The Galerkin Wick-Ordered $Phi^4$ Potential and Its Wick Constant on the Torus

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, so that $X=H^{-m}(\mathbb{T}^{n})$, with noise weights $a_{j}$ ($j\in\mathbb{N}$), and $\kappa$ is the fixed enumeration of $\mathbb{Z}^{n}$, that is, a bijection $\kappa:\mathbb{N}\to\mathbb{Z}^{n}$ as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families, let $a_{j}^{-1/2}$ be as in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads, and let $x(k)$ be 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. In this item the trigonometric system is used through its maps: since $e_{j}$ denotes the basis of $X$, the map written $e_{k}$ in the trigonometric system on $\mathbb{T}^{n}$ is written $\psi_{k}:\mathbb{R}^{n}\to\mathbb{R}$ ($k\in\mathbb{Z}^{n}$); it belongs to $C_{\mathrm{per}}$, and $E_{k}=[\,\psi_{k}|_{Q}\,]$, by The Trigonometric System on the Torus is Orthonormal §classes. Integrals over the cell $Q$ are taken with respect to the measure $\lambda_{Q}$ of The Flat Torus: Standing Notation §measure. For $k\in\mathbb{Z}^{n}$, $\mu_{k}^{-1}$ is the multiplicative inverse of the Fourier weight $\mu_{k}$, which is positive by Summability of the Negative Powers of the Fourier Weights of the Torus §product. $\mathbb{N}$ carries the order of $\mathbb{R}$, a total order, and $[M]$ is the initial segment determined by $M\in\mathbb{N}$.

Let $N\in\mathbb{N}$ (the cutoff), let $\Gamma_{N}$ be the cube of The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points, and let $\varkappa,\mathfrak{m}\in\mathbb{R}$ satisfy $0<\varkappa$ (the coupling and the mass).

(a) (The Galerkin head dimension)¶ The set of those $j\in\mathbb{N}$ with $\kappa(j)\in\Gamma_{N}$ has a greatest element. Indeed, $\Gamma_{N}$ is a nonempty finite set by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite, so it has $M$ elements for some $M\in\mathbb{N}$, and there is a bijection $\theta_{0}:[M]\to\Gamma_{N}$; since $\kappa$ is a bijection, the indices $j\in\mathbb{N}$ with $\kappa(j)\in\Gamma_{N}$ are exactly the components of the $M$-tuple $\bigl(\kappa^{-1}(\theta_{0}(1)),\dots,\kappa^{-1}(\theta_{0}(M))\bigr)$ in $\mathbb{N}$, which has a greatest component by Greatest Element of a Finite Family in a Totally Ordered Set. This greatest element, the largest $j\in\mathbb{N}$ with $\kappa(j)\in\Gamma_{N}$, is the Galerkin head dimension $d_{N}\in\mathbb{N}$ at cutoff $N$; thus $\kappa(j)\notin\Gamma_{N}$ for every $j\in\mathbb{N}$ with $d_{N}<j$. For $j\in[d_{N}]$ put $\chi_{N}(j)=1$ if $\kappa(j)\in\Gamma_{N}$ and $\chi_{N}(j)=0$ otherwise, and let $J_{N}$ be the set of those $j\in[d_{N}]$ with $\kappa(j)\in\Gamma_{N}$, the set of active indices at cutoff $N$. For real numbers, or vectors of a real vector space, $t_{j}$ indexed by $j\in[d_{N}]$, the notation $\sum_{j\in J_{N}}t_{j}$ stands for $\sum_{j=1}^{d_{N}}\chi_{N}(j)\,t_{j}$.

(b) (The Galerkin field)¶ For $x\in X$ and $u\in\mathbb{R}^{d_{N}}$, $\Phi_{N}[x],\varphi_{N}[u]:\mathbb{R}^{n}\to\mathbb{R}$ are the maps

$$\Phi_{N}[x](z)=\sum_{j=1}^{d_{N}}\chi_{N}(j)\,x(\kappa(j))\,\psi_{\kappa(j)}(z),\qquad\varphi_{N}[u](z)=\sum_{j=1}^{d_{N}}\chi_{N}(j)\,a_{j}^{-1/2}u_{j}\,\psi_{\kappa(j)}(z)\qquad(z\in\mathbb{R}^{n}),$$

the Galerkin field of $x$ at cutoff $N$ and the Galerkin field with coordinates $u$. Both belong to $C_{\mathrm{per}}$, by Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus §member used with $M=d_{N}$ and the maps $\psi_{\kappa(j)}$.

(c) (The Wick constant)¶ The Wick constant at cutoff $N$ is the real number

$$C_{N}=\sum_{j=1}^{d_{N}}\chi_{N}(j)\,\mu_{\kappa(j)}^{-1},$$

a finite sum; it depends on $N$ only.

(d) (The Wick polynomial)¶ $f_{N}:\mathbb{R}\to\mathbb{R}$ is the polynomial

$$f_{N}(s)=\frac{\varkappa}{4}\bigl(s^{4}-6C_{N}s^{2}+3C_{N}^{2}\bigr)+\frac{\mathfrak{m}}{2}\bigl(s^{2}-C_{N}\bigr)\qquad(s\in\mathbb{R}),$$

the Wick-ordered $\phi^{4}$ polynomial with Wick constant $C_{N}$, coupling $\varkappa$ and mass $\mathfrak{m}$.

(e) (The potential and its profile)¶ For $x\in X$ and $u\in\mathbb{R}^{d_{N}}$ the maps $f_{N}\circ\Phi_{N}[x]$ and $f_{N}\circ\varphi_{N}[u]$ belong to $C_{\mathrm{per}}$, being built from members of $C_{\mathrm{per}}$ and constants by sums and products, by Elementary Properties of Lattice-Periodic Functions §algebra; so their restrictions to $Q$ are $\lambda_{Q}$-integrable by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member with the exponent $1$. The Galerkin Wick-ordered $\phi^{4}$ potential with cutoff $N$, coupling $\varkappa$ and mass $\mathfrak{m}$ is the function $V_{N}:X\to\mathbb{R}$, and its profile is the function $v_{N}:\mathbb{R}^{d_{N}}\to\mathbb{R}$, given by

$$V_{N}(x)=\int_{Q}f_{N}\bigl(\Phi_{N}[x](z)\bigr)\,\lambda_{Q}(dz),\qquad v_{N}(u)=\int_{Q}f_{N}\bigl(\varphi_{N}[u](z)\bigr)\,\lambda_{Q}(dz).$$

The dependence of $f_{N}$, $V_{N}$ and $v_{N}$ on $\varkappa$ and $\mathfrak{m}$ is not shown in the notation.