A set of questions on continuous Gaussian Free Fields (GFF)

Question

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of this term provides important simplifications to the theory and we can give meaning to the theory when this term is not included using functional integrals and Gaussian measures on functional spaces. My intention with this post is to understand the problems involved in the continuous limit of this theory. I know one usually discretizes the theory to define the objects of interest, but I'm trying to understand the origin of these problems starting from the continuous limit. It's very difficult to find such a complete analysis in books or articles, and I usually find myself having to built the whole picture from little pieces of it, so my intention here is to fill the gaps led by this process.

In what follows, I ask 5 questions and try to answer some of them, but I don't know if my answers and my reasoning are correct. I'd appreciate if you could correct me if needed and add more information, if necessary.

First of all, the idea is to give precise meaning to the probability measure: \begin{eqnarray} \frac{1}{Z}\exp\bigg{(}-\int_{\mathbb{R}^{d}}\frac{1}{2}\varphi(x)(-\Delta+m^{2})\varphi(x)dx \bigg{)}\mathcal{D}\varphi \tag{1}\label{1} \end{eqnarray} Where $\mathcal{D}\varphi$ is a "Lebesgue measure" in the space of fields. Here, the space of fields will be simply $\mathcal{S}'(\mathbb{R}^{d})$. In what follows, $\mathcal{S}'(\mathbb{R}^{d})$ is equipped with the strong topology and its associate Borel $\sigma$-algebra, i.e. the $\sigma$-algebra generated by its open sets.

Question 1: As I said before, I know that it is usual to discretize the theory and define (\ref{1}) by means of thermodynamic + continuous limits. But is it possible to address the problem directly on $\mathbb{R}^{d}$?

My attempted answer: I think that, once you discretized the theory and saw what are the correct limits and objects you need, you can pose the problem directly on $\mathbb{R}^{d}$ at the end of day, but it is by no means obvious, at a first sight, how to properly define (\ref{1}) or even other objects related to it, such as correlations etc.

In what follows, I'll address the problem directly on $\mathbb{R}^{d}$ assuming my answer to the first question is correct and I'm allowed to do it.

Question 2: Is (\ref{1}) a well-defined measure on its own, for all values of $m \ge 0$? How does the ultraviolet divergencies influence the existence of this measure? Does it play any role on its well-definiteness or just on correlation functions?

My attempted answer: I don't think this is well-defined on its own, because I don't think that the "product Lebesgue measure" $\mathcal{D}\varphi$ is well-defined in $\mathcal{S}'(\mathbb{R}^{d})$. However, I know that we can give meaning to (\ref{1}) if we use Minlos-Bochner theorem.

If my answer to question 2 is correct, I must use Minlos-Bochner. Then, (\ref{1}) is the measure $\mu_{G}(\varphi)$ on $\mathcal{S}'(\mathbb{R}^{d})$ induced by $W(f,f):=e^{C(f,f)}$ (using Minlos-Bochner) where: \begin{eqnarray} C(f,g):= \frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\frac{\overline{\hat{f}(\xi)}\hat{g}(\xi)}{|\xi|^{2}+m^{2}}d^{d}\xi \tag{2}\label{2} \end{eqnarray}

Question 3: Intuitivelly, I know that (\ref{2}) is related to (\ref{1}). This is because $\hat{C}(\xi) = 1/(|\xi|^{2}+m^{2})$ is the Fourier transform of the Green's function $G(x)$ of the massive Laplacian $-\Delta+m^{2}$. Informally: Green's functions are inverse operators and, therefore, the measure induced by Minlos-Bochner theorem is a functional analogue of the usual property that Fourier transform of Gaussians are Gaussians. But, apart from the intuition, how can we related (\ref{1}) to $d\mu_{G}$? In other words, does (\ref{1}) have anything to do with the covariance of $d\mu_{G}$?

My attempted answer: I think the only way to realize $d\mu_{G}$ is the corrected Gaussian measure associated to (\ref{1}) (which was not defined as a Gaussian measure in the first place) is by discretizing the space and recovering the theory with thermodynamic + continuous limits. But starting from Minlos-Bochner's theorem, with covariance (\ref{2}), it doesn't seem obvious to me (apart from intuition) that $d\mu_{G}$ has anything to do with (\ref{1}).

