}$ is self-adjoint on $D (K)$ and bounded below by $- ∥ V ∥_{\infty}$ . Moreover $∥ H u ∥ \leq ∥ K u ∥ + ∥ V ∥_{\infty} ∥ u ∥$ and $∥ K u ∥ \leq ∥ H u ∥ + ∥ V ∥_{\infty} ∥ u ∥$ , so the graph norms of $H$ and $K$ are equivalent on $D (K)$ ; hence the core $C_{c}^{\infty}$ of step 2.1 is also a core for $H$ . This proves (a) and (b).

4.1step 3.1step 1.2algebra

[The scalar case with $r$ components] The Hilbert space $L^{2} (R^{3 N}; C) \otimes C^{r}$ is identified with $⨁_{a = 1}^{r} L^{2} (R^{3 N}; C)$ componentwise; under this identification $K \otimes I_{r} + M_{V} \otimes I_{r}$ is the direct sum of $r$ copies of $H$ with domain $D (K)^{\oplus r} = D (K) \otimes C^{r}$ . By step 3.1 and step 1.2 it is self-adjoint and bounded below by $- ∥ V ∥_{\infty}$ , which is (c).

5.1step 1.1step 2.1step 3.1step 1.2step 4.1algebra∎

[Conclusion and scope] Steps 1.1-3.1 prove (a), (b) and (c); the displayed reduction for external and pair potentials only uses the triangle inequality for essential suprema, $∥ V ∥_{\infty} \leq \sum_{i} ∥ V_{i} ∥_{\infty} + \sum_{i < j} ∥ W_{i j} ∥_{\infty} < \infty$ . No HVZ threshold, stability, thermodynamic-limit or physical-interpretation claim is made.

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

Permutation-restricted self-adjointness

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a complex Hilbert space, let $H$ be a self-adjoint operator on $H$ (Symmetric, self-adjoint and essentially self-adjoint operators), let $N \geq 1$ , and let $(U_{π})_{π \in S_{N}}$ be unitary operators on $H$ with $U_{π} U_{σ} = U_{π σ}$ and $U_{π}^{*} = U_{π^{- 1}}$ for all $π, σ \in S_{N}$ (The Hilbert-space adjoint of a bounded operator). Define $P_{+} : = \frac{1}{N!} \sum_{π \in S_{N}} U_{π}, P_{-} : = \frac{1}{N!} \sum_{π \in S_{N}} sgn (π) U_{π} .$ Assume that for every $π \in S_{N}$ , $U_{π} D (H) \subseteq D (H) and U_{π} H ψ = H U_{π} ψ (ψ \in D (H)) .$ Then:

(a) $P_{+}$ and $P_{-}$ are orthogonal projections on $H$ ;

(b) for every $z \in ρ (H)$ one has $U_{π} (H - z)^{- 1} = (H - z)^{- 1} U_{π}$ and hence $P_{\pm} (H - z)^{- 1} = (H - z)^{- 1} P_{\pm}$ ;

(d) if $ran P_{\pm} \neq {0}$ , then the restriction $H_{\pm} : = H ∣_{D (H) \cap ran P_{\pm}}$ is a self-adjoint operator on the Hilbert space $ran P_{\pm}$ (Range criterion for self-adjointness);

(e) application: under the additional Axiom of Choice (The Axiom of Choice), let $N \geq 1$ , $m > 0$ , $ℏ > 0$ , let $r \geq 1$ , let $V_{e x t}, W : R^{3} \to R$ be measurable and bounded, and assume $W (- z) = W (z)$ for every $z \in R^{3}$ . Put $V (X) : = \sum_{i = 1}^{N} V_{e x t} (x_{i}) + \sum_{i < j} W (x_{i} - x_{j})$ on $R^{3 N}$ , let $K$ be the free kinetic operator of Bounded fixed-N operator with all masses $m_{i} = m$ , let $Θ : L^{2} (R^{3})^{\otimes N} \to L^{2} (R^{3 N})$ be the iterated Euclidean product unitary (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Hilbert tensor construction: inner product, completion and product bases), let $U_{π}$ be the permutation unitaries of Permutation sector projections on the n-fold tensor power and set $U_{π}^{p h y s} : = Θ U_{π} Θ^{- 1}, P_{\pm}^{p h y s} : = Θ P_{\pm}^{(N)} Θ^{- 1} .$ Then $H : = K + M_{V}$ satisfies the hypotheses above for the representation $U_{π}^{p h y s}$ , so $P_{\pm}^{p h y s}$ reduce $H$ and the restrictions of $H$ to the symmetric and antisymmetric sectors are self-adjoint on those sectors.

No spin-statistics statement, no identification of a physical species with a sector, and no claim about bound states or spectra is made.

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 complex Hilbert space $H$ , the self-adjoint operator $H$ , the unitary representation $(U_{π})$ of $S_{N}$ , its averages $P_{\pm}$ , numbers $z \in ρ (H)$ and vectors $ψ, φ \in H$ .

[F1]

By The finite symmetric group $S_{n}$ , one-line notation, and cycle notation, $S_{N}$ consists of the bijections of an $N$ -element set, so it has $N!$ elements by choosing successive images in $N, N - 1, \dots, 1$ ways. Its sign map is a homomorphism and is surjective for $N \geq 2$ (The sign is a homomorphism $S_{n} \to {+ 1, - 1}$ , surjective exactly when $n \geq 2$ ); choose $τ$ with $sgn (τ) = - 1$ . Left multiplication by $τ$ pairs the terms of $\sum_{π \in S_{N}} sgn (π)$ with their negatives, so this sum is zero. Group averaging now gives $P_{+}^{*} = P_{+}$ , $P_{+}^{2} = P_{+}$ and likewise $P_{-}^{*} = P_{-}$ , $P_{-}^{2} = P_{-}$ ; for $N = 1$ , both averages equal the identity. The Hilbert adjoint and bounded-operator conventions are The Hilbert-space adjoint of a bounded operator and A bounded linear operator between normed spaces.

[F2]

For a self-adjoint $H$ and $z \in ρ (H)$ the resolvent $R (z) : = (H - z)^{- 1}$ is a bounded bijection of $H$ onto $D (H)$ with inverse $H - z$ ; $D (H)$ is dense (Symmetric, self-adjoint and essentially self-adjoint operators, A bounded linear operator between normed spaces).

[F3]

Range criterion: a densely defined symmetric operator $S$ is self-adjoint if and only if it is closed and $ran (S - i) = ran (S + i) = H$ ; equivalently if it is closed, symmetric and both $S \pm i$ are surjective (Range criterion for self-adjointness).

[F4]

The Euclidean product unitary $Θ : H_{1}^{\otimes N} \to L^{2} (R^{3 N})$ satisfies $Θ (f_{1} \otimes \dots \otimes f_{N}) (x) = f_{1} (x_{1}) \dots f_{N} (x_{N})$ ; the permutation unitaries act by $U_{π} (f_{1} \otimes \dots \otimes f_{N}) = f_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes f_{{\hat{π}}^{- 1} (N)}$ . The equal-mass free kinetic operator is the maximal Fourier multiplier $K = F_{2}^{*} M_{κ} F_{2}$ , $κ (ξ) = \frac{ℏ^{2}}{2 m} \sum_{i} ∣ ξ_{i} ∣^{2}$ , on $D (K) = W^{2, 2} (R^{3 N}; C)$ . With the supplier's $e^{- 2 π i X \cdot ξ}$ convention, it equals $- \frac{ℏ^{2}}{8 π^{2} m} \sum_{i} Δ_{i}$ there. The multiplier $V$ as in (e) is invariant under permutations of the coordinates because $W$ is even (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Hilbert tensor construction: inner product, completion and product bases, Permutation sector projections on the n-fold tensor power, Bounded fixed-N operator).

Proof

1.1F1algebra

[The averages are projections] Since $π \mapsto U_{π}$ is a unitary representation and $U_{π}^{*} = U_{π^{- 1}}$ , $P_{+}^{*} = \frac{1}{N!} \sum_{π} U_{π^{- 1}} = P_{+}$ and $P_{+}^{2} = \frac{1}{(N!)^{2}} \sum_{π, σ} U_{π σ} = \frac{1}{N!} \sum_{τ} U_{τ} = P_{+}$ ; the same computation with the characters gives $P_{-}^{*} = P_{-}$ , $P_{-}^{2} = P_{-}$ . Hence both are orthogonal projections, which is (a).

1.2F2algebra

[Resolvent commutation] Fix $z \in ρ (H)$ and $φ \in H$ ; put $ψ : = R (z) φ \in D (H)$ . By hypothesis $U_{π} ψ \in D (H)$ and $(H - z) U_{π} ψ = U_{π} (H - z) ψ = U_{π} φ$ ; hence $U_{π} R (z) φ = R (z) U_{π} φ$ for every $φ$ , so $U_{π} R (z) = R (z) U_{π}$ ; averaging gives $P_{\pm} R (z) = R (z) P_{\pm}$ , which is (b).

1.3givenF1algebra

[Reduction] Put $χ_{+} (π) = 1$ and $χ_{-} (π) = sgn (π)$ . For $ψ \in D (H)$ , $P_{\pm} ψ = \frac{1}{N!} \sum_{π} χ_{\pm} (π) U_{π} ψ \in D (H)$ as a finite linear combination, and the commutation hypothesis gives $H P_{\pm} ψ = \frac{1}{N!} \sum_{π} χ_{\pm} (π) H U_{π} ψ = \frac{1}{N!} \sum_{π} χ_{\pm} (π) U_{π} H ψ = P_{\pm} H ψ$ , proving (c).

2.1F3step 1.3algebra

[Self-adjointness of the restrictions] Assume $ran P \neq {0}$ for $P = P_{\pm}$ and put $K : = ran P$ and $S : = H ∣_{D (H) \cap K}$ . Then $D (S) = P D (H)$ is dense in $K$ and $S$ is symmetric: for $φ, ψ \in D (S)$ , $⟨ S φ, ψ ⟩ = ⟨ H φ, ψ ⟩ = ⟨ φ, H ψ ⟩ = ⟨ φ, S ψ ⟩$ because $H$ is self-adjoint and $ψ = P ψ$ . $S$ is closed: if $ψ_{k} \in D (S)$ , $ψ_{k} \to ψ$ and $S ψ_{k} \to η$ with $ψ, η \in K$ , then $H$ closed gives $ψ \in D (H)$ and $H ψ = η$ , and $ψ \in K$ because $K$ is closed; hence $ψ \in D (S)$ , $S ψ = η$ . Finally $ran (S - i) = P ran (H - i)$ : indeed $(S - i) P ψ = P (H - i) ψ$ by step 1.3, and conversely every $P χ$ with $χ \in H$ equals $P (H - i) (H - i)^{- 1} χ = (S - i) P (H - i)^{- 1} χ$ . Since $H - i$ is onto $H$ ([F3] applied to the self-adjoint $H$ ), $ran (S - i) = K$ , and the same argument gives $ran (S + i) = K$ . By [F3], $S$ is self-adjoint on $K$ ; this proves (d).

3.1F4step 1.2step 1.3step 2.1algebra

[The permutation-invariant instance] Let $H = K + M_{V}$ as in (e). The unitaries $U_{π}^{p h y s} = Θ U_{π} Θ^{- 1}$ form a unitary representation of $S_{N}$ on $L^{2} (R^{3 N})$ ([F4]). The weight $κ (ξ) = \frac{ℏ^{2}}{2 m} \sum_{i} ∣ ξ_{i} ∣^{2}$ is invariant under permutations of the blocks $ξ_{i}$ , so $Θ U_{π} Θ^{- 1}$ maps $D (K)$ onto itself and intertwines $K$ with itself; and $V (π X) = V (X)$ gives $U_{π}^{p h y s} M_{V} U_{π}^{p h y s, - 1} = M_{V}$ on all of $L^{2}$ . Hence $U_{π}^{p h y s} D (H) = D (H)$ and $U_{π}^{p h y s} H = H U_{π}^{p h y s}$ on $D (H)$ , and step 2.1 applies to $P_{\pm}^{p h y s}$ , proving (e).

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

[Conclusion and scope] Steps 1.1-3.1 prove (a)-(e). The scope paragraph of the statement records that no spin-statistics assignment and no spectral conclusion is asserted; the symmetric and antisymmetric sectors are mathematical invariant subspaces of the specified operator.

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

Fixed-N Coulomb self-adjointness

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $N \geq 1$ and $r \geq 1$ , let $m_{1}, \dots, m_{N} > 0$ , $ℏ > 0$ , let $a_{1}, \dots, a_{M} \in R^{3}$ be fixed points, and let $z_{i m}, z_{i j} \in R$ be real coefficients ( $1 \leq i \leq N$ , $1 \leq m \leq M$ , $1 \leq i < j \leq N$ ). Let $K$ be the free kinetic operator of Bounded fixed-N operator, $D (K) = W^{2, 2} (R^{3 N}; C)$ with equivalent graph norm, and define the singular multiplier $V (X) : = \sum_{i = 1}^{N} \sum_{m = 1}^{M} \frac{z_{i m}}{∣ x_{i} - a_{m} ∣} + \sum_{1 \leq i < j \leq N} \frac{z_{i j}}{∣ x_{i} - x_{j} ∣}, X = (x_{1}, \dots, x_{N}) \in R^{3 N},$ as an a.e.-defined real measurable function (The space $L^{p} (μ)$ as the quotient by null functions). Then:

(a) each of the multipliers $u \mapsto u / ∣ x_{i} - a_{m} ∣$ and $u \mapsto u / ∣ x_{i} - x_{j} ∣$ is $K$ -bounded with $K$ -bound zero: for every $η > 0$ there is a finite constant $C_{η}$ with $∥ \frac{u}{∣ x_{i} - a_{m} ∣} ∥_{2} \leq η ∥ K u ∥_{2} + C_{η} ∥ u ∥_{2}, ∥ \frac{u}{∣ x_{i} - x_{j} ∣} ∥_{2} \leq η ∥ K u ∥_{2} + C_{η} ∥ u ∥_{2}$ for all $u \in D (K)$ (Relative boundedness with respect to an operator, Weak derivative of a locally integrable function);

(b) $D (K) \subseteq D (M_{V})$ , and $M_{V}$ is symmetric on $D (K)$ with $K$ -bound zero; consequently $H : = K + M_{V}, D (H) : = D (K) = W^{2, 2} (R^{3 N}; C),$ is self-adjoint and bounded below, and $C_{c}^{\infty} (R^{3 N}; C)$ is a core for $H$ (Kato-Rellich theorem, Symmetric, self-adjoint and essentially self-adjoint operators);

(c) the operator $K \otimes I_{r} + M_{V} \otimes I_{r}$ on $L^{2} (R^{3 N}; C) \otimes C^{r}$ with domain $D (K) \otimes C^{r}$ is self-adjoint and bounded below;

(d) if $m_{1} = \dots = m_{N} = m$ and the coefficients are permutation invariant ( $z_{i m}$ independent of $i$ and $z_{i j}$ independent of the pair), then the transported permutation unitaries of Permutation-restricted self-adjointness reduce $H$ , and the restrictions of $H$ to the symmetric and antisymmetric sectors are self-adjoint on those sectors.

No Hunziker-van Winter-Zhislin threshold statement, no uniform-in- $N$ stability, no thermodynamic limit, no spectral statement beyond self-adjointness and boundedness from below, and no physical interpretation of $H$ , the masses or the coefficients is asserted.

Facts & Assumptions

Given: the integers $N, r \geq 1$ , masses $m_{i} > 0$ , the constant $ℏ > 0$ , fixed points $a_{m}$ , real coefficients $z_{i m}, z_{i j}$ , the free kinetic operator $K$ with domain $D (K) = W^{2, 2} (R^{3 N})$ , the partial Laplacians $Δ_{i}$ and the operators $K_{i} : = - ℏ^{2} Δ_{i} / (8 π^{2} m_{i})$ on the $i$ -th block, and the multiplier $V$ defined above.

[F1]

$K = \sum_{i} K_{i}$ is self-adjoint, positive, with $D (K) = W^{2, 2} (R^{3 N}; C)$ and graph norm equivalent to the $W^{2, 2}$ norm; for every $u \in D (K)$ : $∥ \nabla_{i} u ∥_{2}^{2} = (8 π^{2} m_{i} / ℏ^{2}) ⟨ K_{i} u, u ⟩ \leq (8 π^{2} m_{i} / ℏ^{2}) ⟨ K u, u ⟩ \leq (8 π^{2} m_{i} / ℏ^{2}) ∥ K u ∥_{2} ∥ u ∥_{2}$ by Cauchy-Schwarz in $L^{2}$ , and $∥ Δ_{i} u ∥_{2} \leq (8 π^{2} m_{i} / ℏ^{2}) ∥ K u ∥_{2}$ (Bounded fixed-N operator, Integer-order W^{k,2} and H^k agree with equivalent norms, Plancherel theorem, Integer-order Sobolev spaces and their norms, Cauchy-Schwarz inequality for $L^{2}$ ).

