The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, so that $n$ is the dimension of the torus, $X=H^{-m}(\mathbb{T}^{n})$ is the Sobolev space of order $-m$, $\kappa$ is the fixed enumeration of $\mathbb{Z}^{n}$ and $x(k)$ is the value at $k\in\mathbb{Z}^{n}$ of the coefficient family $x\in X$, as in The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data, and the numbers $\rho_{k}$ and their natural powers $\rho_{k}^{m}$ are those of The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale, let $M\in\mathbb{N}$, and let the side $L$, the cube $\Gamma_{M}$, the maps $\psi_{k}$, the lattice torus $\mathbb{L}_{M}$, the lattice fields and their real vector space $\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})$, and the discrete Fourier coefficients $\hat{f}(k)$ of a lattice field $f$ be those of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field. The trigonometric system is used here through its maps $\psi_{k}$ of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field. Sums over finite index sets are those of Sum over a Finite Index Set. $L^{n/2}$ is the nonnegative square root of the natural power $L^{n}$, which is positive by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §coefficients; $L^{n/2}$ is nonzero, since $0\cdot0=0$ by Zero Products and Elementary Identities in a Field §annihilation while $L^{n}\neq0$, hence positive, and $L^{-n/2}$ is its multiplicative inverse. $d_{M}\in\mathbb{N}$ is the Galerkin head dimension of The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension read with the cutoff $M$ in place of $N$, a clause not involving the coupling and the mass of that definition.

1. (The white-noise embedding)¶ For a lattice field $f$, $\mathcal{E}_{M}f$ is the coefficient family on $\mathbb{Z}^{n}$ with

$$\mathcal{E}_{M}f(k)=L^{n/2}\,\hat{f}(k)\quad(k\in\Gamma_{M}),\qquad \mathcal{E}_{M}f(k)=0\quad(k\in\mathbb{Z}^{n},\ k\notin\Gamma_{M}).$$

It belongs to $X$: by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §series, applied with the enumeration $\kappa$, a coefficient family $c$ belongs to $X$ if and only if the series $\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{m})^{2}\,c(\kappa(j))^{2}$ converges; for $c=\mathcal{E}_{M}f$ the terms with $d_{M}<j$ vanish, because $\kappa(j)\notin\Gamma_{M}$ for such $j$ by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension, so the partial sums are constant from $d_{M}$ on and the series converges. The map $\mathcal{E}_{M}:\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})\to X$ is the white-noise embedding of lattice fields at cutoff $M$.

2. (The site field)¶ For $x\in X$, the site field of $x$ at cutoff $M$ is the lattice field $\varphi_{x}:\mathbb{L}_{M}\to\mathbb{R}$,

$$\varphi_{x}(z)=L^{-n/2}\sum_{k\in\Gamma_{M}}x(k)\,\psi_{k}(z)\qquad(z\in\mathbb{L}_{M}).$$

The notation $\varphi_{x}$ does not show the cutoff $M$.