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

Each result cited below is universally quantified over the data in its own statement.

Order and arithmetic in $\mathbb{R}$ are handled with Elementary Order Arithmetic in an Ordered Field and Zero Products and Elementary Identities in a Field, whose claims are cited by number. For every mode $k$ the weight $\mu_{k}$ is positive, since $1\le\mu_{k}$ by The Wick-Square Problem on the Torus: Standing Notation §modes and $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field; so $\frac{1}{\mu_{k}}$ is defined by claim 7 of that lemma. Differentiability of a function $\mathbb{R}\to\mathbb{R}$ is that of Single-Variable Calculus on an Interval §derivative with $I=\mathbb{R}$, as fixed in Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes; every point of $\mathbb{R}$ is an interior point of $\mathbb{R}$, and the restriction of a map $\mathbb{R}\to\mathbb{R}$ to $I=\mathbb{R}$ is the map itself, so the rules of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives and Derivative of a Polynomial Function on the Real Line (the ones named in Single-Variable Calculus on an Interval §derivative) apply at every point.

Clause 1. Fix $x\in H^{-1}$ and a mode $k$, and let $t\in\mathbb{R}$. Put $y=x+te_{k}$, which lies in $H^{-1}$ by The Wick-Square Problem on the Torus: Standing Notation §units; the operations are the pointwise ones of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ (The Wick-Square Problem on the Torus: Standing Notation §state-space), so $y(k')=x(k')+te_{k}(k')$ for every mode $k'$. Hence $y(k')=x(k')$ for $k'\ne k$, and $y(k)=x(k)+t$, so that $y(k)^{2}=x(k)^{2}+2t\,x(k)+t^{2}$ by claim 5 of Zero Products and Elementary Identities in a Field. Define the families

$$u(k')=\frac{x(k')^{2}}{\mu_{k'}},\qquad d_{t}(k')=\frac{y(k')^{2}}{\mu_{k'}}-\frac{x(k')^{2}}{\mu_{k'}}\qquad(k'\in\mathbb{Z}^{n}).$$

Then $d_{t}(k')=0$ for $k'\ne k$ and $d_{t}(k)=\frac{2t\,x(k)+t^{2}}{\mu_{k}}$. The set $\{k\}$ is nonempty and finite, so by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support the family $d_{t}$ is cube-summable with lattice sum $\sum_{k'\in\{k\}}d_{t}(k')=d_{t}(k)$, the last equality by claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set. By The Wick-Square Problem on the Torus: Standing Notation §state-space, the family $u$ is cube-summable with lattice sum $|x|_{H^{-1}}^{2}$, and the family $k'\mapsto\frac{y(k')^{2}}{\mu_{k'}}=u(k')+d_{t}(k')$ is cube-summable with lattice sum $|y|_{H^{-1}}^{2}$. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear (and uniqueness of limits, claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences) we get $|y|_{H^{-1}}^{2}=|x|_{H^{-1}}^{2}+d_{t}(k)$. By The Gaussian Penalty of the Wick-Square Problem this means that the section $\sigma=\sigma_{x,k}$ of $P$ at $x$ along $k$ (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes) is

$$\sigma(t)=P(x+te_{k})=\frac{\beta}{2}\Bigl(|x|_{H^{-1}}^{2}+\frac{2t\,x(k)+t^{2}}{\mu_{k}}\Bigr)=A+B\,t+C\,t^{2}\qquad(t\in\mathbb{R}),$$

with the constants $A=P(x)$, $B=\frac{\beta\,x(k)}{\mu_{k}}$ and $C=\frac{\beta}{2\mu_{k}}$ (field arithmetic).

Now $t^{2}=t^{S(1)}$ and $t=t^{1}$ by claim 1 of Properties of Natural Number Powers in a Field. By claim 1 of Derivative of a Polynomial Function on the Real Line, at every $t_{0}\in\mathbb{R}$ the map $t\mapsto t$ is differentiable with derivative $1$ and the map $t\mapsto t^{2}$ is differentiable with derivative $2t_{0}$, where $2$ is the image of $S(1)$ under the canonical map, equal to $1+1$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. By claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives (constants, sums and constant multiples), $\sigma$ is differentiable at every $t_{0}\in\mathbb{R}$ with

$$\sigma'(t_{0})=B+2C\,t_{0} .$$

Applying the same two claims to the function $\sigma':t_{0}\mapsto B+2C\,t_{0}$, it is differentiable at $0$ (indeed everywhere) with derivative $2C=\frac{\beta}{\mu_{k}}$. Since $x$ and $k$ were arbitrary, $P$ is twice differentiable along the modes in the sense of Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice, and by Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives

$$\partial_{k}P(x)=\sigma'(0)=B=\frac{\beta\,x(k)}{\mu_{k}},\qquad\partial_{k}^{2}P(x)=(\sigma')'(0)=\frac{\beta}{\mu_{k}} .$$