[F2]

Three-dimensional Hardy inequality: for $v \in W^{1, 2} (R^{3}; C)$ , $∥ v / ∣ \cdot ∣ ∥_{2} \leq 2 ∥ \nabla v ∥_{2}$ (Three-dimensional Hardy bound (c)).

[F3]

$C_{c}^{\infty} (R^{3 N}; C)$ is dense in $W^{1, 2} (R^{3 N}; C)$ and in $W^{2, 2}$ ; under Countable Choice $L^{2}$ convergence supplies an a.e.-convergent subsequence, and Fatou bounds the integral of the nonnegative pointwise limit by the liminf of integrals (Compactly supported smooth functions are dense in W^{k,p}(R^n), Complex Lp completeness and almost-everywhere subsequences, Fatou's lemma, The space $L^{p} (μ)$ as the quotient by null functions). Tonelli permits the nonnegative slice integrations over the sigma-finite Lebesgue product spaces (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F4]

Translations preserve Lebesgue measure, and the orthogonal map $(x_{i}, x_{j}) \mapsto (y, z) = ((x_{i} - x_{j}) / \sqrt{2}, (x_{i} + x_{j}) / \sqrt{2})$ preserves Lebesgue measure, hence $L^{2}$ norms and integrals; under it, $\partial_{x_{i}} = 2^{- 1 / 2} (\partial_{z} + \partial_{y})$ , $\partial_{x_{j}} = 2^{- 1 / 2} (\partial_{z} - \partial_{y})$ and $∣ \nabla_{y} w ∣^{2} + ∣ \nabla_{z} w ∣^{2} = ∣ \nabla_{i} w ∣^{2} + ∣ \nabla_{j} w ∣^{2}$ (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, Lebesgue measure on $R^{n}$ is invariant under every orthogonal linear map, Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Weak derivative of a locally integrable function).

[F5]

Kato-Rellich: if $A$ is self-adjoint, $B$ symmetric with $D (A) \subseteq D (B)$ and $A$ -bound less than $1$ , then $A + B$ on $D (A)$ is self-adjoint and bounded below when $A$ is; a finite sum of $A$ -bound-zero symmetric operators has $A$ -bound zero (Kato-Rellich theorem, Relative boundedness with respect to an operator).

Proof

1.1F2F3F4algebra

[Global Hardy inequalities] Let $u \in C_{c}^{\infty} (R^{3 N})$ and fix a block index $i$ and a center $a \in R^{3}$ . For each fixed value of the remaining coordinates the slice $x_{i} \mapsto u (\dots, x_{i}, \dots)$ is a compactly supported smooth function on $R^{3}$ ; [F2] gives $\int_{R^{3}} ∣ u ∣^{2} / ∣ x_{i} - a ∣^{2} d x_{i} \leq 4 \int_{R^{3}} ∣ \nabla_{i} u ∣^{2} d x_{i}$ for that slice, and integrating over the remaining coordinates by Tonelli [F3] with [F4] (translation to the center $a$ ) yields $\int ∣ u ∣^{2} / ∣ x_{i} - a ∣^{2} \leq 4 \int ∣ \nabla_{i} u ∣^{2}$ . For a pair replace $(x_{i}, x_{j})$ by the orthogonal coordinates $(y, z)$ of [F4]: then $∣ x_{i} - x_{j} ∣^{2} = 2 ∣ y ∣^{2}$ and, applying the same slice argument in the $y$ -variable to the transformed function, $\int ∣ u ∣^{2} / ∣ x_{i} - x_{j} ∣^{2} = \frac{1}{2} \int ∣ u^{'} ∣^{2} / ∣ y ∣^{2} \leq 2 \int ∣ \nabla_{y} u^{'} ∣^{2} \leq 2 \int (∣ \nabla_{i} u ∣^{2} + ∣ \nabla_{j} u ∣^{2})$ . Both inequalities extend to all $u \in W^{1, 2} (R^{3 N})$ : choose $u_{k} \in C_{c}^{\infty}$ with $u_{k} \to u$ and $\nabla u_{k} \to \nabla u$ in $L^{2}$ ([F3]); the right-hand sides converge to the corresponding norms of $u$ , while the left-hand sides are lower semicontinuous by Fatou along an a.e. convergent subsequence ([F3]).

2.1F1step 1.1algebra

[Operator-norm $K$ -bounds] Let $u \in D (K)$ and put $a : = ∥ K u ∥_{2}$ , $b : = ∥ u ∥_{2}$ . The squared Hardy bound of step 1.1 and [F1] give $∥ u / ∣ x_{i} - a_{m} ∣ ∥_{2}^{2} \leq 4 ∥ \nabla_{i} u ∥_{2}^{2} \leq (32 π^{2} m_{i} / ℏ^{2}) ⟨ K u, u ⟩ \leq (32 π^{2} m_{i} / ℏ^{2}) a b$ . For a pair, the corrected pair bound gives $∥ u / ∣ x_{i} - x_{j} ∣ ∥_{2}^{2} \leq 2 (∥ \nabla_{i} u ∥_{2}^{2} + ∥ \nabla_{j} u ∥_{2}^{2}) \leq (16 π^{2} \max (m_{i}, m_{j}) / ℏ^{2}) ⟨ K u, u ⟩ \leq (16 π^{2} \max (m_{i}, m_{j}) / ℏ^{2}) a b$ : the sum uses $⟨ (K_{i} + K_{j}) u, u ⟩ \leq ⟨ K u, u ⟩$ . For either multiplier let $C$ be the square root of its displayed finite coefficient. For every $η > 0$ , $C \sqrt{a b} \leq η a + C^{2} b / (4 η)$ , obtained by expanding $(\sqrt{η a} - C \sqrt{b} / (2 \sqrt{η}))^{2} \geq 0$ . Thus both actual $L^{2}$ operator norms are at most $η ∥ K u ∥_{2} + C_{η} ∥ u ∥_{2}$ , with finite $C_{η} = C^{2} / (4 η)$ , including $u = 0$ . No form-to-operator-norm inference is used. This proves (a) for the literal Fourier-normalized $K$ of the supplier.

3.1F1step 1.1step 2.1algebra

[Separate quadratic-form bound] The same norm Hardy and kinetic-gradient inequalities give $∥ u / r ∥_{2} \leq C ⟨ K u, u ⟩^{1 / 2}$ for each distance multiplier, so Cauchy–Schwarz gives $∣ ⟨ M_{V} u, u ⟩ ∣ \leq \int ∣ V ∣ ∣ u ∣^{2} \leq C_{V} ⟨ K u, u ⟩^{1 / 2} ∥ u ∥_{2}$ , where $C_{V}$ is the finite sum of the respective Hardy-gradient constants times $∣ z_{i m} ∣, ∣ z_{i j} ∣$ . For every $ϵ > 0$ , Young's square inequality yields $∣ ⟨ M_{V} u, u ⟩ ∣ \leq ϵ ⟨ K u, u ⟩ + C_{V}^{2} ∥ u ∥_{2}^{2} / (4 ϵ)$ . In particular $⟨ (K + M_{V}) u, u ⟩ \geq (1 - ϵ) ⟨ K u, u ⟩ - C_{V}^{2} ∥ u ∥_{2}^{2} / (4 ϵ)$ ; this is a valid form bound and lower estimate, separate from the operator-norm bound of step 2.1.

3.2F5step 2.1algebra

[The full multiplier and Kato-Rellich] Since $V$ is a finite real linear combination of the multipliers of part (a), each with $K$ -bound zero, $V$ is measurable and $M_{V}$ is a symmetric operator with $D (K) \subseteq D (M_{V})$ and $K$ -bound zero ([F5]); in particular the bound $∥ M_{V} u ∥ \leq a ∥ K u ∥ + b ∥ u ∥$ holds on $D (K)$ for some $a < 1$ and finite $b$ (choose $a < 1$ by the zero bound and adjust $b$ ). By [F5] (Kato-Rellich) $H = K + M_{V}$ is self-adjoint on $D (K)$ and bounded below. Since the graph norms of $H$ and $K$ are equivalent on $D (K)$ (as in Bounded fixed-N operator (b)), the core $C_{c}^{\infty}$ of $K$ is a core for $H$ . This proves (b).

4.1step 3.2algebra

[Component extension and permutation sectors] The passage to $L^{2} (R^{3 N}; C) \otimes C^{r}$ is the finite direct sum of $r$ copies of $H$ and is self-adjoint and bounded below by the same argument as part (c) of Bounded fixed-N operator; this proves (c). Under equal masses and permutation-invariant coefficients, both $K$ and the multiplier $M_{V}$ are invariant under the transported permutation unitaries (the weight $κ$ and the multiplier $V$ are permutation invariant, and the domain $D (M_{V})$ is permutation invariant because the a.e.-defined function $V$ is), so $U_{π}^{p h y s} D (H) = D (H)$ and $U_{π}^{p h y s} H = H U_{π}^{p h y s}$ on $D (H)$ ; part (d) then follows from Permutation-restricted self-adjointness (d).

5.1step 1.1step 2.1step 3.2step 4.1algebra∎

[Conclusion and scope] Steps 1.1, 2.1, 3.1, 3.2 and 4.1 prove (a)-(d). The final paragraph of the statement records the excluded claims: no HVZ threshold, no uniform-in- $N$ stability, no thermodynamic limit, no further spectral statement and no physical interpretation is asserted.

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Support of the spectral measure, the spectrum and the values of the scalar law

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space (Hilbert space) and let $A$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E_{A}$ on the Borel $σ$ -algebra of $R$ and spectrum $σ (A)$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Resolvent and spectrum of an unbounded operator). Then:

(a) $E_{A} (R ∖ σ (A)) = 0$ , equivalently $E_{A} (B) = E_{A} (B \cap σ (A))$ for every Borel set $B \subseteq R$ ; and $E_{A} (U) \neq 0$ for every nonempty relatively open subset $U \subseteq σ (A)$ . Thus the support of $E_{A}$ is exactly $σ (A)$ .

(b) For every density operator $ρ$ on $H$ , the scalar law $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ of Density operators and scalar spectral probability clause (b) satisfies $μ_{ρ}^{A} (B) = μ_{ρ}^{A} (B \cap σ (A)) and μ_{ρ}^{A} (σ (A)) = 1$ for every Borel set $B$ : no mass lies outside the spectrum.

(c) More generally, if $C \subseteq R$ is a Borel set with $E_{A} (C) = 0$ , then $μ_{ρ}^{A} (C) = 0$ for every density operator $ρ$ , and if $E_{A} (C) = E_{A} (R)$ then $μ_{ρ}^{A} (C) = 1$ .

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )); a nonzero complex Hilbert space $H$ ; a self-adjoint operator $A$ with spectral PVM $E_{A}$ and spectrum $σ (A)$ ; density operators $ρ$ on $H$ .

[F1]

Spectrum from the calculus. The spectrum of $f (A)$ is the essential range ${z \in C : E_{A} (f^{- 1} (B_{ε} (z))) \neq 0 for every ε > 0}$ of the Borel function $f$ with respect to $E_{A}$ , and for $f (λ) = λ$ the operator $f (A)$ is $A$ with domain $D (A)$ ; consequently, for real $z$ , $z \in σ (A)$ if and only if $E_{A} ((z - ε, z + ε)) \neq 0$ for every $ε > 0$ (Unbounded Borel functional calculus: domains, products, spectral mapping, Spectral theorem for unbounded self-adjoint operators (PVM form)).

[F2]

Projection-valued measure. $E_{A}$ takes values in orthogonal projections, $E_{A} (\emptyset) = 0$ , $E_{A} (R) = I$ , $E_{A} (B \cap C) = E_{A} (B) E_{A} (C)$ , and for pairwise disjoint Borel $B_{n}$ with union $B$ the series $\sum_{n} E_{A} (B_{n}) x$ converges in norm to $E_{A} (B) x$ for every $x$ (Projection valued measure).

[F3]

Monotonicity and vanishing. If $B \subseteq C$ then $E_{A} (B) = E_{A} (B) E_{A} (C) = E_{A} (C) E_{A} (B)$ and $E_{A} (C) - E_{A} (B) = E_{A} (C ∖ B)$ is an orthogonal projection with range orthogonal to the range of $E_{A} (B)$ ; hence $⟨ E_{A} (B) x, x ⟩ \leq ⟨ E_{A} (C) x, x ⟩$ for every $x$ . Consequently, if $E_{A} (C) = 0$ then $E_{A} (B) = 0$ for every Borel $B \subseteq C$ (Projection valued measure, Scalar and complex measures from a pvm).

[F4]

Scalar law. For a density operator $ρ$ and a PVM $E$ the assignment $B \mapsto tr (ρ E (B))$ is a probability measure with $tr (ρ E (B)) = \sum_{n} w_{n} ∥ E (B) u_{n} ∥^{2}$ in every expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ , $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , $(u_{n})$ orthonormal; in particular $E (B) = 0$ implies $tr (ρ E (B)) = 0$ and $E (B) = I$ implies $tr (ρ E (B)) = 1$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F5]

Countable dense subset. The set $Q$ is countable and dense in $R$ ( $Q$ is countably infinite, Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable).

Proof

1.1givenF1F2algebra

[The spectrum is the essential range of the identity] By [F1] applied to $f (λ) = λ$ , $σ (A)$ is the essential range of the identity with respect to $E_{A}$ . For real $z$ this says: $z \in σ (A)$ if and only if $E_{A} ((z - ε, z + ε)) \neq 0$ for every $ε > 0$ ; and for non-real $z$ , taking $0 < ε < ∣ Im z ∣$ gives $B_{ε} (z) \cap R = \emptyset$ , where $B_{ε} (z) : = {w \in C : ∣ w - z ∣ < ε}$ , so $E_{A} (B_{ε} (z) \cap R) = E_{A} (\emptyset) = 0$ by [F2] and no non-real point lies in the essential range.

2.1givenstep 1.1F2F3F5algebra

[The open null set equals the complement of the spectrum] Let $I$ be the set of pairs of rationals $p < q$ , countable by [F5], and put $V : = ⋃ {(p, q) : (p, q) \in I, E_{A} ((p, q)) = 0}$ , a countable union of Borel sets. To see $V \subseteq R ∖ σ (A)$ , suppose $z \in V \cap σ (A)$ ; then $z \in (p, q)$ with $E_{A} ((p, q)) = 0$ for some rationals $p < q$ , and choosing $ε > 0$ with $(z - ε, z + ε) \subseteq (p, q)$ gives $E_{A} ((z - ε, z + ε)) = 0$ by [F3], contradicting $z \in σ (A)$ by step 1.1. Conversely, if $z \notin σ (A)$ , step 1.1 supplies $ε > 0$ with $E_{A} ((z - ε, z + ε)) = 0$ , and by density of the rationals [F5] there are rationals $p < z < q$ with $[p, q] \subseteq (z - ε, z + ε)$ ; then $E_{A} ((p, q)) = 0$ by [F3], so $z \in V$ . Hence $V = R ∖ σ (A)$ .

3.1givenstep 2.1F2F3algebra

[The complement of the spectrum is null] The representation of step 2.1 writes $V = R ∖ σ (A)$ as a countable union of Borel null sets. Enumerate these sets as $(I_{n})$ , repeating $\emptyset$ if there are only finitely many, and put $C_{0} : = I_{0}$ and $C_{n} : = I_{n} ∖ ⋃_{k < n} I_{k}$ for $n \geq 1$ . The $C_{n}$ are pairwise disjoint, have union $V$ , and satisfy $E_{A} (C_{n}) = 0$ by monotonicity [F3]. Strong countable additivity [F2] therefore gives $E_{A} (V) x = \sum_{n} E_{A} (C_{n}) x = 0$ for every $x$ , so $E_{A} (V) = 0$ . By additivity [F2], this is equivalent to $E_{A} (B) = E_{A} (B \cap σ (A)) + E_{A} (B ∖ σ (A)) = E_{A} (B \cap σ (A))$ for every Borel $B$ .

4.1step 1.1step 3.1F2F3algebra

[Nonempty relatively open subsets of the spectrum are not null] Let $U \subseteq σ (A)$ be nonempty and relatively open, so $U = σ (A) \cap W$ for some open $W \subseteq R$ . Choose $z \in U$ and $ε > 0$ with $(z - ε, z + ε) \subseteq W$ . By step 1.1, $E_{A} ((z - ε, z + ε)) \neq 0$ because $z \in σ (A)$ ; by step 3.1 applied to the Borel set $(z - ε, z + ε)$ , $E_{A} ((z - ε, z + ε)) = E_{A} ((z - ε, z + ε) \cap σ (A))$ , and $(z - ε, z + ε) \cap σ (A) \subseteq U$ by the choice of $ε$ . Were $E_{A} (U) = 0$ , monotonicity [F3] would give $E_{A} ((z - ε, z + ε) \cap σ (A)) = 0$ , a contradiction; hence $E_{A} (U) \neq 0$ . This proves clause (a) together with step 3.1.

