The Free-Field Generator with a Mode Cutoff

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $\varphi:H^{-1}\to\mathbb{R}$ be twice differentiable along the modes, with mode derivatives $\partial_{k}\varphi$ and $\partial_{k}^{2}\varphi$, and let $x\in H^{-1}$. Sums over the cube $\Gamma_{N}$ are sums over a finite index set.

1. (Free-field generator)¶ For $N\in\mathbb{N}$, the free-field generator at cutoff $N$ applied to $\varphi$ at $x$ is

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

2. (Uncut generator)¶ If the sequence $(L_{N}\varphi(x))_{N\in\mathbb{N}}$ converges, $L\varphi$ is said to be defined at $x$, and $L\varphi(x)$ denotes the limit, which is unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences.