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Finite-mode Fock to Schrödinger identification

Statement

Assume Countable Choice. Let $r \geq 1$ , fix a positive real action-scale parameter $ℏ > 0$ , and let $H = C^{r}$ with orthonormal basis $e_{1}, \dots, e_{r}$ , and let $F_{s} (H) = ⨁_{n \geq 0} H^{\otimes_{s} n}$ be the bosonic Fock space with vacuum $Ω$ and creation and annihilation operators $a_{j}^{*}, a_{j}$ from Hilbert direct-sum construction and Creation contraction bounds. For $N = (n_{1}, \dots, n_{r}) \in N_{0}^{r}$ , set $∣ N ⟩ : = \prod_{j = 1}^{r} \frac{(a_{j}^{*})^{n_{j}}}{\sqrt{n_{j}!}} Ω .$ Let $g_{N} (x) = \prod_{j = 1}^{r} g_{n_{j}} (x_{j})$ be the normalized product Hermite function from Multidimensional oscillator completeness. Then $(∣ N ⟩)_{N}$ and $(g_{N})_{N}$ are complete orthonormal families, and the map $U ∣ N ⟩ = g_{N}$ extends uniquely to a unitary $U : F_{s} (C^{r}) ⟶ L^{2} (R^{r}) .$ On the algebraic finite-particle span, $U a_{j} U^{- 1}$ and $U a_{j}^{*} U^{- 1}$ are respectively $(x_{j} + \partial_{x_{j}}) / \sqrt{2}$ and $(x_{j} - \partial_{x_{j}}) / \sqrt{2}$ on product Hermite functions. In particular the quadratures $Q_{j} = \sqrt{ℏ / 2} (a_{j} + a_{j}^{*}), P_{j} = - i \sqrt{ℏ / 2} (a_{j} - a_{j}^{*})$ act there as multiplication by $\sqrt{ℏ} x_{j}$ and as $- i \sqrt{ℏ} \partial_{x_{j}}$ , respectively. These are core identities; they do not assert that a finite number cutoff preserves the canonical commutation relations.

Facts & Assumptions

Given: Countable Choice, an integer $r \geq 1$ , a positive real action-scale parameter $ℏ > 0$ , $H = C^{r}$ , its specified orthonormal basis, and the Fock and product Hermite constructions above.

[A1]

Countable Choice is the explicit set-theoretic hypothesis used for the countable Hilbert direct sum and completeness arguments (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The Fock direct sum is a Hilbert space, its finite-particle subspace is dense, and its $n$ th sector is the completed symmetric tensor power (Hilbert direct-sum construction).

[F2]

On finite-particle vectors, creation and annihilation satisfy the canonical commutation relations and the adjoint identity; in a fixed orthonormal basis, $a_{j} ∣ N ⟩ = \sqrt{n_{j}} ∣ N - e_{j} ⟩$ and $a_{j}^{*} ∣ N ⟩ = \sqrt{n_{j} + 1} ∣ N + e_{j} ⟩$ , with a zero term when $n_{j} = 0$ (Creation contraction bounds).

[F3]

The product Hermite functions $g_{N}$ form a complete orthonormal family in $L^{2} (R^{r})$ , and the one-coordinate Hermite recurrence is $x g_{n} = (\sqrt{n + 1} g_{n + 1} + \sqrt{n} g_{n - 1}) / \sqrt{2}$ and $g_{n}^{'} = (\sqrt{n} g_{n - 1} - \sqrt{n + 1} g_{n + 1}) / \sqrt{2}$ (Multidimensional oscillator completeness, Hermite recurrence).

[F4]

A linear combination of the normalized symmetrizations of the elementary tensors in $e_{1}, \dots, e_{r}$ is dense in every symmetric tensor sector; the orthogonal symmetrizations with occupation multiplicities $N$ , $\sum_{j} n_{j} = n$ , have squared norm $\prod_{j} n_{j}! / n!$ (Hilbert direct-sum construction, Creation contraction bounds).

Proof

1.1givenA1F1F2F4algebra

For fixed $n$ , the symmetric tensors are the range of the averaging projection on $(C^{r})^{\otimes n}$ . The elementary tensors in the basis $e_{1}, \dots, e_{r}$ span the tensor power, so their averages span that range. Two such averages with different occupation vectors are orthogonal, because at least one tensor slot has orthogonal basis vectors in every paired term. For occupation vector $N$ the squared norm of the averaged tensor is $\prod_{j} n_{j}! / n!$ by counting the $\prod_{j} n_{j}!$ permutations that preserve the ordered occupation word among $n!$ permutations. Multiplying by $\sqrt{n! / \prod_{j} n_{j}!}$ normalizes it. The CCR in [F2], applied successively to the vacuum, show that this normalized vector is exactly $∣ N ⟩$ ; therefore the vectors $∣ N ⟩$ are an orthonormal basis of each sector. The sector direct sum in [F1] makes their union a complete orthonormal family of Fock space.

2.1F1F3step 1.1algebra

The target family $(g_{N})$ is complete orthonormal by [F3]. Hence the linear map $∣ N ⟩ \mapsto g_{N}$ preserves norms on finite linear combinations, extends by completion to an isometry on Fock space, and is surjective because its range contains the complete family $(g_{N})$ and is closed. Uniqueness follows from density of the finite span.

3.1F2F3step 2.1algebra∎

On a basis vector $∣ N ⟩$ , the ladder formulas [F2] and Hermite recurrences [F3] give $U a_{j} U^{- 1} g_{N} = \sqrt{n_{j}} g_{N - e_{j}} = \frac{x_{j} + \partial_{x_{j}}}{\sqrt{2}} g_{N},$ and $U a_{j}^{*} U^{- 1} g_{N} = \sqrt{n_{j} + 1} g_{N + e_{j}} = \frac{x_{j} - \partial_{x_{j}}}{\sqrt{2}} g_{N} .$ Linearity proves these identities on the finite span. Adding and subtracting the two identities gives the displayed $Q_{j}, P_{j}$ formulas there. The formulas are only asserted on this core, where all terms are defined.

Finite-mode Fock to Schrödinger identification

Statement

Facts & Assumptions

Proof

Depends on

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