4.2step 3.1F4algebra

[Clause (b)] By step 3.1 and additivity of $E_{A}$ , $E_{A} (B) = E_{A} (B \cap σ (A))$ for every Borel $B$ ; applying the scalar law [F4] to both sides gives $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B)) = tr (ρ E_{A} (B \cap σ (A))) = μ_{ρ}^{A} (B \cap σ (A))$ , and with $B = R$ this reads $μ_{ρ}^{A} (σ (A)) = tr (ρ E_{A} (R)) = tr ρ = 1$ by [F4].

5.1givenF4algebra∎

[Clause (c)] If $E_{A} (C) = 0$ then $tr (ρ E_{A} (C)) = 0$ by [F4] for every density operator; if $E_{A} (C) = I$ then $tr (ρ E_{A} (C)) = tr ρ = 1$ by [F4]. Both statements are instances of the scalar law, and clause (b) is the case $C = B ∖ σ (A)$ and $C = σ (A)$ respectively.

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Spectral measure of a real L2 multiplication operator

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $(X, Σ, μ)$ be a $σ$ -finite measure space with $μ (X) > 0$ , let $m : X \to R$ be measurable and finite $μ$ -almost everywhere, and on the complex Hilbert space $H = L^{2} (X, μ; C)$ (The space $L^{p} (μ)$ as the quotient by null functions, $L^{2}$ with the integral pairing is a Hilbert space, Hilbert space) let $D (M_{m}) = {f \in H : \int_{X} ∣ m ∣^{2} ∣ f ∣^{2} d μ < \infty}, M_{m} f = m f .$ Then:

(a) $M_{m}$ is self-adjoint on $D (M_{m})$ .

(b) The assignment $E (B) : = M_{1_{m^{- 1} (B)}}$ on Borel sets $B \subseteq R$ is a regular projection-valued measure on the Borel $σ$ -algebra of $R$ (Projection valued measure), and for a unit vector $ψ \in H$ $∥ E (B) ψ ∥^{2} = \int_{m^{- 1} (B)} ∣ ψ ∣^{2} d μ .$

(c) The operator identity $\int_{R} λ d E (λ) = M_{m}$ holds with domain $D (M_{m})$ (Integral of a measurable function against a projection-valued measure); consequently $E$ is the spectral projection-valued measure of $M_{m}$ (Spectral theorem for unbounded self-adjoint operators (PVM form)).

(d) $E (B) = 0$ if and only if $μ (m^{- 1} (B)) = 0$ , and the spectrum $σ (M_{m})$ equals the essential range of $m$ : $σ (M_{m}) = {t \in R : μ (m^{- 1} (t - ε, t + ε)) > 0 for every ε > 0} .$

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice); a $σ$ -finite measure space $(X, Σ, μ)$ ; a real measurable finite $μ$ -a.e. multiplier $m$ ; the operator $M_{m}$ on $H = L^{2} (X, μ; C)$ ; Borel sets $B, B_{n} \subseteq R$ ; a unit vector $ψ \in H$ .

[F1]

$H = L^{2} (X, μ; C)$ has inner product $⟨ f, g ⟩ = \int_{X} f \overline{g} d μ$ , Cauchy-Schwarz holds in this Hilbert space, and $m$ is real-valued and measurable. The density, symmetry and adjoint-domain checks for $M_{m}$ are supplied locally in step 1.1 ( $L^{2}$ with the integral pairing is a Hilbert space, Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs, The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

Projection-valued measures. A PVM is a map into orthogonal projections with $E (\emptyset) = 0$ , $E (X_{0}) = I$ and $E (B \cap C) = E (B) E (C)$ on its measurable space $X_{0}$ , strongly countably additive on disjoint unions; its scalar measures $E_{x} (B) = ⟨ E (B) x, x ⟩ = ∥ E (B) x ∥^{2}$ are finite positive measures (Projection valued measure, Scalar and complex measures from a pvm).

[F3]

Bounded integrals against a PVM. For bounded Borel $g$ there is a unique operator $Φ_{E} (g)$ with $⟨ Φ_{E} (g) x, y ⟩ = \int g d E_{x, y}$ ; it satisfies $⟨ Φ_{E} (g) x, x ⟩ = \int g d E_{x}$ , $∥ Φ_{E} (g) ∥ \leq ∥ g ∥_{\infty}$ , and it is multiplicative and adjoint preserving (Bounded borel pvm integral).

[F4]

Unbounded integrals. For Borel $f$ the operator $f (E)$ has domain $D (f (E)) = {x : \int ∣ f ∣^{2} d E_{x} < \infty}$ and $f (E) x = \lim_{n} Φ_{E} (f 1_{{∣ f ∣ \leq n}}) x$ (Integral of a measurable function against a projection-valued measure, The unbounded PVM integral is densely defined, closed and normal).

[F5]

Spectral uniqueness. A self-adjoint operator has a unique regular PVM on the Borel sets of $R$ representing it; conversely, if a regular PVM $E^{'}$ satisfies $\int λ d E^{'} (λ) = T$ with domain ${x : \int λ^{2} d E_{x}^{'} < \infty}$ , then $E^{'}$ is the spectral PVM of the self-adjoint operator $T$ (Spectral theorem for unbounded self-adjoint operators (PVM form)).

[F6]

Regularity of finite Borel measures on the line. Every finite Borel measure on the second-countable locally compact Hausdorff space $R$ is regular (Locally finite Borel measures on second-countable LCH spaces are regular, Regular Borel measure on an LCH space).

[F7]

Convergence theorems. If $∣ f_{n} ∣ \leq g$ with $g$ integrable and $f_{n} \to f$ pointwise, then $\int f_{n} \to \int f$ (Dominated convergence).

[F8]

Support of the spectral measure. For a self-adjoint operator with spectral PVM $F$ one has $F (R ∖ σ (T)) = 0$ , and $F (U) \neq 0$ for every nonempty relatively open $U \subseteq σ (T)$ : the support of $F$ is the spectrum (Support of the spectral measure, the spectrum and the values of the scalar law clause (a)).

Proof

1.1givenF2F7algebra

[Self-adjoint multiplication and indicator PVM] Put $X_{n} : = {x : ∣ m (x) ∣ \leq n}$ . For $f \in H$ , $f 1_{X_{n}} \in D (M_{m})$ and $f 1_{X_{n}} \to f$ in $H$ by dominated convergence, so $D (M_{m})$ is dense. For $f, g \in D (M_{m})$ , Cauchy-Schwarz makes the products integrable and real-valuedness of $m$ gives $⟨ M_{m} f, g ⟩ = ⟨ f, M_{m} g ⟩$ , so $M_{m}$ is symmetric. If $h \in D (M_{m}^{*})$ and $k : = M_{m}^{*} h$ , the adjoint identity holds for every $f \in D (M_{m})$ . For each $n$ take $f = (m h - k) 1_{X_{n}}$ ; this vector lies in $D (M_{m})$ because $m$ is bounded on $X_{n}$ , and the identity gives $0 = ⟨ M_{m} f, h ⟩ - ⟨ f, k ⟩ = \int_{X_{n}} ∣ m h - k ∣^{2} d μ$ . Thus $m h = k$ a.e. on every $X_{n}$ , and since $⋃_{n} X_{n}$ has full measure, $m h = k \in H$ . Therefore $h \in D (M_{m})$ and $M_{m}^{*} h = M_{m} h$ , so $M_{m} = M_{m}^{*}$ ; this proves (a). [F1, F7, algebra] For a Borel set $B$ the class $1_{m^{- 1} (B)}$ is real measurable and bounded, so multiplication by it is a bounded self-adjoint operator on $H$ whose square is multiplication by $1_{m^{- 1} (B)}^{2} = 1_{m^{- 1} (B)}$ : hence $E (B) = M_{1_{m^{- 1} (B)}}$ is an orthogonal projection. Moreover $E (\emptyset) = 0$ and $E (R) = I$ because $m$ is finite $μ$ -a.e., and $E (B) E (C) = M_{1_{m^{- 1} (B)} 1_{m^{- 1} (C)}} = E (B \cap C)$ because $m^{- 1} (B) \cap m^{- 1} (C) = m^{- 1} (B \cap C)$ up to a $μ$ -null set. For pairwise disjoint Borel $B_{n}$ with union $B$ the multiplicativity of classes gives $∥ E (B) h - \sum_{n \leq N} E (B_{n}) h ∥^{2} = \int ∣ (1_{B} - \sum_{n \leq N} 1_{B_{n}}) \circ m ∣^{2} ∣ h ∣^{2} d μ$ for every $h \in H$ , and the integrand is bounded by $∣ h ∣^{2}$ and tends to $0$ pointwise, so dominated convergence [F7] makes the truncated sums converge in norm to $E (B) h$ ; thus $E$ is strongly countably additive and is a PVM.

2.1step 1.1F2F6algebra

[Regularity] For $x \in H$ the scalar measure of [F2] is $E_{x} (B) = ⟨ E (B) x, x ⟩ = \int_{m^{- 1} (B)} ∣ x ∣^{2} d μ$ , a finite Borel measure on $R$ of total mass $∥ x ∥^{2}$ ; $R$ is locally compact, Hausdorff and second countable, so [F6] makes $E_{x}$ regular. As this holds for every $x$ , $E$ is a regular PVM.

2.2step 1.1F1F2F3F4F7algebra

[The integral of the identity is $M_{m}$ ] Let $g$ be bounded Borel. For indicators $g = 1_{B}$ the pairing identity $⟨ Φ_{E} (g) x, y ⟩ = \int g d E_{x, y}$ and the scalar measures of [F2] give $⟨ Φ_{E} (1_{B}) x, y ⟩ = \int_{m^{- 1} (B)} x \overline{y} d μ = ⟨ M_{1_{B} \circ m} x, y ⟩$ , so $Φ_{E} (1_{B}) = M_{1_{B} \circ m}$ ; by linearity of $Φ_{E}$ and [F3] the identity passes to simple functions, and uniform approximation with $∥ Φ_{E} (g) ∥ \leq ∥ g ∥_{\infty}$ [F3] gives $Φ_{E} (g) = M_{g \circ m}$ for every bounded Borel $g$ . For the unbounded identity function $f (λ) = λ$ the scalar measures give $\int λ^{2} d E_{x} = \int ∣ m ∣^{2} ∣ x ∣^{2} d μ$ , so by [F4] the domain of $f (E)$ is exactly $D (M_{m})$ , and for $x \in D (M_{m})$ the truncations satisfy $f 1_{{∣ f ∣ \leq n}} (E) x = M_{m 1_{{∣ m ∣ \leq n}}} x$ by the bounded case; these converge to $m x$ in $H$ by dominated convergence [F7] applied to $∣ m 1_{{∣ m ∣ \leq n}} - m ∣^{2} ∣ x ∣^{2} \leq 4 ∣ m ∣^{2} ∣ x ∣^{2}$ . Hence $f (E) x = M_{m} x$ on $D (M_{m})$ .

3.1step 1.1step 2.1step 2.2F1F5algebra

[The spectral PVM and the scalar law] By step 2.2 the regular PVM $E$ of step 1.1 satisfies $\int λ d E (λ) = M_{m}$ with the displayed domain, and $M_{m}$ is self-adjoint by [F1]; the converse uniqueness clause of [F5] therefore identifies $E$ as the spectral projection-valued measure of $M_{m}$ . For a unit vector $ψ$ the scalar law is $∥ E (B) ψ ∥^{2} = \int_{m^{- 1} (B)} ∣ ψ ∣^{2} d μ$ by step 2.1.

4.1step 1.1step 2.1step 3.1F2F8algebra∎

[Vanishing sets and the spectrum] If $E (B) = 0$ then $0 = ∥ E (B) h ∥^{2} = \int_{m^{- 1} (B)} ∣ h ∣^{2} d μ$ for every $h \in H$ ; were $μ (m^{- 1} (B)) > 0$ , $σ$ -finiteness would give a subset $A \subseteq m^{- 1} (B)$ of positive finite measure, and $h = 1_{A}$ would give $\int_{m^{- 1} (B)} ∣ h ∣^{2} d μ = μ (A) > 0$ , a contradiction; hence $μ (m^{- 1} (B)) = 0$ . Conversely, if $μ (m^{- 1} (B)) = 0$ then the multiplier $1_{m^{- 1} (B)}$ vanishes as an $L^{2}$ class and $E (B) = 0$ . By step 3.1, $E$ is the spectral PVM of $M_{m}$ , so [F8] identifies its support with $σ (M_{m})$ ; by the vanishing criterion just proved the support is the set of $t$ with $μ (m^{- 1} (t - ε, t + ε)) > 0$ for every $ε > 0$ , namely the essential range of $m$ . This proves (d) and completes the argument.

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Transverse continuum free Fock Hamiltonian

Statement

Assume the Axiom of Countable Choice, the Axiom of Dependent Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice). Fix a parameter $α > 0$ , define the transverse subspace $h : = {f \in L^{2} (R^{3}; C^{3}) : k \cdot f (k) = 0 for a.e. k}$ with the $L^{2}$ inner product, define the bounded multiplier-valued function $Π (k) : = i d - ∣ k ∣^{- 2} k k^{T}$ for $k \neq 0$ and $Π (0) : = 0$ , and let $M_{Π}$ be multiplication by $Π$ on $L^{2} (R^{3}; C^{3})$ (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions). Write $ε (k) : = α ∣ k ∣$ and let $M_{ε}$ be multiplication by $ε$ , with $ε_{h} : = M_{ε} ∣_{h}$ . Let $F (h)$ be the Fock space of Hilbert direct-sum construction with sectors $ran P_{+}^{(n)}$ , and for $n \geq 1$ define $T_{n} ψ : = ε_{n} ψ, ε_{n} (k_{1}, \dots, k_{n}) : = \sum_{i = 1}^{n} α ∣ k_{i} ∣, D (T_{n}) : = {ψ \in ran P_{+}^{(n)} : ε_{n} ψ \in ran P_{+}^{(n)}},$ and $T_{0} : = 0$ on $C$ . Finally put $H_{f} : = ⨁_{n \geq 0} T_{n}, D (H_{f}) : = {ψ \in F (h) : ψ_{n} \in D (T_{n}) for every n, \sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty} .$ Then:

(a) $h = ran M_{Π}$ is a closed separable subspace of $L^{2} (R^{3}; C^{3})$ , invariant under $M_{ε}$ , and $ε_{h}$ is a positive self-adjoint operator on $h$ with $D (ε_{h}) = {f \in h : ε f \in h}$ (Real multiplication operators and free plus bounded-potential Schrödinger operators, The Hilbert orthogonal projection onto a closed subspace, Separability: the existence of an at most countable dense subset);

(b) each $T_{n}$ is a positive self-adjoint operator on the Hilbert space $ran P_{+}^{(n)}$ with the displayed domain (Hilbert space, Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Permutation sector projections on the n-fold tensor power);

(c) $H_{f}$ is self-adjoint and positive with domain $D (H_{f})$ ; it commutes with every sector projection $Π_{n}$ (i.e. $T_{n}$ maps its sector into itself);

(d) for every $β > 0$ the bounded operator $e^{- β H_{f}}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure) is not compact and in particular not trace class (Compact linear operator, Trace class operator); more precisely its restriction to the one-particle sector equals the multiplication operator $M_{e^{- β ε}}$ on $h$ , which is not compact, and the restriction of a compact operator to a closed invariant subspace is compact.

The identification of $h$ , $ε$ and $H_{f}$ with physical transverse photons in a chosen frame is a separate model assumption and is not asserted here; no thermal interpretation of $e^{- β H_{f}}$ is claimed either.

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 parameter $α > 0$ , the transverse subspace $h$ , the projection-valued multiplier $M_{Π}$ , the function $ε (k) = α ∣ k ∣$ , the sector operators $T_{n}$ , and the direct sum $H_{f}$ .

[F1]

$Π (k)$ is for $k \neq 0$ the orthogonal projection of $C^{3}$ onto the orthogonal complement of $k$ ; a measurable, bounded projection-valued function defines a bounded self-adjoint projection $M_{Π}$ on $L^{2} (R^{3}; C^{3})$ , with range a closed subspace (The Hilbert orthogonal projection onto a closed subspace, Real multiplication operators and free plus bounded-potential Schrödinger operators, The space $L^{p} (μ)$ as the quotient by null functions).