Question 4: As I mentioned before, $d\mu_{G}$ is a Gaussian measure on $\mathcal{S}'(\mathbb{R}^{d})$ while (\ref{1}) seems to be just induced by a bilinear form on $\mathcal{S}(\mathbb{R}^{d})\subset \mathcal{S}'(\mathbb{R}^{d})$. Is (\ref{1}) well-defined only as a subset $\mathcal{S}(\mathbb{R}^{d})$ of $\mathcal{S}'(\mathbb{R}^{d})$? Or is it actually a quadratic form on $\mathcal{S}'(\mathbb{R}^{d})$ (in which case I don't seem to understand it correctly)?

Question 5: If I can, in fact, work the theory directly in the infinite/continuous setup, and all the Gaussian measures are properly defined, is it possible to calculate correlations, say, by using properties of Gaussian measures?

Note: I said, right from the beginning, that the space of fields is $\mathcal{S}'(\mathbb{R}^{d})$ but I know it because I already studied some models before and I knew what was the properly funcional space to consider. However, I believe (not sure) that physicists interpret fields as proper functions e.g. on $\mathcal{S}(\mathbb{R}^{d})$ and (\ref{1}) would be something like a quadratic form $\langle \varphi, (-\Delta+m^{2})\varphi\rangle$ on $\mathcal{S}(\mathbb{R}^{d})$. Then, because of Minlos-Bochner theorem, one notices that $\varphi$ must actually be considered as an element of a larger space $\mathcal{S}'(\mathbb{R}^{d})$ in which (\ref{1}) have no meaning unless $\varphi \in \mathcal{S}(\mathbb{R}^{d})$. This is what I think, but I don't know if I'm completely wrong and fields have physical reasons to be tempered distributions right from the beginning.

Answer 1

Essentially, what is asked is the continuation of my previous MO answer

Reformulation - Construction of thermodynamic limit for GFF

and the solution of the exercise I mentioned at the end of that answer.

There, I explained the construction of Gaussian Borel measures $\mu_m$ on the space $s'(\mathbb{Z}^d)$ of temperate multisequences indexed by the unit lattice in $d$ dimensions. The measure $\mu_m$ is specified by its characteristic function $$ p\longmapsto\exp\left(-\frac{1}{2}\sum_{x,y\in\mathbb{Z}^d}p(x)G_m(x,y)p(y)\right) $$ for $p=(p(x))_{x\in\mathbb{Z}^d}$ in $s(\mathbb{Z}^d)$, the space of multisequences with fast decay. The discrete Green's function $G_m(x,y)$ is defined on $\mathbb{Z}^d\times\mathbb{Z}^d$ by $$ G_m(x,y)=\frac{1}{(2\pi)^d}\int_{[0,2\pi]^d}d^d\xi\ \frac{e^{i\xi\cdot(x-y)}}{m^2+2\sum_{j=1}^{d}(1-\cos \xi_j)}\ . $$ Here we will assume $m\ge 0$ for $d\ge 3$, and $m>0$ if $d$ is $1$ or $2$.

