The Wick Square with a Mode Cutoff and the Wick Domain

In the setting of The Wick-Square Problem on the Torus: Standing Notation, sums over the cube $\Gamma_{N}$ are sums over a finite index set.

Let $c_{k}$ be the free-field variance of the mode $k$.

1. (Cutoff Wick square)¶ For $x\in H^{-1}$ and $N\in\mathbb{N}$, the Wick square of $x$ at cutoff $N$ is

$${:}x^{2}{:}_{N}=\sum_{k\in\Gamma_{N}}\bigl(x(k)^{2}-c_{k}\bigr).$$

2. (Wick domain)¶ The Wick domain $\mathcal{W}$ is the set of those $x\in H^{-1}$ for which the sequence $({:}x^{2}{:}_{N})_{N\in\mathbb{N}}$ converges; for $x\in\mathcal{W}$ its limit, unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, is the Wick square ${:}x^{2}{:}$ of $x$.