[F2]

For a real measurable finite-valued function $a$ on a Euclidean space, the scalar multiplier $M_{a}$ with domain ${f : a f \in L^{2}}$ is densely defined and self-adjoint (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (a)). Its finite-component extension and its restriction to a reducing closed subspace are proved locally in step 1.1, using the adjoint definition Adjoint of a densely defined operator. No restriction theorem is attributed to the scalar supplier.

[F3]

Put $G = L^{2} (R^{3}; C^{3})$ . The completed tensor construction and iterated Euclidean product unitary give $Θ_{n} : G^{\otimes n} ⟶ L_{n} = L^{2} (R^{3 n}; (C^{3})^{\otimes n}), Θ_{n} (f_{1} \otimes \dots \otimes f_{n}) (k) = f_{1} (k_{1}) \otimes \dots \otimes f_{n} (k_{n}) .$ The elementary tensor span is dense, and permutation unitaries act on tensor factors by $U_{π} (f_{1} \otimes \dots \otimes f_{n}) = f_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes f_{{\hat{π}}^{- 1} (n)}$ ; their average is the symmetric projection (Hilbert tensor construction: inner product, completion and product bases, Permutation sector projections on the n-fold tensor power, Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})). The corresponding transverse and symmetric projections in $L_{n}$ , including the polarization-factor permutations, are constructed in step 3.1.

[F4]

For self-adjoint $T_{n}$ on every sector $ran P_{+}^{(n)}$ , including $T_{0}$ on $C$ , the direct sum is self-adjoint on ${ψ : ψ_{n} \in D (T_{n}), \sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty}$ (Hilbert direct-sum construction (f)). Here the vacuum summand is $T_{0} = 0$ .

[F5]

Each self-adjoint $T_{n}$ has a unique regular spectral PVM $E_{n}$ , its domain is ${u : \int t^{2} d (E_{n})_{u} (t) < \infty}$ , and $T_{n} = \int t d E_{n} (t)$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, Integral of a measurable function against a projection-valued measure). For bounded Borel $h$ , $h (E_{n}) = Φ_{E_{n}} (h)$ and its norm and quadratic identities hold (Bounded borel pvm integral). For a real multiplier on a Euclidean $L^{2}$ space, the spectral PVM is the inverse-image indicator multiplier; this applies componentwise to $C^{3}$ -valued functions (Spectral measure of a real L2 multiplication operator).

[F6]

Sequential characterization of compactness: a bounded operator $T$ is compact if and only if every bounded sequence has a subsequence whose images converge in norm; hence an orthonormal sequence whose images have pairwise distances bounded below by a positive constant obstructs compactness (Sequential characterization of compact operators, Compact linear operator, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F7]

Bump functions: for every $c \in R^{3}$ and $ρ > 0$ there is $φ \in C_{c}^{\infty} (R^{3})$ with $0 \leq φ \leq 1$ , $φ = 1$ on the ball of radius $ρ / 2$ about $c$ and $φ = 0$ outside the ball of radius $ρ$ about $c$ (A Euclidean bump for a compact set inside an open set, Complex Lp classes and Euclidean test-function conventions).

[F8]

For nonnegative measurable $g_{n} ↑ g$ , $\int g_{n} d μ ↑ \int g d μ$ (Monotone convergence for the integral).

[F9]

Cauchy-Schwarz in each sector Hilbert space gives $∣ ⟨ T_{n} u, u ⟩ ∣ \leq ∥ T_{n} u ∥ ∥ u ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F10]

The real space $L^{2} (R^{3})$ has a countable dense family of rational box-step functions (Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$ ); six copies of this countable family are countable by finite iteration of A product of two at most countable sets is at most countable and give a dense family in $G = L^{2} (R^{3}; C^{3})$ by approximating the real and imaginary parts of each component. Since $M_{Π}$ is a bounded projection, the image of this dense family under $M_{Π}$ is countable and dense in $h = ran M_{Π}$ (Finite, countably infinite, countable, uncountable, Separability: the existence of an at most countable dense subset).

Proof

1.1F2algebra

[Finite components and reducing restrictions] The scalar self-adjointness in [F2] extends to any finite number of components: the multiplication domain is the finite product of scalar domains, hence dense, and testing the adjoint identity with a vector supported in one component shows that the adjoint domain and action are exactly the componentwise multiplication domain and action. Now let $A$ be self-adjoint, let $Q$ be a bounded orthogonal projection with $Q D (A) \subseteq D (A)$ and $A Q x = Q A x$ on $D (A)$ , and put $K = ran Q$ . The domain $D (A) \cap K$ is dense in $K$ , since applying $Q$ to approximants in the dense $D (A)$ gives approximants in this intersection. The restriction $A_{K}$ is symmetric. If $y \in D (A_{K}^{*})$ and $h = A_{K}^{*} y \in K$ , then for every $x \in D (A)$ , [F2, algebra] $⟨ A x, y ⟩ = ⟨ Q A x, y ⟩ = ⟨ A Q x, y ⟩ = ⟨ Q x, h ⟩ = ⟨ x, h ⟩ .$ Thus $y \in D (A^{*}) = D (A)$ and $A y = h$ , so $y \in D (A_{K})$ and $A_{K}^{*} y = A_{K} y$ . Conversely symmetry gives $A_{K} \subseteq A_{K}^{*}$ . Therefore $A_{K}$ is self-adjoint on exactly $D (A) \cap K$ . This argument uses the adjoint definition, not a missing clause of the scalar multiplier theorem.

2.1F1F2F10step 1.1algebra

[The transverse space] For $k \neq 0$ the matrix $Π (k)$ is the orthogonal projection onto ${v : k \cdot v = 0}$ : it is symmetric, $Π (k)^{2} = Π (k)$ and $Π (k) v = v$ exactly for $k \cdot v = 0$ . Extending by $Π (0) = 0$ gives a measurable, bounded, projection-valued function, so $M_{Π}$ is a bounded self-adjoint projection and $h = ran M_{Π}$ is a closed subspace. By [F10], the image under $M_{Π}$ of a countable dense subset of $G$ is countable and dense in its range, so $h$ is separable. Since $ε$ is radial, $M_{ε} M_{Π} = M_{Π} M_{ε}$ , so $M_{ε}$ preserves $h$ ; by step 1.1 the restriction $ε_{h}$ is self-adjoint with domain ${f \in h : ε f \in h}$ , and it is positive because $ε \geq 0$ . This proves (a).

3.1F2F3F9step 1.1step 2.1algebra

[Transverse symmetric sectors and their energy domains] Fix $n \geq 1$ . The isometric inclusion $h \to G$ induces an isometric inclusion of the completed tensor powers, since the elementary inner products agree and their spans are dense. Under $Θ_{n}$ of [F3], its closed range is that of multiplication by [F2, F3, step 1.1, step 2.1, algebra] $Q_{n} (k) = Π (k_{1}) \otimes \dots \otimes Π (k_{n}) .$ Indeed this bounded orthogonal projection sends every elementary product to the product of its transverse projections, so its range is the closure of the transverse elementary span. Let $τ_{π}$ permute the factors of $(C^{3})^{\otimes n}$ by the same convention as $U_{π}$ . The ambient permutation is $(C_{π} f) (k_{1}, \dots, k_{n}) = τ_{π} f (k_{\hat{π} (1)}, \dots, k_{\hat{π} (n)}),$ as follows first on elementary products and then everywhere by density. Hence $S_{n} = (n!)^{- 1} \sum_{π} C_{π}$ is the orthogonal symmetric projection. The transverse tensor projection commutes with every permutation on elementary tensors, and hence everywhere; thus $R_{n} = M_{Q_{n}} S_{n}$ is an orthogonal projection whose range $K_{n}$ is exactly the image of $ran P_{+}^{(n)} \subseteq h^{\otimes n}$ . Denote the resulting isometry onto $K_{n}$ by $I_{n}$ . Since each $τ_{π}$ is unitary, every sector vector has $∥ I_{n} u (k_{\hat{π} (1)}, \dots, k_{\hat{π} (n)}) ∥ = ∥ I_{n} u (k_{1}, \dots, k_{n}) ∥$ almost everywhere. This is the norm symmetry used in sector energy integrals; it does not assert invariance of each polarization component without permuting its tensor factors. The ambient finite-component multiplier $A_{n} = M_{ε_{n}}$ is self-adjoint on its maximal multiplication domain by step 1.1. Its domain is preserved by $M_{Q_{n}}$ , since this pointwise projection is contractive and commutes with the scalar multiplier. It is preserved by every $C_{π}$ , and $A_{n} C_{π} = C_{π} A_{n}$ on that domain, because $ε_{n} (k) = \sum_{i} α ∣ k_{i} ∣$ is permutation invariant. Therefore $R_{n}$ preserves the domain and commutes with $A_{n}$ there. Step 1.1 makes its restriction to $K_{n}$ self-adjoint on precisely ${f \in K_{n} : ε_{n} f \in L_{n}}$ . Commutation implies that the product lies in $K_{n}$ whenever it lies in $L_{n}$ . Transport by $I_{n}$ gives exactly $T_{n}$ and the displayed domain of the Statement. Its pairing with a domain vector is the integral of $ε_{n} ∥ I_{n} ψ ∥^{2}$ , which is nonnegative and finite by Cauchy-Schwarz [F9]. Thus $T_{n} \geq 0$ , proving (b). For $n = 0$ , $T_{0} = 0$ on all of $C$ is self-adjoint and positive.

4.1F4F9step 3.1algebra

[The free Fock Hamiltonian] By step 3.1 each $T_{n}$ , including $T_{0} = 0$ , is self-adjoint on its sector, so [F4] gives that $H_{f} = ⨁_{n} T_{n}$ is self-adjoint on the displayed domain. For $ψ \in D (H_{f})$ , $⟨ H_{f} ψ, ψ ⟩ = \sum_{n} ⟨ T_{n} ψ_{n}, ψ_{n} ⟩ \geq 0$ ; the sum is absolutely convergent by [F9] and finite-sequence Cauchy–Schwarz applied to $(∥ T_{n} ψ_{n} ∥)_{n}$ and $(∥ ψ_{n} ∥)_{n}$ . Since each $T_{n}$ maps $ran P_{+}^{(n)}$ into itself, $H_{f}$ commutes with every sector projection $Π_{n}$ . This proves (c).

5.1F2F3F4F5F8step 2.1step 3.1step 4.1algebra

[The direct-sum spectral PVM and the exponential] For each $n \geq 0$ let $E_{n}$ be the regular spectral PVM of $T_{n}$ from [F5], with $E_{0} (B) = 1_{B} (0) I_{C}$ because $T_{0} = 0$ , and define $E (B) : = ⨁_{n \geq 0} E_{n} (B)$ . Each $E (B)$ is an orthogonal projection, $E (R) = I$ , and the PVM product law holds coordinatewise. Strong countable additivity follows by first restricting a vector to finitely many sectors, where it is the strong additivity of the $E_{n}$ , and then bounding the omitted squared norm by the tail $\sum_{n > N} ∥ ψ_{n} ∥^{2}$ . Its scalar measure is $E_{ψ} (B) = \sum_{n \geq 0} (E_{n})_{ψ_{n}} (B)$ ; this finite measure given by a countable sum of sector measures is regular, since one first discards a tail of total mass $\sum_{n > N} ∥ ψ_{n} ∥^{2}$ and then uses inner compact and outer open approximations for the finitely many remaining regular measures. For every nonnegative simple $g$ , $\int g d E_{ψ} = \sum_{n} \int g d (E_{n})_{ψ_{n}}$ by the definition of the sum measure, and [F8] extends this identity to $g (t) = t^{2}$ . By [F5], this second moment is finite exactly when each $ψ_{n} \in D (T_{n})$ and $\sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty$ , so the domain of $\int t d E (t)$ is $D (H_{f})$ . For $ψ \in D (H_{f})$ the bounded truncations $t_{R} (t) : = t 1_{∣ t ∣ \leq R}$ satisfy $Φ_{E} (t_{R}) = ⨁_{n} Φ_{E_{n}} (t_{R})$ : this follows from the definition of $E$ for simple functions and uniform approximation for bounded Borel functions. As $R \to \infty$ , these vectors converge to $(T_{n} ψ_{n})_{n}$ in norm, first on finitely many sectors by the spectral integral and then on the sector tail using $\sum_{n} ∥ T_{n} ψ_{n} ∥^{2} < \infty$ . Therefore $\int t d E (t) = H_{f}$ on precisely $D (H_{f})$ ; regularity and the uniqueness clause of [F5] give $E = E_{H_{f}}$ . For $n = 1$ , the full-space multiplier PVM of $ε (k) = α ∣ k ∣$ commutes with the transverse projection $M_{Π}$ (scalar multiplication commutes pointwise with $Π (k)$ ), so its restriction to $h$ is a regular PVM. Its scalar second moment is $\int ε (k)^{2} ∥ f (k) ∥^{2} d^{3} k$ , and bounded spectral truncations act as multiplication by $ε 1_{{ε \leq R}}$ , converging in norm on exactly $D (ε_{h})$ . Thus its unbounded identity integral is $T_{1} = ε_{h}$ , and spectral uniqueness identifies this restricted PVM with $E_{1}$ . The bounded function $h (t) : = e^{- β t} 1_{[0, \infty)} (t)$ is $e^{- β t}$ on the nonnegative spectrum of every $T_{n}$ ; bounded PVM integration is coordinatewise, hence $e^{- β H_{f}} = ⨁_{n} e^{- β T_{n}}$ and its one-particle restriction is $M_{e^{- β ε}}$ on $h$ .

6.1F6F7step 5.1algebra

[Non-compactness of the one-particle multiplier] For $m \geq 1$ put $c_{m} : = 2^{- m - 2} e_{1} \in R^{3}$ and $ρ_{m} : = 2^{- m - 5}$ , and let $φ_{m}$ be a bump as in [F7] with centre $c_{m}$ and radius $ρ_{m}$ ; the balls $B (c_{m}, ρ_{m})$ are pairwise disjoint and contained in ${∣ k ∣ \leq 1 / 2}$ . Define $g_{m} : = Π φ_{m} e_{3} \in h$ ; on $B (c_{m}, ρ_{m})$ one has $Π (k) e_{3} = e_{3} - (k_{3} / ∣ k ∣^{2}) k$ with norm $\sqrt{k_{1}^{2} + k_{2}^{2}} / ∣ k ∣ \geq ∣ k_{1} ∣ / ∣ k ∣ \geq (1 - \frac{1}{8}) / (1 + \frac{1}{8}) > 1 / 2$ , so $∥ g_{m} ∥ \geq \frac{1}{2} ∥ φ_{m} ∥_{2} > 0$ ; the $g_{m}$ have pairwise disjoint supports and hence are pairwise orthogonal. Put $f_{m} : = g_{m} / ∥ g_{m} ∥$ ; then $(f_{m})$ is orthonormal and $∣ e^{- β ε (k)} ∣ \geq e^{- β α}$ on every $B (c_{m}, ρ_{m})$ (since $∣ k ∣ \leq 1$ there), so $∥ M_{e^{- β ε}} f_{m} ∥_{2} \geq e^{- β α}$ and $∥ M_{e^{- β ε}} f_{m} - M_{e^{- β ε}} f_{m^{'}} ∥_{2}^{2} = ∥ M f_{m} ∥_{2}^{2} + ∥ M f_{m^{'}} ∥_{2}^{2} \geq 2 e^{- 2 β α}$ for $m \neq m^{'}$ . The bounded (indeed orthonormal) sequence $(f_{m})$ therefore has no subsequence with norm-convergent images, so $M_{e^{- β ε}}$ is not compact by [F6].

7.1F6step 4.1step 5.1step 6.1algebra

[Non-compactness and non-trace-class of $e^{- β H_{f}}$ ] Suppose $e^{- β H_{f}}$ were compact. The one-particle sector $ran Π_{1}$ is a closed subspace invariant under $e^{- β H_{f}}$ (by step 4.1 and [F5]); if $(x_{m})$ is a bounded sequence in it, compactness in $F (h)$ yields a norm-convergent subsequence of $(e^{- β H_{f}} x_{m})$ , whose limit lies again in the closed invariant subspace; hence the restriction is compact. By step 5.1 the restriction is $M_{e^{- β ε}}$ , contradicting step 6.1. Therefore $e^{- β H_{f}}$ is not compact; since every trace-class operator is compact (Trace class operator), it is not trace class. This proves (d).

8.1step 2.1step 3.1step 4.1step 5.1step 6.1step 7.1algebra∎

[Conclusion and scope] Step 2.1 proves (a), step 3.1 proves (b), step 4.1 proves (c), and steps 5.1, 6.1 and 7.1 prove (d). The closing paragraphs of the statement record that the physical photon identification and any thermal interpretation are separate model assumptions, not asserted here.

