The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $P$ be the Gaussian penalty, $c_{k}$ the free-field variances and ${:}x^{2}{:}_{N}$ the cutoff Wick square; mode derivatives are those of Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives, and $L_{N}$ and $|D_{N}\cdot|^{2}$ are the free-field generator and gradient energy at cutoff $N$. Then the following hold.

1. (Mode derivatives of the penalty)¶ $P$ is twice differentiable along the modes, and for every $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$

$$\partial_{k}P(x)=\frac{\beta\,x(k)}{\mu_{k}},\qquad\partial_{k}^{2}P(x)=\frac{\beta}{\mu_{k}}.$$

2. (The corrector identity)¶ For every $N\in\mathbb{N}$ and $x\in H^{-1}$,

$$L_{N}P(x)=-\beta\,{:}x^{2}{:}_{N}.$$

3. (Penalised form of the cutoff operators)¶ Let $g:H^{-1}\to\mathbb{R}$ and $b:\mathbb{Z}^{n}\to\mathbb{R}$, let $F_{N}^{b}$ and $F_{N}$ be the cutoff operators with counterterm $b$ and the Wick-ordered cutoff operators for $g$, and let $\varphi:H^{-1}\to\mathbb{R}$ be twice differentiable along the modes. Then $\varphi-P$ is twice differentiable along the modes, and for every $N\in\mathbb{N}$ and $x\in H^{-1}$

$$F_{N}^{b}[\varphi](x)=\gamma\,\varphi(x)-L_{N}(\varphi-P)(x)+\tfrac12\,|D_{N}\varphi(x)|^{2}-\beta\sum_{k\in\Gamma_{N}}\bigl(c_{k}-b(k)\bigr)-g(x);$$

in particular

$$F_{N}[\varphi](x)=\gamma\,\varphi(x)-L_{N}(\varphi-P)(x)+\tfrac12\,|D_{N}\varphi(x)|^{2}-g(x).$$