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In the past months, I've been trying to understand the so-called Sine-Gordon transformation, so I've posted some questions here about this topic. I also did an extensive research about this subject, so I came to some conclusions. Still, I have some questions that I'd like to share with you. First, I'll draw the general picture and some of my conclusions.

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 $j=1,...,N$. We can prove 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 $:e^{iT(f)}:_{V} := e^{iT(f)}e^{\frac{1}{2}B(f,f)}$. Let's introduce the notation: \begin{eqnarray} 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} \label{2} \end{eqnarray} The right hand side of (\ref{2}) does not make sense, once $T$ cannot be evaluated pointwise. However, this is just a notation. Now, the partition function of a system in the grand-canonical ensemble is given by: \begin{eqnarray} \Xi_{\Lambda}(\beta, z) = 1+\sum_{N=1}^{\infty}\frac{z^{N}}{N!2^{N}}\sum_{\substack{\epsilon_{j}=\pm 1 \\ j=1,...,N}}\int_{\Lambda^{N}}dx_{1}\cdots dx_{N}e^{-\beta \sum_{1\le i<j\le N}\epsilon_{i}\epsilon_{j}V(x_{i},x_{j})} \tag{3}\label{3} \end{eqnarray} Thus, we can rewrite (\ref{3}) using the notation in (\ref{2}): \begin{eqnarray} \Xi_{\Lambda}(\beta,z) = 1+\sum_{N=1}^{\infty}\frac{z^{N}}{N!2^{N}}\sum_{\substack{\epsilon_{j}=\pm 1 \\ j=1,...,N}}\int_{\Lambda^{N}}dx_{1}\cdots dx_{N}\bigg{\langle}\prod_{j=1}^{N}:e^{i\sqrt{\beta}\epsilon_{j}T(x_{j})}:_V\bigg{\rangle}_{V} \tag{4}\label{4} \end{eqnarray} This motivates another notation simplification. We interpret the integrand in (\ref{4}) as a $N$ iterated integrals, so that we write: \begin{eqnarray} \frac{1}{2^{N}}\sum_{\substack{\epsilon_{j}=\pm 1 \\ j=1,...,N}}\int_{\Lambda^{N}}dx_{1}\cdots dx_{N}\bigg{\langle}\prod_{j=1}^{N}:e^{i\sqrt{\beta}\epsilon_{j}T(x_{j})}:_V\bigg{\rangle}_{V} \equiv \bigg{(}\frac{1}{2}\bigg{\langle}\sum_{\epsilon = \pm 1}\int_{\Lambda} dx :e^{i\sqrt{\beta}\epsilon T(x)}:_{V}\bigg{\rangle}_{V}\bigg{)}^{N} \equiv \bigg{(}\bigg{\langle}\int_{\Lambda}:\cos \sqrt{\beta}T(x):_{V}dx\bigg{\rangle}_{V}\bigg{)}^{N} :=\langle C_{\Lambda,\beta}\rangle_{V}^{N} \tag{5}\label{5} \end{eqnarray} Finally, we have: \begin{eqnarray} \Xi_{\Lambda}(\beta,z) = \sum_{N=0}^{\infty}\frac{z^{N}}{N!}\langle C_{\Lambda,\beta}\rangle_{V}^{N} \equiv \langle\exp(z C_{\Lambda, \beta})\rangle_{V} \tag{6}\label{6} \end{eqnarray} Relation (\ref{6}) is called the Sine-Gordon transformation. Now, this being said, I'd like to raise some questions.

(1) If my reasoning is correct, the Sine-Gordon representation is formal, in the sense that it does not represent a proper Gaussian integral; instead, it is just a matter of notation. If this is the case, I'm ok with that, but I don't get the point here. If (\ref{6}) is just a matter of notation, why is it useful? If I draw any conclusion having (\ref{6}) as a starting point, why should it actually hold if all this is formal written? I know Gaussian integrals are useful tools but this is not a proper Gaussian integral, right?

(2) Is it possible to give precise meaning to (\ref{6})? Is there any construction where $\Xi_{\Lambda}(\beta,z)$ is an actual Gaussian measure and, in case there is, how to proceed? (It's not unusual to come across some paper where the Sine-Gordon transformation is treated as a real mathematically meaningful representation, so I wonder if this I'm just misreading it or if there is actually have a meaningful version of it).

(3) In practice, notation (\ref{2}) is useful because it allows us to perform some formal operations such as interchanging integrals and products as in (\ref{5}) and (\ref{6}). Can some of these operations be properly justified? In other words, to what extent does (\ref{6}) holds only as a formal series?

Remark: For completeness, I've based this post mainly on Fröhlich's work and Fröhlich and Park's work. Other good references are Brydges and Federbush's work and Dimock's work.

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  • $\begingroup$ is this question really different from mathoverflow.net/q/350040/11260 ? $\endgroup$ – Carlo Beenakker Feb 16 at 14:45
  • $\begingroup$ Yes. Let me elaborate. In the first question I tried to understand the mathematical meaning of (2). This is because I thought there might be such meaning. At the moment, I think the Sine-Gordon transform is meant to be formal, as many papers state. But this raises other questions such as (1) what's the point of such a formal definiton, (2) is this the only construction? $\endgroup$ – IamWill Feb 16 at 15:29
  • $\begingroup$ Of Course, there are some intersections between these two questions. The answer I got in the prévios question was great, but the authors seem to understand these topics in a different way (none of them, for instance, make explicit use of formal series) so I'm trying to understand what am I missing and how should I read all these properly. $\endgroup$ – IamWill Feb 16 at 15:33
  • $\begingroup$ A possible approach to make sense of the formula which is not just formal: take your random $T$, convolve it with a rescaled mollifier and see if the result converges as a random variable on $\Omega=S'$ with values in some Besov or other function space made of true functions inside $S'$. If $V$ is sufficiently regular, you may be able to show in this way the measure $d\mu_V$ is supported on a space of function instead of all of $S'$. See arxiv.org/abs/1502.07335 and arxiv.org/abs/1506.05740 $\endgroup$ – Abdelmalek Abdesselam Feb 16 at 16:31
  • $\begingroup$ Thanks for your comments and references. Let me ask you something: in general, the sine-gordon is used as I stated in only formal way? I hardly never see different approaches. $\endgroup$ – IamWill Feb 16 at 16:41

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