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

Slater direct and exchange calculation

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $H$ be a complex Hilbert space with completed tensor powers $H^{\otimes n}$ and symmetric projections $P_{\pm}^{(n)}$ (Hilbert tensor construction: inner product, completion and product bases, Permutation sector projections on the n-fold tensor power). Then:

(a) Slater vectors: for $n \geq 2$ and orthonormal $u_{1}, \dots, u_{n} \in H$ , the Slater vector $S (u_{1}, \dots, u_{n}) : = \frac{1}{\sqrt{n!}} \sum_{σ \in S_{n}} sgn (σ) u_{\hat{σ} (1)} \otimes \dots \otimes u_{\hat{σ} (n)}$ satisfies $∥ S (u_{1}, \dots, u_{n}) ∥ = 1$ and $S (u_{1}, \dots, u_{n}) = \sqrt{n!} P_{-}^{(n)} (u_{1} \otimes \dots \otimes u_{n})$ ;

(b) exclusion: if $u_{i} = u_{j}$ for some $i \neq j$ , then $P_{-}^{(n)} (u_{1} \otimes \dots \otimes u_{n}) = 0$ (Permutation sector projections on the n-fold tensor power (d));

(c) two orbitals, bounded pair interaction: let $u, v \in L^{2} (R^{3}; C)$ be normalized and orthogonal, let $W : R^{3} \to R$ be measurable and bounded with $W (- z) = W (z)$ for a.e. $z$ , let $Θ : L^{2} (R^{3}) \otimes L^{2} (R^{3}) \to L^{2} (R^{6})$ be the Euclidean product unitary (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})), put $φ_{\pm} : = (u \otimes v \pm v \otimes u) / \sqrt{2}$ and $D : = \int_{R^{6}} ∣ u (x) ∣^{2} ∣ v (y) ∣^{2} W (x - y) d x d y, X : = \int_{R^{6}} \overline{u (x)} v (x) u (y) \overline{v (y)} W (x - y) d x d y,$ integrals of absolutely integrable functions. Then $D, X \in R$ and the pair expectation of the multiplier $M_{W}$ in the two spatial states is $⟨ M_{W} (Θ φ_{\pm}), Θ φ_{\pm} ⟩ = D \pm X .$

(d) repulsive Coulomb exchange: let $κ > 0$ , $W (z) = κ / ∣ z ∣$ , let $u, v \in W^{1, 2} (R^{3}; C) \cap L^{2}$ be normalized and orthogonal (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions). The vectors $Θ φ_{\pm}$ belong to $D (M_{W})$ : Hardy translated in the $y$ variable and Tonelli give $\iint ∣ u (x) v (y) ∣^{2} / ∣ x - y ∣^{2} \leq 4 ∥ u ∥_{2}^{2} ∥ \nabla v ∥_{2}^{2} < \infty$ , and the swapped product has the same property. Then the integrals $D$ and $X$ converge absolutely, $0 \leq D < \infty$ , $X \in R$ with $X \geq 0$ ; consequently $D - X \leq D + X$ , i.e. the antisymmetric spatial pair state has the lower expectation of the repulsive Coulomb pair interaction.

The exchange calculation is a statement about the indicated spatial vectors; it does not deduce the spin-statistics assignment of any species, and combining the spatial states with spin requires a separately stipulated symmetry rule. No physical interpretation of $D$ , $X$ or the states is asserted.

Facts & Assumptions

Given: orthonormal vectors $u_{1}, \dots, u_{n} \in H$ , normalized orthogonal $u, v \in L^{2} (R^{3}; C)$ , a bounded measurable even $W$ , the pair multiplier $M_{W}$ on $L^{2} (R^{6})$ , the numbers $D, X$ , and $κ > 0$ .

[F1]

The permutation unitaries satisfy $U_{π} (v_{1} \otimes \dots \otimes v_{n}) = v_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes v_{{\hat{π}}^{- 1} (n)}$ , $P_{-}^{(n)} = \frac{1}{n!} \sum_{π} sgn (π) U_{π}$ , and $P_{-}^{(n)} (u_{1} \otimes \dots \otimes u_{n}) = 0$ whenever two of the $u_{i}$ coincide; elementary tensors have norm $∥ v_{1} \otimes \dots \otimes v_{n} ∥ = ∥ v_{1} ∥ \dots ∥ v_{n} ∥$ (Permutation sector projections on the n-fold tensor power, Hilbert tensor construction: inner product, completion and product bases).

[F2]

Cauchy-Schwarz and the $L^{2}$ product structure: for $f, g \in L^{2}$ , $∥ f g ∥_{1} \leq ∥ f ∥_{2} ∥ g ∥_{2}$ , and for bounded measurable $W$ , $∣ W (x - y) ∣ \leq ∥ W ∥_{\infty}$ (Cauchy-Schwarz inequality for $L^{2}$ , The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

[F3]

Hardy for $W^{1, 2} (R^{3})$ : $∥ w / ∣ \cdot ∣ ∥_{2} \leq 2 ∥ \nabla w ∥_{2}$ (Three-dimensional Hardy bound (c)).

[F4]

Gaussian integrals and transforms: $\int_{R} e^{- s^{2}} d s = \sqrt{π}$ ; for $t > 0$ the Euclidean Fourier transform of $G_{t} (x) : = e^{- t ∣ x ∣^{2}}$ is $\hat{G_{t}} (ξ) = (π / t)^{3 / 2} e^{- π^{2} ∣ ξ ∣^{2} / t} > 0$ ; the identity $1 / r = π^{- 1 / 2} \int_{0}^{\infty} s^{- 1 / 2} e^{- s r^{2}} d s$ for $r > 0$ follows by the substitution $s = u^{2} / r^{2}$ (The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ , Euclidean Gaussian transform with the 2π normalization, Plancherel theorem).

[F5]

Tonelli for $σ$ -finite product spaces permits the interchange of the order of integration for nonnegative measurable integrands (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F6]

The Euclidean product unitary $Θ$ satisfies $Θ (f \otimes g) (x, y) = f (x) g (y)$ ; because $M_{W}$ acts on $L^{2} (R^{6})$ , the spatial pairing is $⟨ M_{W} Θ (f \otimes g), Θ (f^{'} \otimes g^{'}) ⟩ = \int f (x) g (y) W (x - y) \overline{f^{'} (x) g^{'} (y)} d x d y$ , in the inner product linear in the first argument (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})). Once Tonelli establishes absolute integrability, Fubini applies separately to real and imaginary parts (Fubini's theorem for L^1 functions on a sigma-finite product).

Proof

1.1F1algebra

[Slater norm and identity] For $σ, π \in S_{n}$ , the tensor inner product is $⟨ u_{\hat{σ} (1)} \otimes \dots \otimes u_{\hat{σ} (n)}, u_{\hat{π} (1)} \otimes \dots \otimes u_{\hat{π} (n)} ⟩ = \prod_{i = 1}^{n} ⟨ u_{\hat{σ} (i)}, u_{\hat{π} (i)} ⟩$ by [F1], which equals $1$ if $σ = π$ and $0$ otherwise because the $u_{i}$ are orthonormal. Hence the $n!$ summands of $\sqrt{n!} S$ are mutually orthogonal of norm $1$ , so $∥ \sqrt{n!} S ∥^{2} = n!$ and $∥ S ∥ = 1$ . Moreover $P_{-}^{(n)} (u_{1} \otimes \dots \otimes u_{n}) = \frac{1}{n!} \sum_{π} sgn (π) u_{\hat{π} (1)} \otimes \dots \otimes u_{\hat{π} (n)} = \frac{1}{\sqrt{n!}} S$ , using $U_{π} (u_{1} \otimes \dots \otimes u_{n}) = u_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes u_{{\hat{π}}^{- 1} (n)}$ and reindexing $π \mapsto π^{- 1}$ . This proves (a).

1.2F1

[Exclusion] The assertion (b) is exactly [F1]'s vanishing statement for $P_{-}^{(n)}$ on a tensor with two equal factors; in particular for $n = 2$ , $P_{-}^{(2)} (u \otimes u) = 0$ .

1.3F2F6algebra

[Two-orbital expansion] Assume the hypotheses of (c). Absolute integrability of the four products $∣ u (x) ∣^{2} ∣ v (y) ∣^{2} ∣ W (x - y) ∣$ , $∣ \overline{u (x)} v (x) u (y) \overline{v (y)} W (x - y) ∣, \dots$ follows from [F2]: e.g. $\int \int ∣ u (x) ∣^{2} ∣ v (y) ∣^{2} ∣ W ∣ \leq ∥ W ∥_{\infty} ∥ u ∥_{2}^{2} ∥ v ∥_{2}^{2} = ∥ W ∥_{\infty}$ and $\int \int ∣ \overline{u} v ∣ ∣ u \overline{v} ∣ ∣ W ∣ \leq ∥ W ∥_{\infty} (\int ∣ u v ∣)^{2} \leq ∥ W ∥_{\infty} ∥ u ∥_{2}^{2} ∥ v ∥_{2}^{2}$ . Expanding the four terms in the tensor representation of [F6] gives $⟨ M_{W} Θ φ_{\pm}, Θ φ_{\pm} ⟩ = \frac{1}{2} [D \pm 2 Re X + D^{'}]$ , where $D^{'} = \int \int ∣ v (x) ∣^{2} ∣ u (y) ∣^{2} W (x - y)$ ; replacing $(x, y)$ by $(y, x)$ and using $W (x - y) = W (y - x)$ gives $D^{'} = D$ , so the expectation is $D \pm Re X$ . Since $W$ is real and even, exchanging the two integration variables in $X$ shows $X = \overline{X}$ , so $X \in R$ and $Re X = X$ ; the same symmetry with real $W$ gives $D \in R$ . This proves (c).

1.4F3F5algebra

[Absolute convergence for the Coulomb kernel] Assume the hypotheses of (d). For the direct term, by [F3] applied in the $y$ -variable for a.e. $x$ and Cauchy-Schwarz, $\int ∣ v (y) ∣^{2} / ∣ x - y ∣ d y \leq 2 ∥ v ∥_{2} ∥ \nabla v ∥_{2} < \infty$ , so with Tonelli [F5] $\int \int \frac{∣ u (x) ∣^{2} ∣ v (y) ∣^{2}}{∣ x - y ∣} = \int ∣ u (x) ∣^{2} (\int \frac{∣ v (y) ∣^{2}}{∣ x - y ∣} d y) d x \leq 2 ∥ v ∥_{2} ∥ \nabla v ∥_{2} ∥ u ∥_{2}^{2} < \infty$ . For the crossed term use $2 ∣ u (x) v (y) v (x) u (y) ∣ \leq ∣ u (x) v (y) ∣^{2} + ∣ v (x) u (y) ∣^{2}$ and the same bound to conclude $\iint \frac{∣ \overline{u (x)} v (x) u (y) \overline{v (y)} ∣}{∣ x - y ∣} < \infty$ . Hence $D, X$ are absolutely convergent and $D = κ \int \int ∣ u ∣^{2} ∣ v ∣^{2} / ∣ x - y ∣ \geq 0$ .

2.1F2F4F5F6step 1.4algebra

[Gaussian positivity with only $L^{1}$ data] Put $f = \overline{u} v \in L^{1}$ by [F2], with no assumption that $f \in L^{2}$ . Step 1.4 gives $\iint ∣ f (x) f (y) ∣ / ∣ x - y ∣ < \infty$ . The Gaussian representation [F4], Tonelli on absolute values [F5] and complex Fubini [F6] therefore give $X = κ π^{- 1 / 2} \int_{0}^{\infty} s^{- 1 / 2} I_{s} d s$ , where $I_{s} : = \iint f (x) \overline{f (y)} e^{- s ∣ x - y ∣^{2}} d x d y$ . For $s > 0$ completion of the square and the three-dimensional Gaussian integral give $e^{- s ∣ x - y ∣^{2}} = (4 s / π)^{3 / 2} \int_{R^{3}} e^{- 2 s ∣ x - z ∣^{2}} e^{- 2 s ∣ y - z ∣^{2}} d z$ . The triple integral of its absolute integrand is $(π / (4 s))^{3 / 2} \iint ∣ f (x) f (y) ∣ e^{- s ∣ x - y ∣^{2}} d x d y \leq (π / (4 s))^{3 / 2} ∥ f ∥_{1}^{2}$ , finite by Tonelli. Complex Fubini then yields $I_{s} = (4 s / π)^{3 / 2} \int_{R^{3}} ∣ \int f (x) e^{- 2 s ∣ x - z ∣^{2}} d x ∣^{2} d z \geq 0$ . The outer integral is absolutely convergent by the first bound from step 1.4 and the Gaussian kernel identity, so $X \geq 0$ and is real. Since $D \geq 0$ , the two absolutely convergent interaction expectations satisfy $D - X \leq D + X$ under the full original $W^{1, 2}$ hypotheses.

3.1step 1.1step 1.2step 1.3step 1.4step 2.1algebra∎

[Conclusion and scope] Steps 1.1 and 1.2 prove (a) and (b), steps 1.3 and 1.4 prove (c), and step 2.1 proves (d). The final paragraph of the statement records that the spin-statistics assignment and any physical interpretation are separate assumptions: the symmetric ( $+$ ) and antisymmetric ( $-$ ) spatial vectors are mathematical vectors, and combining them with spin requires a stipulated symmetry rule. This completes the proof.

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

Cutoff dipole relative bound

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $α > 0$ , let $h$ be the transverse space and $H_{f} = ⨁_{n} T_{n}$ the free transverse Fock Hamiltonian of Transverse continuum free Fock Hamiltonian, and let $A_{\pm} (g)$ , $g \in h$ , be the maximal creation and annihilation operators on $F (h)$ of Closed adjoint sector domains. Fix a cutoff $Λ > 0$ and vectors $g_{1}, \dots, g_{J} \in h$ ( $J \geq 1$ ) with $supp g_{j} \subseteq {k : 0 < ε (k) \leq Λ} and \frac{g_{j}}{\sqrt{ε}} \in h, ε (k) = α ∣ k ∣,$ and define the symmetric operator $ϕ (g) : = A_{-} (g) + A_{+} (g), D (ϕ (g)) : = D (H_{f}) .$ Let $d \geq 1$ , let $H_{a}$ be a Hermitian $d \times d$ matrix, and let $D_{1}, \dots, D_{J}$ be Hermitian $d \times d$ matrices with operator norms $∥ D_{j} ∥$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum). Then:

(a) for every $Ψ \in D (H_{f})$ , $∥ A_{-} (g) Ψ ∥^{2} \leq ∥ \frac{g}{\sqrt{ε}} ∥^{2} ⟨ H_{f} Ψ, Ψ ⟩, ∥ A_{+} (g) Ψ ∥^{2} = ∥ A_{-} (g) Ψ ∥^{2} + ∥ g ∥^{2} ∥ Ψ ∥^{2};$ in particular $Ψ \in D (A_{+} (g)) \cap D (A_{-} (g))$ and for every $η > 0$ there is a finite $C_{η}$ with $∥ ϕ (g) Ψ ∥ \leq η ∥ H_{f} Ψ ∥ + C_{η} ∥ Ψ ∥;$

(b) $ϕ (g)$ is symmetric on $D (H_{f})$ and has $H_{f}$ -bound zero (Relative boundedness with respect to an operator, Symmetric, self-adjoint and essentially self-adjoint operators);

(c) the free part $H_{0} : = H_{a} \otimes I + I \otimes H_{f}$ with domain $C^{d} \otimes D (H_{f})$ is self-adjoint on $C^{d} \otimes F (h)$ (Hilbert space); for every $η_{0} \in R$ the operator $H : = H_{0} + η_{0} \sum_{j = 1}^{J} D_{j} \otimes ϕ (g_{j}), D (H) : = C^{d} \otimes D (H_{f}),$ is self-adjoint on $C^{d} \otimes F (h)$ and bounded below (Kato-Rellich theorem, Adjoint of a densely defined operator).

The product $D_{j} \otimes ϕ (g_{j})$ and the coupling $η_{0}$ carry the dimension of energy in a physical normalisation; this normalisation is a model choice and no physical statement about radiation or spontaneous emission is made here.

Facts & Assumptions

Given: the cutoff $Λ > 0$ and vectors $g_{j}$ with $supp g_{j} \subseteq {0 < ε \leq Λ}$ , the maps $A_{\pm} (g_{j})$ and $ϕ (g_{j})$ on $F (h)$ , the free Hamiltonian $H_{f}$ with sector multipliers $T_{n}$ and number projections $Π_{n}$ , Hermitian matrices $H_{a}, D_{j}$ on $C^{d}$ , and $Ψ, Φ \in D (H_{f})$ .