For any integer $N\ge 1$, define the discrete sampling map $\theta_N:\mathscr{S}(\mathbb{R}^d)\rightarrow s(\mathbb{Z}^d)$ which sends a Schwartz function $f$ to the multisequence $$ \left(f\left(\frac{x}{N}\right)\right)_{x\in\mathbb{Z}^d}\ . $$ This map is well defined and linear continuous. Indeed, $$ \langle Nx\rangle^2=1+\sum_{j=1}^{d} (Nx_j)^2\le N^2\langle x\rangle^2 $$ because $N\ge 1$. So $$ ||\theta_N(f)||_k:= \sup_{x\in\mathbb{Z}^d} \langle x\rangle^k \left|f\left(\frac{x}{N}\right)\right| \le \sup_{z\in\mathbb{R}^d}\langle Nz\rangle^k|f(z)|\ \le N^k\ ||f||_{0,k} $$ where we used the standard seminorms $$ ||f||_{\alpha,k}=\sup_{z\in\mathbb{R}^d}\langle z\rangle^k|\partial^{\alpha}f(z)| $$ for Schwartz functions. Now consider the transpose map $\Theta_N=\theta_N^{\rm T}$ from $s'(\mathbb{Z}^d)$ to $\mathscr{S}'(\mathbb{R}^d)$. It is defined by $$ \langle \Theta_N(\psi),f\rangle=\langle\psi,\theta_N(f)\rangle=\sum_{x\in\mathbb{Z}^d}\psi(x)f\left(\frac{x}{N}\right) $$ for all discrete temperate fields $\psi$ and continuum test functions $f$. Essentially, $$ \Theta_N(\psi)=\sum_{x\in\mathbb{Z}^d}\psi(x)\ \delta_{\frac{x}{N}} $$ where $\delta_z$ denotes the $d$-dimensional Dirac Delta Function located at the point $z$. Now $\Theta_N$ is continuous for the strong topologies. Indeed if $A$ is a bounded subset of Schwartz space $$ ||\Theta_N(\psi)||_A=\sup_{f\in A}|\langle \Theta_N(\psi),f\rangle|= \sup_{p\in \theta_N(A)}|\langle \psi,p\rangle| $$ and $\theta_N(A)$ is bounded in $s(\mathbb{Z}^d)$ (because a continuous linear map sends bounded sets to bounded sets). Suppose we are given sequences $m_N$ and $\alpha_N$ dependent on the UV cutoff $N$. Define the Borel measure $$ \nu_N=(\alpha_N\Theta_N)_{\ast}\mu_{m_N} $$ on $\mathscr{S}'(\mathbb{R}^d)$. Its characteristic function is $$ W_N(f)=\int_{\mathscr{S}'(\mathbb{R}^d)}d\nu_N(\phi)\ e^{i\langle\phi,f\rangle} =\int_{s'(\mathbb{Z}^d)}d\mu_{m_N}(\psi)\ e^{i\langle\psi,\alpha_N\theta_N(f)\rangle} $$ by the abstract change of variable theorem. We then get $W_N(f)=\exp\left(-\frac{1}{2}Q_N(f)\right)$ where $$ Q_N(f)=\frac{\alpha_N^2}{(2\pi)^d}\sum_{x,y\in\mathbb{Z}^d} f\left(\frac{x}{N}\right)f\left(\frac{y}{N}\right) \int_{[0,2\pi]^d}d^d\xi\ \frac{e^{i\xi\cdot(x-y)}}{m^2+2\sum_{j=1}^{d}(1-\cos \xi_j)} $$ $$ =\frac{N^{2-d}\alpha_N^2}{(2\pi)^d}\sum_{x,y\in\mathbb{Z}^d} f\left(\frac{x}{N}\right)f\left(\frac{y}{N}\right) \int_{[-N\pi,N\pi]^d}d^d\zeta\ \frac{e^{i\zeta\cdot(\frac{x}{N}-\frac{y}{N})}}{N^2 m_N^2+2N^2\sum_{j=1}^{d}\left(1-\cos \left(\frac{\zeta_j}{N}\right)\right)} $$ after changing $[0,2\pi]^d$ to $[-\pi,\pi]^d$ by periodicity, then changing variables to $\zeta=N\xi$, and finally some algebraic rearrangement.

Pointwise in $\zeta\in\mathbb{R}^d$, we have $$ \lim\limits_{N\rightarrow\infty} 2N^2\sum_{j=1}^{d}\left(1-\cos \left(\frac{\zeta_j}{N}\right)\right) =\zeta^2 $$ and this is why I put an $N^2$ in the denominator. Finally, we can pick the right choice for the sequences $m_N$ and $\alpha_N$. For a fixed $m\ge 0$ (or strictly positive if $d=1,2$) we let $m_N=\frac{m}{N}$. Now we pick $\alpha_N$ so that the prefactor $N^{2-d}\alpha_N^2$ becomes the volume element $N^{-2d}$ for a Riemann sum approximation of a double integral on $\mathbb{R}^d\times\mathbb{R}^d$. Namely, we pick $\alpha_N=N^{-\frac{d}{2}-1}$. Equivalently, going back to $\alpha_N\Theta_N(\psi)$, that means choosing $$ \alpha_N\sum_{x\in\mathbb{Z}^d}\psi(x)\ \delta_{\frac{x}{N}}=\left(\frac{1}{N}\right)^{d-[\phi]} \sum_{x\in\mathbb{Z}^d}\psi(x)\ \delta_{\frac{x}{N}} $$ where $[\phi]=\frac{d-2}{2}$ is the (canonical) scaling dimension of the free field. I wrote the last equation in a way to explicitly display the lattice spacing $\frac{1}{N}$.

Now an excellent exercise, for graduate students in analysis, is to show that $$ \lim\limits_{N\rightarrow \infty}Q_N(f)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d} d^d\zeta\ \frac{|\widehat{f}(\zeta)|^2}{\zeta^2+m^2} $$ where the Fourier transform is normalized as $\widehat{f}(\zeta)=\int_{\mathbb{R}^d}d^dx\ e^{-i\zeta\cdot x} f(x)$. Finally, Fernique's version of the Lévy Continuity Theorem for $\mathscr{S}'(\mathbb{Z}^d)$, shows that the Borel measures $\nu_N$ converge weakly to the one obtained directly in the continuum using the Bochner-Minlos Theorem.