Clause 2. Let $N\in\mathbb{N}$ and $x\in H^{-1}$. By clause 1, $L_{N}P(x)$ is defined, and by The Free-Field Generator with a Mode Cutoff §generator and clause 1,

$$L_{N}P(x)=\sum_{k\in\Gamma_{N}}\Bigl(\frac{\nu}{2}\cdot\frac{\beta}{\mu_{k}}-\mu_{k}\,x(k)\,\frac{\beta\,x(k)}{\mu_{k}}\Bigr).$$

For each mode $k$, field arithmetic gives $\frac{\nu}{2}\cdot\frac{\beta}{\mu_{k}}=\beta\,\frac{\nu}{2\mu_{k}}=\beta c_{k}$ by The Free-Field Variances of the Fourier Modes §variances, and $\mu_{k}\,x(k)\,\frac{\beta x(k)}{\mu_{k}}=\beta\,x(k)^{2}$ because $\mu_{k}\frac{1}{\mu_{k}}=1$; so, by claim 2 of Zero Products and Elementary Identities in a Field and distributivity, the summand equals $-\beta\bigl(x(k)^{2}-c_{k}\bigr)$. By claim 4 of Properties of a Sum over a Finite Index Set (homogeneity, with $\lambda=-\beta$) and The Wick Square with a Mode Cutoff and the Wick Domain §cutoff,

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

Clause 3 (differentiability of $\varphi-P$). Let $\varphi$ be twice differentiable along the modes, and fix $x\in H^{-1}$ and a mode $k$. The section of $\varphi-P$ at $x$ along $k$ is $t\mapsto\varphi(x+te_{k})-P(x+te_{k})$, that is $\sigma^{\varphi}+(-1)\sigma^{P}$, where $\sigma^{\varphi}$ and $\sigma^{P}$ are the sections of $\varphi$ and $P$. Both are differentiable at every point of $\mathbb{R}$ (by hypothesis, and by clause 1), so by claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives the section of $\varphi-P$ is differentiable at every point with derivative $(\sigma^{\varphi})'+(-1)(\sigma^{P})'$. Both derivatives are differentiable at $0$ (by hypothesis, and by clause 1), so by the same claim this derivative is differentiable at $0$. Hence $\varphi-P$ is twice differentiable along the modes (Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice), and by Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives

$$\partial_{k}(\varphi-P)(x)=\partial_{k}\varphi(x)-\partial_{k}P(x),\qquad\partial_{k}^{2}(\varphi-P)(x)=\partial_{k}^{2}\varphi(x)-\partial_{k}^{2}P(x).$$

Clause 3 (the penalised form). Let $N\in\mathbb{N}$, $x\in H^{-1}$ and $b:\mathbb{Z}^{n}\to\mathbb{R}$. By The Free-Field Generator with a Mode Cutoff §generator, the last display, distributivity, and claims 3 and 4 of Properties of a Sum over a Finite Index Set (additivity, and homogeneity with $\lambda=-1$),

$$L_{N}(\varphi-P)(x)=\sum_{k\in\Gamma_{N}}\Bigl(\frac{\nu}{2}\partial_{k}^{2}\varphi(x)-\mu_{k}x(k)\partial_{k}\varphi(x)\Bigr)-\sum_{k\in\Gamma_{N}}\Bigl(\frac{\nu}{2}\partial_{k}^{2}P(x)-\mu_{k}x(k)\partial_{k}P(x)\Bigr)=L_{N}\varphi(x)-L_{N}P(x).$$

With clause 2 this gives $-L_{N}\varphi(x)=-L_{N}(\varphi-P)(x)+\beta\sum_{k\in\Gamma_{N}}\bigl(x(k)^{2}-c_{k}\bigr)$, using The Wick Square with a Mode Cutoff and the Wick Domain §cutoff. Substituting into The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm §operator,

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

By claims 3 and 4 of Properties of a Sum over a Finite Index Set, the two sums combine to

$$\beta\sum_{k\in\Gamma_{N}}\Bigl(\bigl(x(k)^{2}-c_{k}\bigr)-\bigl(x(k)^{2}-b(k)\bigr)\Bigr)=\beta\sum_{k\in\Gamma_{N}}\bigl(b(k)-c_{k}\bigr)=-\beta\sum_{k\in\Gamma_{N}}\bigl(c_{k}-b(k)\bigr),$$

where the summands were simplified by field arithmetic and claim 2 of Zero Products and Elementary Identities in a Field. This is the asserted formula for $F_{N}^{b}[\varphi](x)$. Finally, by The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm §wick, $F_{N}[\varphi]=F_{N}^{b}[\varphi]$ with $b(k)=c_{k}$; then every summand $c_{k}-b(k)$ is $0$, so the sum vanishes by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, and the formula reduces to

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

This completes the proof.