[F1]

$A_{\pm} (g)$ are closed operators with $A_{+} (g) = A_{-} (g)^{*}$ and $A_{-} (g) = A_{+} (g)^{*}$ ; on the core $F_{0}$ the sector formulas $(A_{+} (g) ψ)_{n} = \sqrt{n} P_{+}^{(n)} (g \otimes ψ_{n - 1})$ and $(A_{-} (g) ψ)_{n} = \sqrt{n + 1} Ψ_{g}^{(n)} (ψ_{n + 1})$ hold, where $Ψ_{g}^{(n)}$ is the contraction in the first tensor factor with coefficient $⟨ \cdot, g ⟩$ (Closed adjoint sector domains, Creation contraction bounds).

[F2]

$H_{f} = ⨁_{n} T_{n}$ is self-adjoint and positive with $T_{n} = ε_{n} ∣_{P_{+}^{(n)}}$ , $ε_{n} (k_{1}, \dots, k_{n}) = \sum_{i \leq n} ε (k_{i})$ , and $⟨ H_{f} Ψ, Ψ ⟩ = \sum_{n} ⟨ ε_{n} Ψ_{n}, Ψ_{n} ⟩ < \infty$ for $Ψ \in D (H_{f})$ (Transverse continuum free Fock Hamiltonian).

[F3]

Under the iterated Euclidean product unitary, $h^{\otimes m}$ is identified with a closed subspace of $L^{2} ((R^{3})^{m}; (C^{3})^{\otimes m})$ ; elements of $ran P_{+}^{(m)}$ correspond to tensor-valued functions $F$ invariant under simultaneous momentum and polarization-factor permutations, so $∥ F ∥$ is invariant under momentum permutations, and the energy multiplier is $ε_{m} (k_{1}, \dots, k_{m}) = \sum_{i = 1}^{m} ε (k_{i})$ . For $F$ in its operator domain, $⟨ ε_{m} F, F ⟩ = \sum_{i = 1}^{m} \int ε (k_{i}) ∥ F (k_{1}, \dots, k_{m}) ∥^{2}$ , and symmetry makes each integral equal to $m^{- 1} ⟨ ε_{m} F, F ⟩$ (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Hilbert tensor construction: inner product, completion and product bases, Permutation sector projections on the n-fold tensor power).

[F4]

Cauchy-Schwarz for the weighted $L^{2}$ integrals in step 1.1 follows from Cauchy-Schwarz inequality for $L^{2}$ ; Cauchy-Schwarz in the Fock Hilbert space, used for $⟨ H_{f} Ψ, Ψ ⟩$ , follows from Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs.

[F5]

For $Ψ \in F_{0}$ the identity $∥ A_{+} (g) Ψ ∥^{2} = ∥ A_{-} (g) Ψ ∥^{2} + ∥ g ∥^{2} ∥ Ψ ∥^{2}$ holds (Closed adjoint sector domains (d)).

[F6]

Kato-Rellich: if $A$ is self-adjoint and $B$ is symmetric with $D (A) \subseteq D (B)$ and $A$ -bound less than $1$ , then $A + B$ on $D (A)$ is self-adjoint and bounded below when $A$ is (Kato-Rellich theorem, Relative boundedness with respect to an operator).

[F7]

Tonelli applies to the nonnegative squared norms on the sigma-finite Lebesgue product spaces; Fubini applies to absolutely integrable scalar component pairings (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product).

Proof

1.1F1F2F3F4F7algebra

[Contraction estimate] Let $Ψ \in D (H_{f})$ and $n \geq 0$ ; represent $Ψ_{n + 1} \in ran P_{+}^{(n + 1)}$ by its symmetric function $F$ under [F3]. For $v \in C^{3}$ define the first-factor contraction $C_{v} : (C^{3})^{\otimes (n + 1)} \to (C^{3})^{\otimes n}$ by $C_{v} (a \otimes b) = ⟨ a, v ⟩ b$ , extended linearly; finite orthonormal coordinates give $∥ C_{v} ∥ \leq ∥ v ∥$ . The contraction is represented by the tensor-valued function $G (k^{'}) = \int C_{g (k_{1})} F (k_{1}, k^{'}) d k_{1}$ : unweighted Cauchy–Schwarz gives integrable slices almost everywhere, Fubini identifies the pairing first for elementary tensors and then by density, and the bounded contraction formula [F1] extends the identity. Weighted Cauchy–Schwarz, tested against unit vectors of the finite-dimensional target tensor space, gives $∥ G (k^{'}) ∥^{2} \leq (\int ∥ g (k_{1}) ∥^{2} / ε (k_{1}) d k_{1}) (\int ε (k_{1}) ∥ F (k_{1}, k^{'}) ∥^{2} d k_{1})$ ; the null point $k_{1} = 0$ is omitted. Integrating over $k^{'}$ and using the symmetry computation of [F3] gives $∥ Ψ_{g}^{(n)} Ψ_{n + 1} ∥^{2} \leq ∥ g / \sqrt{ε} ∥^{2} \cdot \frac{1}{n + 1} ⟨ ε_{n + 1} F, F ⟩ = \frac{1}{n + 1} ∥ g / \sqrt{ε} ∥^{2} ⟨ ε_{n + 1} Ψ_{n + 1}, Ψ_{n + 1} ⟩$ . Hence $∥ (A_{-} (g) Ψ)_{n} ∥^{2} = (n + 1) ∥ Ψ_{g}^{(n)} Ψ_{n + 1} ∥^{2} \leq ∥ g / \sqrt{ε} ∥^{2} ⟨ ε_{n + 1} Ψ_{n + 1}, Ψ_{n + 1} ⟩$ ; summing over $n$ and using [F2] gives $∥ A_{-} (g) Ψ ∥^{2} \leq ∥ g / \sqrt{ε} ∥^{2} ⟨ H_{f} Ψ, Ψ ⟩$ .

1.2F2F3F4F6algebra

[The free part] Fix the standard orthonormal basis $(u_{a})_{a = 1}^{d}$ of $C^{d}$ . The map $(Ψ_{a})_{a = 1}^{d} \mapsto \sum_{a} u_{a} \otimes Ψ_{a}$ is an isometry from the finite Hilbert sum $F (h)^{d}$ onto $C^{d} \otimes F (h)$ , since elementary tensors span a dense subspace and the finite sum is complete. Under this identification $T : = I \otimes H_{f}$ has dense domain $D (H_{f})^{d}$ and acts componentwise. It is symmetric by [F2]. If $Φ \in D (T^{*})$ with $T^{*} Φ = Z$ , testing the adjoint identity on a vector with only component $a$ nonzero gives $Φ_{a} \in D (H_{f}^{*}) = D (H_{f})$ and $Z_{a} = H_{f} Φ_{a}$ for each $a$ . Conversely symmetry gives $T \subseteq T^{*}$ , so $T = T^{*}$ and $T \geq 0$ . The matrix amplification $R : = H_{a} \otimes I$ is bounded and self-adjoint: its entries act by scalar multiples of the identity, its adjoint entries are the conjugate transposed entries, and finite-dimensional Cauchy–Schwarz bounds its norm. It has $T$ -bound zero; hence [F6] makes $H_{0} = T + R$ self-adjoint on $D (H_{f})^{d} = C^{d} \otimes D (H_{f})$ and bounded below by $- ∥ R ∥$ , also directly from $T \geq 0$ and $∣ ⟨ R Ψ, Ψ ⟩ ∣ \leq ∥ R ∥ ∥ Ψ ∥^{2}$ . This componentwise adjoint argument establishes the general amplification here without applying the fixed- $N$ kinetic-operator interface.

2.1F1F5step 1.1algebra

[The norm identity on $D (H_{f})$ ] Let $Ψ \in D (H_{f})$ and let $Ψ^{(N)}$ be its truncations to sectors $\leq N$ . Step 1.1 gives $A_{-} (g) Ψ^{(N)} \to A_{-} (g) Ψ$ and $Ψ^{(N)} \to Ψ$ . For $N > M$ , $χ : = Ψ^{(N)} - Ψ^{(M)}$ is in $F_{0}$ , so [F5] gives $∥ A_{+} (g) χ ∥^{2} = ∥ A_{-} (g) χ ∥^{2} + ∥ g ∥^{2} ∥ χ ∥^{2} \to 0$ as $M, N \to \infty$ . Thus $A_{+} (g) Ψ^{(N)}$ is Cauchy; closedness of $A_{+} (g)$ from [F1] implies $Ψ \in D (A_{+} (g))$ and $A_{+} (g) Ψ^{(N)} \to A_{+} (g) Ψ$ . Passing to the limit in [F5] for $Ψ^{(N)}$ now proves $∥ A_{+} (g) Ψ ∥^{2} = ∥ A_{-} (g) Ψ ∥^{2} + ∥ g ∥^{2} ∥ Ψ ∥^{2}$ .

3.1F4step 1.1step 2.1algebra

[The relative bound] From steps 1.1 and 2.1, $∥ ϕ (g) Ψ ∥ \leq ∥ A_{-} (g) Ψ ∥ + ∥ A_{+} (g) Ψ ∥ \leq 2 ∥ g / \sqrt{ε} ∥ \sqrt{⟨ H_{f} Ψ, Ψ ⟩} + ∥ g ∥ ∥ Ψ ∥$ for $Ψ \in D (H_{f})$ ; since $⟨ H_{f} Ψ, Ψ ⟩ \leq ∥ H_{f} Ψ ∥ ∥ Ψ ∥$ by [F4], the inequality $2 \sqrt{a b} \leq η a + η^{- 1} b$ with $a = ∥ H_{f} Ψ ∥$ , $b = ∥ g / \sqrt{ε} ∥^{2} ∥ Ψ ∥$ gives, for every $η > 0$ , $∥ ϕ (g) Ψ ∥ \leq η ∥ H_{f} Ψ ∥ + C_{η} ∥ Ψ ∥$ with $C_{η} = η^{- 1} ∥ g / \sqrt{ε} ∥^{2} + ∥ g ∥$ . This proves (a) and shows the $H_{f}$ -bound zero.

4.1F1step 1.1step 2.1step 3.1algebra

[Symmetry of $ϕ (g)$ ] For $Ψ, Φ \in D (H_{f})$ , steps 1.1 and 2.1 put both vectors in the domains of the two closed operators, and the mutual adjointness $A_{+} (g) = A_{-} (g)^{*}$ of [F1] gives $⟨ ϕ (g) Ψ, Φ ⟩ = ⟨ A_{-} (g) Ψ, Φ ⟩ + ⟨ A_{+} (g) Ψ, Φ ⟩ = ⟨ Ψ, A_{+} (g) Φ ⟩ + ⟨ Ψ, A_{-} (g) Φ ⟩ = ⟨ Ψ, ϕ (g) Φ ⟩$ . By step 3.1, $ϕ (g)$ is symmetric on the dense domain $D (H_{f})$ with $H_{f}$ -bound zero, which is (b).

5.1F6step 3.1step 4.1step 1.2algebra

[Kato-Rellich] Let $B : = η_{0} \sum_{j} D_{j} \otimes ϕ (g_{j})$ with domain $C^{d} \otimes D (H_{f})$ . Since the $D_{j}$ are bounded and each $ϕ (g_{j})$ is symmetric with $H_{f}$ -bound zero (step 4.1, step 3.1), $B$ is symmetric and for every $η > 0$ there are finite constants with $∥ B Ψ ∥ \leq η ∥ H_{f} Ψ ∥ + C_{η} ∥ Ψ ∥$ ; as $H_{0}$ differs from $I \otimes H_{f}$ by the bounded operator $H_{a} \otimes I$ , the same estimate holds with $H_{f}$ replaced by $H_{0}$ up to constants, so the $H_{0}$ -bound of $B$ is zero and in particular less than $1$ . As $H_{0}$ is self-adjoint (step 1.2), [F6] gives that $H = H_{0} + B$ is self-adjoint on $C^{d} \otimes D (H_{f})$ and bounded below. This proves (c).

6.1step 1.1step 2.1step 3.1step 4.1step 1.2step 5.1algebra∎

[Conclusion and scope] Steps 1.1 and 2.1 prove (a), step 4.1 proves (b), and steps 1.2 and 5.1 prove (c). The final paragraph of the statement records that the normalisation of the coupling and of the dipole matrices is a model choice, and no physical radiation or emission statement is asserted. This completes the proof.

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

Vacuum emission Dyson convergence

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Adopt the objects and hypotheses of Cutoff dipole relative bound: $d \geq 1$ , Hermitian matrices $H_{a}, D_{1}, \dots, D_{J}$ on $C^{d}$ , cutoff $Λ > 0$ , transverse vectors $g_{j}$ with $supp g_{j} \subseteq {0 < ε \leq Λ}$ and $g_{j} / \sqrt{ε} \in h$ , coupling $η_{0} \in R$ , $H_{0} : = H_{a} \otimes I + I \otimes H_{f}, B : = η_{0} \sum_{j = 1}^{J} D_{j} \otimes ϕ (g_{j}), H : = H_{0} + B,$ all with domain $C^{d} \otimes D (H_{f})$ so that $H_{0}$ and $H$ are self-adjoint on that domain. Fix $ℏ > 0$ , a unit vector $v_{a} \in C^{d}$ and $Ψ_{0} : = v_{a} \otimes Ω$ , and define the interaction-picture family $V (t) : = e^{i t H_{0} / ℏ} B e^{- i t H_{0} / ℏ} (t \in R) .$ For $n \geq 0$ and $t \geq 0$ let $Ψ_{n} (t) : = (\frac{1}{i ℏ})^{⁣ n} \int_{0 < t_{1} < \dots < t_{n} < t} V (t_{n}) \dots V (t_{1}) Ψ_{0} d t_{1} \dots d t_{n}$ (iterated Bochner integral in $C^{d} \otimes D (H_{f})$ ; $Ψ_{0} (t) : = Ψ_{0}$ ). Put $C : = 2 \sum_{j = 1}^{J} ∥ D_{j} ∥ ∥ g_{j} ∥ / ℏ$ and $C^{'} : = C Λ$ . Then:

(a) for all $n \geq 0$ and $t \geq 0$ , $∥ Ψ_{n} (t) ∥ \leq \frac{(∣ η_{0} ∣ C t)^{n}}{\sqrt{n!}};$

(b) the vectors $Ψ_{n} (t)$ lie in $C^{d} \otimes D (H_{f})$ , the series $\sum_{n \geq 0} Ψ_{n} (t)$ converges in the graph norm of $H_{f}$ uniformly for $t$ in compact subsets of $[0, \infty)$ , and its sum $Ψ (t)$ satisfies $i ℏ \partial_{t} Ψ (t) = V (t) Ψ (t), Ψ (0) = Ψ_{0};$

(c) the Dyson sum solves the actual self-adjoint evolution: $e^{- i t H_{0} / ℏ} Ψ (t) = e^{- i t H / ℏ} Ψ_{0}$ for every $t \geq 0$ ;

(d) explicit tail bound: for $0 \leq t \leq T$ and $N \geq 1$ , $∥ e^{- i t H / ℏ} Ψ_{0} - e^{- i t H_{0} / ℏ} \sum_{n = 0}^{N - 1} Ψ_{n} (t) ∥ \leq \sum_{n \geq N} \frac{(∣ η_{0} ∣ C T)^{n}}{\sqrt{n!}} .$

The convergence is proved for the actual self-adjoint $H$ on the stated domain; this is not a formal manipulation of divergent series, and no all-time weak-coupling, Markov or exponential-decay statement is made.

Facts & Assumptions

Given: the cutoff data, self-adjoint $H_{0}, H$ on $G : = C^{d} \otimes D (H_{f})$ , $F : = I \otimes H_{f}$ , $B : = η_{0} \sum_{j} D_{j} \otimes ϕ (g_{j})$ , $U_{0} (t) : = e^{i t H_{0} / ℏ}$ , $V (t) : = U_{0} (t) B U_{0} (- t)$ , and the vacuum initial vector. Put $∥ x ∥_{G} : = ∥ x ∥ + ∥ F x ∥$ ; let $K_{m}$ be the closed subspace with photon numbers at most $m$ , including the atomic factor, and $G_{m} : = G \cap K_{m}$ . Define the coupling-independent constant $γ : = 2 \sum_{j} ∥ D_{j} ∥ ∥ g_{j} ∥$ , so $C = γ / ℏ$ .

[F1]

The cutoff relative-bound theorem gives symmetry of $B$ , its relative bound zero, and self-adjointness of $H_{0}, H$ on $G$ (Cutoff dipole relative bound (a)-(c)). In particular $B : G \to K$ is bounded for the graph norm, where $K = C^{d} \otimes F (h)$ ; since $H_{0} - F$ is bounded, the two graph norms are equivalent.

