Finite Euclidean quartic interacting measure

Assume the Axiom of Choice. Let $N\ge 1$ be finite and let $C$ be a real symmetric positive-semidefinite $N\times N$ matrix; positive-semidefinite means $u^{\mathsf{T}}Cu\ge 0$ for every $u\in {\mathbb{R}}^N$. The centered multivariate normal Borel probability law with covariance $C$ is ${\gamma }_C:=N_N(0,C)$ as defined by Multivariate normal law, including singular covariance: every linear projection $q\mapsto u^{\mathsf{T}}q$ has scalar normal law $N(0,u^{\mathsf{T}}Cu)$, and the law exists even when $C$ is singular. Here $N(0,{\sigma }^2)$ is the law specified in Standard normal and normal laws, with density $e^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$ for the standard case.

Let $U:{\mathbb{R}}^N\to [0,\mathrm{\infty })$ be a real homogeneous quartic polynomial, meaning $U$ is a polynomial and $U(sq)=s^{4}U(q)$ for all $s\in \mathbb{R}$ and $q\in {\mathbb{R}}^N$. For a coupling $\lambda \ge 0$, define the partition integral and its weighted measure by

$Z_{C,U}(\lambda ):={\int }_{{\mathbb{R}}^N}{e}^{-\lambda U(q)}d{\gamma }_C(q),{\nu }_{C,U,\lambda }(A):={\int }_A{e}^{-\lambda U(q)}d{\gamma }_C(q)$

for Borel $A\subseteq {\mathbb{R}}^N$. Whenever $0<Z_{C,U}(\lambda )<\mathrm{\infty }$, the finite Euclidean interacting measure is the Borel probability measure

$d{\mu }_{C,U,\lambda }(q):=Z_{C,U}(\lambda {)}^{-1}{e}^{-\lambda U(q)}d{\gamma }_C(q)\mathrm{.}$

For a polynomial $P:{\mathbb{R}}^N\to \mathbb{R}$ with finite integral, write $⟨P{⟩}_{C,U,\lambda }:=\int Pd{\mu }_{C,U,\lambda }$. The exact supplier Actual finite Euclidean polynomial measure and moments ↗ proves that $Z_{C,U}(\lambda )$ is positive and finite and that every polynomial insertion is integrable. This is a Borel measure on finite-dimensional ${\mathbb{R}}^N$; it is not a measure on distributions on continuum Euclidean spacetime and does not by itself assert reflection positivity or reconstruction of a continuum field.