I've been trying to understand the so-called Sine-Gordon Transformation which occurs in both classical and quantum statistical mechanics. One of the most cited references on this topic seems to be Fröhlich's article, which is my main reference at the moment. We consider a function $V: \mathbb{R}^{n}\times \mathbb{R}^{n}$ which is continuously differentiable, satisfies $\sup_{x,y \in \mathbb{R}^{n}}|V(x,y)| \le K$ and $$ \langle f,Vg \rangle := \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}f(x)V(x,y)g(y)ddy \ge 0 \tag{1} $$ for every $f,g \in L^{2}(\mathbb{R}^{n})$. If we define $B: \mathcal{S}(\mathbb{R}^{n})\times \mathcal{S}(\mathbb{R}^{n})$ to be $B(f,g) \equiv \langle f, Vg\rangle$, the associate quadratic form $f \mapsto B(f,f)$ is non-negative, so that, by Minlos' Theorem, there exists some (Gaussian) measure $\mu_{V}$ on $\mathcal{S}'(\mathbb{R}^{n})$ such that $$ W(f) := e^{-\frac{1}{2}B(f,f)} = \int_{\mathcal{S}'(\mathbb{R}^{n})}d\mu_{V}(T)e^{iT(f)}$$ Because $\mathcal{S}(\mathbb{R}^{n})\subset \mathcal{S}'(\mathbb{R}^{n})$, each $f \in \mathcal{S}(\mathbb{R}^{n})$ induces a distribution in $\mathcal{S}'(\mathbb{R}^{n})$. Thus, if we fix $\epsilon_{1},...,\epsilon_{N}\in \mathbb{R}$ and $x_{1},...,x_{N}\in \mathbb{R}^{n}$, we can choose sequences $\{f_{l}^{(j)}\}_{l\in \mathbb{N}}$ such that $f_{l}^{(j)} \to \epsilon_{j}\delta_{x_{j}}$, for each $j=1,...,N$. Fröhlich proves that $$\lim_{l\to \infty}\int_{\mathcal{S}'(\mathbb{R}^{n})}d\mu_{V}(T)\prod_{j=1}^{N}:e^{iT(f_{l}^{(j)})}:_{V} = e^{-\sum_{1\le i< j \le N}\epsilon_{i}\epsilon_{j}V(x_{i},x_{j})}$$ where $:e^{iT(f)}:_{V} := e^{iT(f)}e^{\frac{1}{2}B(f,f)}$. Everything looks fine till now. Fröhlich introduces the notation: $$lim_{l\to \infty}\int_{\mathcal{S}'(\mathbb{R}^{n})}d\mu_{V}(T)\prod_{j=1}^{N}:e^{iT(f_{l}^{(j)})}:_{V} \equiv \bigg{\langle}\prod_{j=1}^{N}:e^{i\epsilon_{j}T(x_{j})}:_V\bigg{\rangle}_{V} \tag{2} $$ The right hand side of (2) does not make sense, once $T$ cannot be evaluated pointwise. However, this seems to be defined only as a notation, which is fine to me. The problem is that the Sine-Gordon transformation, which follows these calculations, is obtained by means of manipulations of (2). For instance, one can write the partition function of the system as a Gaussian integral with respect to the measure $\mu_{V}$, but this seems to be ill-defined to me.

**Question:** Is it possible to give mathematical meaning to this Sine-Gordon transformation? Am I missing something? Or this version of the Sine-Gordon transformation can only be defined in a formal way? What is the purpose of defining the partition function in terms of a Gaussian measure if it is formal and does not have mathematical meaning?

**EDIT:** I should clarify what I mean by 'Sine-Gordon Transformation' once Fröhlich does not use this term in his article. The Sine-Gordon Transformation is equation (2.24) in Fröhlich's work, which is a way of writing the partition function $\Xi_{V}(z)$ in terms of a Gaussian measure.