[F2]

The sector formulas give $∥ A_{\pm} (g) x ∥ \leq \sqrt{m + 1} ∥ g ∥ ∥ x ∥$ on $K_{m}$ , and each changes photon number by one; their sum has bound $2 \sqrt{m + 1} ∥ g ∥$ (Closed adjoint sector domains (b),(c), Creation contraction bounds (a),(b)). Contraction is conjugate-linear in its vector argument and satisfies $Ψ_{g} (u \otimes z) = ⟨ u, g ⟩ z$ .

[F3]

The free field operator is the positive closed sector sum with multiplier $\sum_{i} ε (k_{i})$ , and $H_{f} Ω = 0$ . Its sector projections commute with its unitary group (Transverse continuum free Fock Hamiltonian (a)-(c)). The Euclidean product identification, tensor density and permutation formulas are Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Hilbert tensor construction: inner product, completion and product bases and Permutation sector projections on the n-fold tensor power. AC supplies any DC prerequisite of the free-field supplier (AC implies DC implies countable choice).

[F4]

A finite Hermitian matrix has an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely). A self-adjoint operator's unitary group is strongly continuous, preserves its domain and commutes with it, and is differentiable on domain vectors (A self-adjoint operator generates a strongly continuous unitary group).

[F5]

A continuous Banach-valued curve on a compact interval is Bochner integrable; its primitive has the curve as derivative, including one-sided endpoint derivatives, and the integral obeys the norm bound and linearity (Fundamental theorem of calculus for Banach-valued continuous curves, Bochner integrability criterion, Bochner integral norm inequality, Linearity of the Bochner integral). Bounded linear maps commute with such integrals: this holds for simple approximants by linearity and passes to their $L^{1}$ limits by the map's norm bound.

Proof

technique · direct; prove finite-photon graph vertex bounds, integrate vertices in chronological order, sum in the graph norm and identify the resulting strong solution with the self-adjoint evolution

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

1.1F1F2F3F4algebra

[Free flow and vertex bounds] Diagonalizing the finite Hermitian $H_{a}$ and using [F3], [F4] gives $U_{0} (t) = e^{i t H_{a} / ℏ} \otimes e^{i t H_{f} / ℏ}$ . Thus $U_{0} (t)$ preserves each $K_{m}$ , commutes with $F$ on $G$ , and is isometric and strongly continuous for the graph norm: both $U_{0} (t) x$ and $F U_{0} (t) x = U_{0} (t) F x$ are continuous. On $K_{m}$ , the finite-sector extension of $B$ from [F2] satisfies $∥ B x ∥ \leq ∣ η_{0} ∣ γ \sqrt{m + 1} ∥ x ∥$ and takes values in $K_{m + 1}$ . Its free conjugate therefore satisfies $∥ V (t) x ∥ \leq ∣ η_{0} ∣ γ \sqrt{m + 1} ∥ x ∥$ . The factor $∣ η_{0} ∣$ is included exactly once in this bound; $γ$ and $C$ contain no coupling.

1.2F2F3algebraconstruct

[Cutoff commutator and graph-domain extension] For $g$ supported in ${0 < ε \leq Λ}$ , $ε g \in h$ and $∥ ε g ∥ \leq Λ ∥ g ∥$ . On symmetric elementary tensors with factors in $D (ε)$ , the sector multiplier and creation formula give $[H_{f}, A_{+} (g)] = A_{+} (ε g)$ . For contraction the removed factor contributes $- ⟨ ε u, g ⟩ = - ⟨ u, ε g ⟩$ , so $[H_{f}, A_{-} (g)] = - A_{-} (ε g)$ . These identities extend to every finite-photon graph vector, with no claim that all of $F_{0}$ lies in $D (H_{f})$ . Indeed, in sector $n$ first cut each mode to $ε \leq R$ ; dominated convergence in the Euclidean representation [F3] gives convergence of both the vector and its energy multiplier. On that cutoff subspace, energy is bounded by $n R$ , so finite sums of elementary cutoff tensors, symmetrized, approximate in graph norm by tensor density. This supplies a graph core in each $G_{m}$ . The field and its candidate commutator are bounded on $K_{m}$ by [F2], with respective bounds $2 \sqrt{m + 1} ∥ g ∥$ and $2 Λ \sqrt{m + 1} ∥ g ∥$ . Applying the core identity to a graph-approximating sequence makes the output and its energy image converge; closedness of $H_{f}$ proves graph-domain preservation and the identity on the limit. Hence for $x \in G_{m}$ , $B x \in G_{m + 1}$ and $F B x = B F x + Q_{B} x, Q_{B} : = η_{0} \sum_{j} D_{j} \otimes (A_{+} (ε g_{j}) - A_{-} (ε g_{j})), ∥ Q_{B} x ∥ \leq ∣ η_{0} ∣ γ Λ \sqrt{m + 1} ∥ x ∥ .$

2.1F1F2F4step 1.1step 1.2algebra

[Conjugated commutator and graph continuity] Since $F$ commutes with $U_{0} (t)$ , step 1.2 gives on $G_{m}$ $F V (t) x = V (t) F x + Q (t) x, Q (t) : = U_{0} (t) Q_{B} U_{0} (- t),$ with $∥ Q (t) x ∥ \leq ∣ η_{0} ∣ γ Λ \sqrt{m + 1} ∥ x ∥$ . The conjugations in $Q (t)$ are retained. Equivalently, the sector formulas give $D_{j} (t) = e^{i t H_{a} / ℏ} D_{j} e^{- i t H_{a} / ℏ}$ and $g_{j} (t) = e^{i t ε / ℏ} g_{j}$ , with $V (t) = η_{0} \sum_{j} D_{j} (t) \otimes ϕ (g_{j} (t)), Q (t) = η_{0} \sum_{j} D_{j} (t) \otimes (A_{+} (ε g_{j} (t)) - A_{-} (ε g_{j} (t))) .$ The same positive phase occurs in the annihilation vector argument because that argument is conjugate-linear, as in [F2]. The resulting graph bound is $∥ V (t) x ∥_{G} \leq ∣ η_{0} ∣ γ \sqrt{m + 1} (∥ F x ∥ + (1 + Λ) ∥ x ∥) .$ Thus $V (t) : G_{m} \to G_{m + 1}$ is uniformly bounded in $t$ for these graph norms. Strong graph continuity of $U_{0}$ , the fixed graph-bounded $B$ between these spaces, and the expression $U_{0} (t) B U_{0} (- t) x$ show that $t \mapsto V (t) x$ is graph continuous. On the full $G$ , [F1] and the graph isometry of $U_{0}$ additionally give $∥ V (t) x ∥ \leq K ∥ x ∥_{G}$ for one finite $K$ independent of $t$ , and strong continuity as a map $G \to K$ . No global map $V (t) : G \to G$ is asserted.

3.1F3step 1.1step 2.1algebrainduction

[Chronological products and their graph bounds] Define $P_{n} (t_{1}, \dots, t_{n}) : = V (t_{n}) \dots V (t_{1}) Ψ_{0}$ , with $P_{0} = Ψ_{0}$ . At the $k$ -th vertex, the earlier product is in $G_{k - 1}$ , so steps 1.1 and 2.1 give a factor $∣ η_{0} ∣ γ \sqrt{k}$ in its Hilbert bound. Consequently $P_{n} \in G_{n}$ and $∥ P_{n} ∥ \leq (∣ η_{0} ∣ γ)^{n} \sqrt{n!}$ . The graph commutator gives recursively $∥ F P_{k} ∥ \leq ∣ η_{0} ∣ γ \sqrt{k} (∥ F P_{k - 1} ∥ + Λ ∥ P_{k - 1} ∥)$ ; since $F Ψ_{0} = 0$ , induction yields $∥ F P_{n} ∥ \leq n Λ (∣ η_{0} ∣ γ)^{n} \sqrt{n!}$ . Each product is jointly continuous in its time variables for the graph norm: at each stage use strong graph continuity of the next $V$ on a fixed vector and its uniform graph bound to control the changing previous vector. For $n = 0$ the Hilbert bound is $1$ and the energy bound is $0$ , including when $∣ η_{0} ∣ γ = 0$ .

4.1F3F5step 3.1algebra

[Graph Bochner terms and the genuine recurrence] The graph space $G$ is Banach: a graph-Cauchy sequence has limits of both the vectors and their $F$ images in the Hilbert space, and closedness of $F$ identifies them as a graph limit. Each $G_{n}$ is closed in it. Step 3.1 and [F5] therefore make every iterated chronological integral a Bochner integral in $G_{n}$ . Its graph and Hilbert norm bounds follow by integrating the preceding product bounds over the simplex; the volume is $t^{n} / n!$ , obtained recursively from $\int_{0}^{t} s^{n - 1} / (n - 1)! d s = t^{n} / n!$ . Each factor of the integral contributes $1 / ℏ$ , giving $∥ Ψ_{n} (t) ∥ \leq \frac{(∣ η_{0} ∣ C t)^{n}}{\sqrt{n!}}, ∥ F Ψ_{n} (t) ∥ \leq \frac{n Λ (∣ η_{0} ∣ C t)^{n}}{\sqrt{n!}} .$ For $n = 0$ these bounds mean $1$ and $0$ . Integrating the latest vertex last gives the actual left-vertex recurrence $Ψ_{n + 1} (t) = \frac{1}{i ℏ} \int_{0}^{t} V (s) Ψ_{n} (s) d s .$ For fixed $s$ , move $V (s)$ through the inner integral as a bounded linear map $G_{n} \to G_{n + 1}$ by [F5]; the earlier vertices have times below $s$ , with $V (s)$ on the left. This also proves graph continuity of each term, and proves (a) and graph-domain membership in (b).

5.1step 4.1algebra

[Uniform graph convergence] Fix $T < \infty$ and put $a : = ∣ η_{0} ∣ C T$ . The graph majorant is $\sum_{n \geq 0} (1 + n Λ) a^{n} / \sqrt{n!}$ . For $a > 0$ the ratio of consecutive scalar terms tends to zero: it is $a (1 + (n + 1) Λ) / ((1 + n Λ) \sqrt{n + 1})$ . It is eventually at most $1 / 2$ , proving convergence by a geometric tail comparison. If $a = 0$ , every positive-order term is zero and only the zeroth term remains. Thus the series converges uniformly for $0 \leq t \leq T$ in the Banach graph space, and its sum $Ψ (t)$ is graph continuous and belongs to $D (H) = G$ .

6.1F5step 2.1step 4.1step 5.1algebra

[Integral equation and Hilbert derivative] The uniform bound $∥ V (s) x ∥ \leq K ∥ x ∥_{G}$ from step 2.1 makes $V (s) \sum_{n \leq M} Ψ_{n} (s)$ converge uniformly in Hilbert norm on $[0, T]$ to $V (s) Ψ (s)$ . This limiting curve is continuous by graph continuity of $Ψ$ , strong continuity of $V : G \to K$ , and the same uniform bound. Sum the recurrences of step 4.1 and use the Bochner norm inequality [F5] to pass to the limit in their Hilbert integrals. The result is $Ψ (t) = Ψ_{0} + (i ℏ)^{- 1} \int_{0}^{t} V (s) Ψ (s) d s$ . The Banach-valued fundamental theorem now gives $Ψ \in C^{1} ([0, T]; K)$ , with the right derivative at zero, and $i ℏ Ψ^{'} (t) = V (t) Ψ (t)$ , $Ψ (0) = Ψ_{0}$ . Along with step 5.1 this proves (b); graph continuity is used for domain control, while the differential equation is a Hilbert-space derivative.

7.1F1F4step 5.1step 6.1algebra

[Identification with the self-adjoint dynamics] Put $Y (t) : = U_{0} (- t) Ψ (t)$ . It is graph continuous and belongs to $D (H)$ . The difference quotient splits into the difference of the free group on the fixed domain vector $Ψ (t)$ and the free group times the difference quotient of $Ψ$ , so [F4] and step 6.1 justify the product derivative in Hilbert norm. Thus $i ℏ Y^{'} = H_{0} Y + U_{0} (- t) V (t) Ψ = H_{0} Y + B Y = H Y$ and $Y (0) = Ψ_{0}$ . By [F1], [F4], the Stone curve $e^{- i t H / ℏ} Ψ_{0}$ is another strong domain-valued solution because $Ψ_{0} \in G = D (H)$ . For their difference $w$ , symmetry of $H$ gives $\frac{d}{d t} ∥ w (t) ∥^{2} = 2 Re ⟨ w^{'} (t), w (t) ⟩ = 2 Re (- \frac{i}{ℏ} ⟨ H w (t), w (t) ⟩) = 0.$ Both initial values agree, so $w = 0$ , proving (c) for every finite $t \geq 0$ .

8.1F3F4step 4.1step 7.1algebra∎

[Tail, vacuum and zero branches] By step 7.1 and unitarity, the truncation error at time $t$ equals in norm the interaction-picture tail, so it is at most $\sum_{n \geq N} (∣ η_{0} ∣ C t)^{n} / \sqrt{n!}$ ; for $0 \leq t \leq T$ this is at most the stated $T$ tail, proving (d). The vacuum zeroth term has $F Ψ_{0} = 0$ and norm one. At $t = 0$ all positive-order integrals vanish. If $η_{0} = 0$ , or if $γ = 0$ (all contributing $D_{j}$ or $g_{j}$ zero), then $B = V = 0$ , all positive-order terms vanish, the interaction-picture sum is exactly $Ψ_{0}$ , and the tail is zero for every $N \geq 1$ . The argument supplies finite-time convergence and the actual self-adjoint evolution, with the original scope limitations retained.

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

Golden-rule coefficient approximate identity

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $ℏ > 0$ .

(a) The golden-rule kernel. For $t > 0$ define $K_{t} (x) : = \frac{1}{ℏ^{2} t} ∣ \int_{0}^{t} e^{i x s / ℏ} d s ∣^{2}, x \in R .$ Then $K_{t}$ is continuous, nonnegative and even, and for $x \neq 0$ $K_{t} (x) = \frac{4 \sin^{2} (x t / (2 ℏ))}{x^{2} t}, K_{t} (0) = \frac{t}{ℏ^{2}};$ moreover for every $δ > 0$ and every $x$ with $∣ x ∣ \geq δ$ , $K_{t} (x) \leq \frac{4}{x^{2} t} \leq \frac{4}{δ^{2} t}, and \int_{R} K_{t} (x) d x = \frac{2 π}{ℏ} .$

(b) Approximate identity. Let $J \in L^{1} (R; C)$ (including real-valued functions) (Integrable real and complex functions, and their integrals, The space $L^{p} (μ)$ as the quotient by null functions) and let $Δ \in R$ . Choose a measurable representative $J : R \to C$ . Continuity at $Δ$ means that for every $ϵ > 0$ there is $δ > 0$ such that $∣ J (y) - J (Δ) ∣ < ϵ$ whenever $∣ y - Δ ∣ < δ$ ; for real-valued $J$ this is Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, and for complex-valued $J$ it is equivalent to continuity of its real and imaginary parts. If this condition holds, then $\lim_{t \to \infty} \int_{R} K_{t} (x) J (Δ + x) d x = \frac{2 π}{ℏ} J (Δ)$ (For a complex-valued function $Q : (0, \infty) \to C$ , $Q (t) \to L$ means that for every $ϵ > 0$ there is $T > 0$ with $∣ Q (t) - L ∣ < ϵ$ for all $t > T$ ; equivalently its real and imaginary parts have the real limits in Limits at $+ \infty$ and $- \infty$ , and infinite limits at a point.)

(c) First-order emission coefficient. Adopt the objects and hypotheses of Cutoff dipole relative bound: $α > 0$ , the transverse space $h = ran M_{Π}$ of Transverse continuum free Fock Hamiltonian with $ε (k) = α ∣ k ∣$ , the cutoff $Λ > 0$ , the vectors $g_{1}, \dots, g_{J} \in h$ ( $J \geq 1$ ) with $supp g_{j} \subseteq {k : 0 < ε (k) \leq Λ}, \frac{g_{j}}{\sqrt{ε}} \in h,$ the Hermitian $d \times d$ matrices $H_{a}, D_{1}, \dots, D_{J}$ and the coupling $η_{0} \in R$ . Let $v_{a}, v_{b}$ be orthonormal eigenvectors of $H_{a}$ with real eigenvalues $E_{a}, E_{b}$ satisfying $H_{a} v_{a} = E_{a} v_{a}$ , $H_{a} v_{b} = E_{b} v_{b}$ , and let $Δ : = E_{a} - E_{b} > 0$ . Put $c_{j} : = ⟨ D_{j} v_{a}, v_{b} ⟩ (C^{d} inner product), M : = \sum_{j = 1}^{J} c_{j} g_{j} \in h,$ and define the finite Borel measure $ν_{b} (A) : = \int_{R^{3}} 1_{A} (ε (k)) ∥ M (k) ∥^{2} d^{3} k, A \subseteq R Borel,$ and for $t \geq 0$ the amplitude $A_{b} (k, t) : = - \frac{i η_{0}}{ℏ} M (k) \int_{0}^{t} e^{i (ε (k) - Δ) s / ℏ} d s .$ Then:

(i) $ν_{b}$ is a finite Borel measure on $R$ with $ν_{b} (R ∖ (0, Λ]) = 0$ , and $\int_{R} f d ν_{b} = \int_{R^{3}} f (ε (k)) ∥ M (k) ∥^{2} d^{3} k$ for every Borel $f \geq 0$ ;

(ii) $A_{b} (\cdot, t) \in h$ for every $t \geq 0$ , and for $t > 0$ , $∥ A_{b} (\cdot, t) ∥_{h}^{2} = η_{0}^{2} t \int_{R} K_{t} (E - Δ) d ν_{b} (E) .$ At $t = 0$ the amplitude is zero, so its squared norm is zero; no $K_{0}$ is used.

