Finite Euclidean quartic interacting measure — Examples

Given: Assume the Axiom of Choice. Let $Q\sim N(0,1)$ be the one-dimensional centered Gaussian and put $U(q)=q^4\mathrm{/}4!$.

The Axiom of Choice is the explicit hypothesis of the finite-dimensional Gaussian-law suppliers. The Axiom of Choice

Finite Euclidean quartic interacting measure defines the Borel Gaussian reference law ${\gamma }_C$ and the partition integral $Z(\lambda )={\mathbb{E}}_{{\gamma }_C}[e^{-\lambda U}]$.

For a finite Gaussian covariance, polynomial insertions have all Wick-pairing moments. In one dimension with variance one, $\mathbb{E}[Q^{2m}]=(2m-1)!!$ and odd moments vanish. Finite Gaussian normalization and Wick moments

The finite Euclidean partition integral is positive and finite for every $\lambda \ge 0$, and its order-$M$ coefficients are $(-1{)}^k\mathbb{E}[U^k]\mathrm{/}k!$. Finite interacting asymptotic Wick expansion with remainder

Given: Assume the Axiom of Choice and Countable Choice. Let $q\in \mathbb{R}$, $C\ge 0$, $\lambda >0$, and let $a>0$ be a kinetic coefficient.

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

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 (${\mathrm{A}\mathrm{C}}_{\omega }$)

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

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: $N\ge 1$, coefficients $a_j>0$, and a real polynomial $V$ satisfying $V(q)\ge c\mathrm{\mid }q{\mathrm{\mid }}^2-C_0$ for $c>0$ and $C_0\ge 0$. Put $W=V+C_0+1$. It asserts that the completion of $C_c^{\mathrm{\infty }}({\mathbb{R}}^N)$ under $\mathrm{\parallel }u{\mathrm{\parallel }}_Q^2={\sum }_j{a}_j\mathrm{\parallel }{\mathrm{\partial }}_{j}u{\mathrm{\parallel }}_2^2+{\int }_{{\mathbb{R}}^N}(W+1)\mathrm{\mid }u{\mathrm{\mid }}^{2}dq$ 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].

Given: Assume the Axiom of Choice and let $Q\sim N(0,C)$ be a centered scalar Gaussian with variance $C\ge 0$.

The finite-dimensional normal-law suppliers explicitly assume the Axiom of Choice. The Axiom of Choice

A centered scalar normal with covariance $C$ is a one-dimensional instance of the multivariate normal law. Multivariate normal law, including singular covariance

The finite covariance Wick definition gives $:Q^2{:}_C=Q^2-C$. Finite covariance Wick normal powers

The Gaussian pairing formula gives $\mathbb{E}[Q^2]=C$ and $\mathbb{E}[Q^4]=3C^2$. Finite Gaussian normalization and Wick moments