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Explicit stability of a Wick-subtracted quartic

Facts & Assumptions

Given: Assume the Axiom of Choice and Countable Choice. Let $q \in R$ , $C \geq 0$ , $λ > 0$ , and let $a > 0$ be a kinetic coefficient.

[A1]

The Axiom of Choice is carried by the finite Gaussian Wick-power definition. The Axiom of Choice

[A2]

Countable Choice is the explicit set-theoretic hypothesis used by the current coercive-form draft supplier; it is stated here rather than hidden. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The finite covariance Wick definition gives $: q^{4} :_{C} = q^{4} - 6 C q^{2} + 3 C^{2}$ . Finite covariance Wick normal powers

[L1]

Current batch-59 draft carrier; independent review remains pending. The current batch-59 manifest lists Coercive polynomial form completion and injection at item level $1$ with design state “Authored draft with complete local argument; independent review remains pending.” Its actual statement assumes Countable Choice and the finite data of def-qft-regulated-polynomial-hamiltonian: $N \geq 1$ , coefficients $a_{j} > 0$ , and a real polynomial $V$ satisfying $V (q) \geq c ∣ q ∣^{2} - C_{0}$ for $c > 0$ and $C_{0} \geq 0$ . Put $W = V + C_{0} + 1$ . It asserts that the completion of $C_{c}^{\infty} (R^{N})$ under $∥ u ∥_{Q}^{2} = \sum_{j} a_{j} ∥ \partial_{j} u ∥_{2}^{2} + \int_{R^{N}} (W + 1) ∣ u ∣^{2} d q$ injects continuously into $L^{2}$ with dense image, has the indicated weak derivatives and weighted norm, and satisfies the displayed norm identity. I inspected its complete current proof and transitive suppliers: its direct dependencies are the current draft definition def-qft-regulated-polynomial-hamiltonian, published lem-l-two-with-the-integral-pairing-is-a-hilbert-space, published thm-c-c-infinity-rn-is-dense-in-l-p-of-rn, published thm-cauchy-schwarz-in-an-inner-product-space, and published def-countable-choice. The carrier exists in the current manifest and its interface matches the use below, but it remains an unreviewed draft; this batch keeps the form-domain consequence explicitly conditional on [L1].

Example

For the one-coordinate Wick potential $V (q) : = λ : q^{4} :_{C}$ , every prescribed $α > 0$ has an explicit constant $C_{0} = 15 λ C^{2} + α^{2} / (2 λ)$ such that $V (q) \geq α q^{2} - C_{0}$ for all $q$ . Therefore the potential satisfies the coercivity hypothesis of [L1]. Conditional on the current QFT-IR01 draft argument and its stated hypotheses, the shifted kinetic-plus-potential form has the dense, continuously injected form completion stated in [L1]. This example establishes a lower bound and a conditional form-domain consequence; it does not assert self-adjointness or an interacting Hamiltonian here.

Verification

technique · complete the square, then apply the current coercive-form draft interface conditionally

Given: $C \geq 0$ , $λ > 0$ , $α > 0$ , $a > 0$ , and [A1]–[A2].

1.1F1algebra

Put $y = q^{2} \geq 0$ . By [F1], $: q^{4} :_{C} - \frac{1}{2} q^{4} = \frac{1}{2} y^{2} - 6 C y + 3 C^{2} = \frac{1}{2} (y - 6 C)^{2} - 15 C^{2} \geq - 15 C^{2}$ . Thus $: q^{4} :_{C} \geq \frac{1}{2} q^{4} - 15 C^{2}$ .

2.1step 1.1algebra

The identity $\frac{λ}{2} q^{4} - α q^{2} + \frac{α^{2}}{2 λ} = \frac{λ}{2} {(q^{2} - \frac{α}{λ})}^{2} \geq 0$ gives $\frac{λ}{2} q^{4} \geq α q^{2} - \frac{α^{2}}{2 λ}$ . Multiplying the inequality of step 1.1 by $λ > 0$ and combining yields $V (q) = λ : q^{4} :_{C} \geq α q^{2} - (15 λ C^{2} + \frac{α^{2}}{2 λ}) .$

3.1A1A2L1step 2.1given∎

Take $N = 1$ , the positive kinetic coefficient $a$ , $c = α$ , and $C_{0} = 15 λ C^{2} + α^{2} / (2 λ) \geq 0$ . Step 2.1 verifies the exact polynomial lower bound required by [L1]. The current draft statement’s hypotheses therefore match this example, and its stated completion applies conditionally on that supplier argument. The batch does not represent the draft as independently approved.

Explicit stability of a Wick-subtracted quartic

Facts & Assumptions

Example

Verification

Depends on

Dependency tree · 2 levels

Sources