(iii) if $ν_{b} = J_{b} λ$ for some $J_{b} \in L^{1} (λ)$ with $J_{b} = 0$ $λ$ -a.e. on $R ∖ (0, Λ]$ and the representative $J_{b}$ continuous at $Δ$ , then for $0 < Δ < Λ$ $\lim_{t \to \infty} \frac{1}{t} ∥ A_{b} (\cdot, t) ∥_{h}^{2} = \frac{2 π η_{0}^{2}}{ℏ} J_{b} (Δ);$ if instead $Δ > Λ$ , then the same limit exists and equals $0$ . The boundary case $Δ = Λ$ is not asserted.

Motivation and scope. In the cutoff dipole model of Cutoff dipole relative bound the vector $A_{b} (\cdot, t)$ is the explicit single-vertex amplitude with the atom passing from $v_{a}$ to $v_{b}$ and one photon emitted; $∥ A_{b} (\cdot, t) ∥^{2}$ is the corresponding first-order photon coefficient, and (iii) is the golden-rule limit of that coefficient divided by $t$ . This item computes only the limit of the explicit coefficient, under the stated hypotheses on the spectral density $J_{b}$ . It asserts no approximation of the exact model evolution, no all-time or weak-coupling limit, no Markov property and no exact exponential decay, and it reports no measurement or empirical rate. The parameter $ℏ$ , the coupling $η_{0}$ and the matrices $D_{j}$ enter as mathematical normalisation data of the model; their physical identification is a separate assumption.

Facts & Assumptions

Given: the coupling data $(ℏ, α, Λ, η_{0}, g_{j}, H_{a}, D_{j}, v_{a}, v_{b})$ of the cutoff model, the kernel $K_{t}$ , a function $J \in L^{1} (λ)$ continuous at $Δ$ , and the pushforward measure $ν_{b}$ with its possible density $J_{b}$ .

[F1]

For real $θ$ , $∣ e^{i θ} - 1 ∣^{2} = 4 \sin^{2} (θ / 2)$ , $∣ e^{i θ} ∣ = 1$ , and $∣ \sin ∣ \leq 1$ ; the derivative of $\sin$ at $0$ gives $\lim_{a \to 0} \sin (a) / a = 1$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The derivatives of sine and cosine are cosine and minus sine).

[F2]

Fourier/Plancherel package: for $g \in L^{1} (R)$ the integral transform $\hat{g} (ξ) = \int g (u) e^{- 2 π i ξ u} d u$ is defined, on $L^{1} \cap L^{2}$ it agrees with the $L^{2}$ transform, and $F_{2} : L^{2} \to L^{2}$ is a unitary isometry preserving the norm (Fourier transform on complex L1 classes, Agreement of the integral and L2 transforms, Plancherel theorem).

[F3]

Every interval $[0, T]$ , $T \geq 0$ , has Lebesgue measure $T$ , so $∥ 1_{[0, T]} ∥_{2}^{2} = T$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F4]

Change of variables: if $U, V \subseteq R$ are open, $T : U \to V$ is a $C^{1}$ diffeomorphism and $f \in L^{1} (λ)$ , then $\int_{V} f d λ = \int_{U} (f \circ T) ∣ T^{'} ∣ d λ$ (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions); this applies in particular to the translations $x \mapsto Δ + x$ and the dilations $x \mapsto - 2 π x$ of $R$ (both with $∣ T^{'} ∣$ constant). The primitive $- 1 / x$ and monotone convergence on bounded intervals give $\int_{∣ x ∣ \geq δ} x^{- 2} d x = 2 / δ$ for $δ > 0$ (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ , Monotone convergence for the integral).

[F5]

Measure and integral calculus: monotone convergence; additivity and linearity of the Lebesgue integral ( $L^{1}$ linearity, nonnegative additivity); for a nonnegative measurable $f$ the set function $A \mapsto \int_{A} f d μ$ is a measure; a measure with density $h$ is given by $(h μ) (A) = \int_{A} h d μ$ (Monotone convergence for the integral, Additivity of the nonnegative Lebesgue integral, The Lebesgue integral is linear on $L^{1} (μ)$ , The indefinite integral of a nonnegative measurable function is a measure, The measure with density $f$ relative to $μ$ ).

[F6]

Model objects: $h = {f \in L^{2} (R^{3}; C^{3}) : k \cdot f (k) = 0 a.e.}$ is a closed linear subspace, $ε (k) = α ∣ k ∣$ is continuous, $g_{j} \in h$ with $supp g_{j} \subseteq {0 < ε \leq Λ}$ and $g_{j} / \sqrt{ε} \in h$ ; $H_{a}, D_{j}$ are Hermitian with $v_{a}, v_{b}$ orthonormal eigenvectors (Transverse continuum free Fock Hamiltonian, Cutoff dipole relative bound).

[F7]

Cauchy-Schwarz in the finite-dimensional Hilbert space $C^{J}$ gives $∣ \sum_{j} c_{j} g_{j} (k) ∣^{2} \leq (\sum_{j} ∣ c_{j} ∣^{2}) (\sum_{j} ∥ g_{j} (k) ∥^{2})$ for a.e. $k$ , hence $∥ M ∥_{L^{2}}^{2} \leq (\sum_{j} ∣ c_{j} ∣^{2}) (\sum_{j} ∥ g_{j} ∥^{2}) < \infty$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs, The space $L^{p} (μ)$ as the quotient by null functions).

[F8]

Use the modulus definitions of continuity and limits stated in (b); they agree with Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point and Limits at $+ \infty$ and $- \infty$ , and infinite limits at a point for real functions. For complex functions the inequalities $∣ Re z ∣, ∣ Im z ∣ \leq ∣ z ∣ \leq ∣ Re z ∣ + ∣ Im z ∣$ prove equivalence with the two real-component conditions.

Proof

1.1F1algebra

[Explicit kernel and pointwise bounds] For $x \neq 0$ , integrating the exponential gives $\int_{0}^{t} e^{i x s / ℏ} d s = ℏ (e^{i x t / ℏ} - 1) / (i x)$ , so by [F1] $K_{t} (x) = ℏ^{2} ∣ e^{i x t / ℏ} - 1 ∣^{2} / (x^{2} ℏ^{2} t) = 4 \sin^{2} (x t / (2 ℏ)) / (x^{2} t)$ ; at $x = 0$ the defining integral equals $t$ and $K_{t} (0) = t / ℏ^{2}$ . With $a = x t / (2 ℏ)$ the identity $4 \sin^{2} (x t / (2 ℏ)) / (x^{2} t) = (t / ℏ^{2}) (\sin a / a)^{2}$ and [F1] give continuity at $0$ and on $R$ ; $K_{t} \geq 0$ and $K_{t}$ is even by [F1]. For $∣ x ∣ \geq δ > 0$ the same formula and $∣ \sin ∣ \leq 1$ give $K_{t} (x) \leq 4 / (x^{2} t) \leq 4 / (δ^{2} t)$ .

1.2F1F2F3F4algebra

[Kernel mass] Put $T : = t / ℏ$ and $g : = 1_{[0, T]} \in L^{1} \cap L^{2}$ , $∥ g ∥_{2}^{2} = T$ by [F3]. By [F2], $\hat{g} (ξ) = \int_{0}^{T} e^{- 2 π i ξ u} d u$ satisfies $\int_{R} ∣ \hat{g} ∣^{2} = ∥ g ∥_{2}^{2} = T$ . For every real $x$ , $\int_{0}^{T} e^{i x u} d u = \hat{g} (- x / (2 π))$ ; applying [F4] to the dilation $ξ \mapsto - 2 π ξ$ of the $L^{1}$ function $ξ \mapsto ∣ \hat{g} (ξ) ∣^{2}$ gives $\int_{R} ∣ \hat{g} (- x / (2 π)) ∣^{2} d x = 2 π \int_{R} ∣ \hat{g} (ξ) ∣^{2} d ξ = 2 π T$ . Substituting $s = ℏ u$ in the elementary integral gives $\int_{0}^{t} e^{i x s / ℏ} d s = ℏ \int_{0}^{T} e^{i x u} d u$ , hence $\int_{R} K_{t} d x = (1 / (ℏ^{2} t)) \cdot ℏ^{2} \cdot 2 π T = 2 π / ℏ$ .

1.3F5F6F7algebra

[The pushforward measure] With $c_{j} : = ⟨ D_{j} v_{a}, v_{b} ⟩$ and $M = \sum_{j} c_{j} g_{j}$ , [F7] gives $∥ M ∥_{L^{2}}^{2} < \infty$ and $M \in h$ by [F6], and $supp M \subseteq {0 < ε \leq Λ}$ . Hence for every Borel $A \subseteq R$ the function $k \mapsto 1_{A} (ε (k)) ∥ M (k) ∥^{2}$ is nonnegative and measurable, so $A \mapsto ν_{b} (A) : = \int_{R^{3}} 1_{A} (ε (k)) ∥ M (k) ∥^{2} d^{3} k$ is a finite measure by [F5], with $ν_{b} (R) = ∥ M ∥_{L^{2}}^{2} < \infty$ and $ν_{b} (R ∖ (0, Λ]) = 0$ . The identity $\int f d ν_{b} = \int f (ε (k)) ∥ M (k) ∥^{2} d^{3} k$ holds for indicators by definition, for nonnegative simple $f$ by additivity, and for all Borel $f \geq 0$ by monotone convergence [F5]. This proves (i).

2.1F4F5F8step 1.1step 1.2algebra

[Approximate identity] Let $J \in L^{1} (λ; C)$ have the representative continuous at $Δ$ specified in (b). For fixed $t > 0$ , $K_{t} \leq t / ℏ^{2}$ by the defining time integral, so $K_{t} (x) J (Δ + x)$ is integrable by translation invariance, and the subtraction below is licensed by $L^{1}$ linearity [F5]. Fix $δ > 0$ . By step 1.2, $\int K_{t} (x) J (Δ + x) d x - (2 π / ℏ) J (Δ) = \int K_{t} (x) (J (Δ + x) - J (Δ)) d x$ , and we split the right side at ${∣ x ∣ \geq δ}$ . On $∣ x ∣ < δ$ the bound $K_{t} \geq 0$ and step 1.2 give at most $(2 π / ℏ) \sup_{∣ x ∣ < δ} ∣ J (Δ + x) - J (Δ) ∣$ ; on $∣ x ∣ \geq δ$ , separate the two terms to obtain at most $(4 / (δ^{2} t)) \int_{∣ x ∣ \geq δ} ∣ J (Δ + x) ∣ d x + ∣ J (Δ) ∣ \int_{∣ x ∣ \geq δ} 4 / (x^{2} t) d x \leq 4 ∥ J ∥_{1} / (δ^{2} t) + 8 ∣ J (Δ) ∣ / (δ t)$ , since $\int_{∣ x ∣ \geq δ} x^{- 2} d x = 2 / δ$ , where $\int_{R} ∣ J (Δ + x) ∣ d x = ∥ J ∥_{1}$ by [F4] applied to the translation $x \mapsto Δ + x$ . Given $ε > 0$ , continuity of $J$ at $Δ$ [F8] yields $δ > 0$ with $(2 π / ℏ) \sup_{∣ x ∣ < δ} ∣ J (Δ + x) - J (Δ) ∣ \leq ε / 3$ , and then $t$ large enough that $4 ∥ J ∥_{1} / (δ^{2} t) + 8 ∣ J (Δ) ∣ / (δ t) < ε / 3$ ; hence the modulus of the displayed difference is $< ε$ for all large $t$ , which is the asserted limit $\frac{2 π}{ℏ} J (Δ)$ [F8].

2.2step 1.1step 1.3F6algebra

[Coefficient computation] For $t = 0$ the defining time integral is zero and $A_{b} (\cdot, 0) = 0$ . For $t > 0$ let $φ_{t} (E) : = \int_{0}^{t} e^{i (E - Δ) s / ℏ} d s$ , so that $A_{b} (k, t) = - (i η_{0} / ℏ) M (k) φ_{t} (ε (k))$ and $∣ φ_{t} ∣ \leq t$ . The function $k \mapsto - (i η_{0} / ℏ) φ_{t} (ε (k))$ is bounded and measurable, so $A_{b} (\cdot, t) \in L^{2} (R^{3}; C^{3})$ with $∣ A_{b} (k, t) ∣ \leq (∣ η_{0} ∣ t / ℏ) ∥ M (k) ∥$ , and pointwise scalar multiplication preserves the transversality condition $k \cdot f (k) = 0$ a.e.; hence $A_{b} (\cdot, t) \in h$ by [F6]. Then $∥ A_{b} (\cdot, t) ∥_{h}^{2} = (η_{0}^{2} / ℏ^{2}) \int_{R^{3}} ∥ M (k) ∥^{2} ∣ φ_{t} (ε (k)) ∣^{2} d^{3} k = (η_{0}^{2} / ℏ^{2}) \int_{R} ∣ φ_{t} (E) ∣^{2} d ν_{b} (E)$ by step 1.3 with $f = ∣ φ_{t} ∣^{2}$ , and $∣ φ_{t} (E) ∣^{2} = ∣ \int_{0}^{t} e^{i (E - Δ) s / ℏ} d s ∣^{2} = ℏ^{2} t K_{t} (E - Δ)$ for every $E$ by the definition of $K_{t}$ in (a) and step 1.1; this proves (ii).

3.1F4F5step 1.1step 2.1step 1.3step 2.2algebra

[Density and the limit] Assume $ν_{b} = J_{b} λ$ with $J_{b} \in L^{1} (λ)$ , $J_{b} = 0$ $λ$ -a.e. on $R ∖ (0, Λ]$ , and $J_{b}$ continuous at $Δ$ . As in step 1.3, the identity $\int f d ν_{b} = \int f J_{b} d λ$ holds for every Borel $f \geq 0$ (indicators are the hypothesis, simple functions additivity, general $f$ monotone convergence, [F5]). Hence by step 2.2, for every $t > 0$ , $\frac{1}{t} ∥ A_{b} (\cdot, t) ∥_{h}^{2} = η_{0}^{2} \int K_{t} (E - Δ) d ν_{b} (E) = η_{0}^{2} \int K_{t} (E - Δ) J_{b} (E) d E = η_{0}^{2} \int K_{t} (x) J_{b} (Δ + x) d x$ , the last equality by the change of variables $E = Δ + x$ [F4], the integrand $K_{t} (\cdot - Δ) J_{b}$ being bounded times $L^{1}$ . If $0 < Δ < Λ$ , then step 2.1 gives $\lim_{t} \frac{1}{t} ∥ A_{b} (\cdot, t) ∥^{2} = η_{0}^{2} \frac{2 π}{ℏ} J_{b} (Δ) = \frac{2 π η_{0}^{2}}{ℏ} J_{b} (Δ)$ . If instead $Δ > Λ$ , then $J_{b} (Δ + x) \neq 0$ for $λ$ -a.e. $x$ only when $Δ + x \leq Λ$ , i.e. $x \leq Λ - Δ < 0$ ; on the support up to a null set one has $∣ x ∣ \geq Δ - Λ = : δ > 0$ , so step 1.1 and $\int ∣ J_{b} (Δ + x) ∣ d x = ∥ J_{b} ∥_{1}$ [F4] give $∣ \int K_{t} (x) J_{b} (Δ + x) d x ∣ \leq (4 / (δ^{2} t)) ∥ J_{b} ∥_{1} \to 0$ , and the limit is $0$ . This proves (iii); the case $Δ = Λ$ is excluded, consistently with the statement.

4.1step 1.1step 1.2step 2.1step 1.3step 2.2step 3.1∎

[Conclusion and scope] Steps 1.1 and 1.2 prove (a), step 2.1 proves (b), and steps 1.3, 2.2 and 3.1 prove (i)-(iii). The scope paragraph of the statement records that only the explicit single-vertex coefficient is treated and that no all-time, Markov, exponential-decay or empirical claim is made. This completes the proof.

5 · Examples, experiments and counterexamples

None yet.