Mathematical meaning for the (continuous) Sine-Gordon transformation

Question

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.

Answer 1

I'm not sure what exactly the "Sine-Gordon transformation" is (Frölich's article doesn't use that terminology), but I guess your question is specifically about the meaning of the symbol ${:} e^{i \epsilon T(x)} {:}_V$, when not inside the expectation value $\langle - \rangle_V$, and what algebraic manipulations are allowed with it.

Ideally, it would be an element of the algebra of functions on $\mathcal{S}'(\mathbb{R})$, so that its expectation value $\langle - \rangle_V$ could be taken by integrating against the Gaussian measure $d\mu_V$. Of course, as you point out, that is not possible. But it is very easy to take the algebra of functions on $\mathcal{S}'(\mathbb{R})$ and extend it to formal series in the new formal variables ${:} T^k(x) {:}_V$, for various $x$ and $k$. Then $${:} e^{i\epsilon T(x)} {:}_V := \sum_{k=0}^\infty \frac{(i\epsilon)^k}{k!} {:} T^k(x) {:}_V$$ is a well-defined element of this extended algebra, which can be thought of as a generating function for the elements ${:} T^k(x) {:}_V$. So expressions involving ${:} e^{i\epsilon T(x)} {:}_V$ can be manipulated using the usual algebraic operations. The expectation value $\langle - \rangle_V$ can be uniquely extended to the larger algebra by requiring that your (2) holds as an identity of corresponding generating functions, where the limit on the left-hand side has already been computed.

Strictly speaking, at this stage, applying $\langle - \rangle_V$ to an element of the formal series algebra only gives a formal series of real (or complex) numbers. But formal numerical series can also be manipulated using the usual algebraic operations. Moreover, if manipulating absolutely convergent series in such a way, the result is again absolutely convergent. And indeed, the left-hand side of (2) is given by an absolutely convergent series. So you can presume that Frölich implicitly sums all such formal numerical series. You will notice that only such (implicitly summed) expectation values are subject to inequalities in Frölich's article.

Frölich is also using smeared generating functions, which can be defined by $$\int dx\, g(x) {:} e^{i\epsilon T(x)} {:}_V := \sum_{k=0}^\infty \frac{(i\epsilon)^k}{k!} {:} T^k(g) {:}_V,$$ where ${:} T^k(g) {:}_V$ are yet new formal variables, whose expectation values are required to satisfy $$ \langle {:} T^k(g) {:}_V \rangle_V = \int dx \, g(x) \langle {:} T^k(x) {:}_V \rangle_V, $$ so I think you get the idea.

So, why use formal series? First of all, they absolutely have a precise and rigorous mathematical meaning, just as I have explained above. Second, the use of generating functions, even formal ones, can dramatically simplify and shorten various formulas. As an exercise, try to imagine how much more complicated the formulas in Frölich's article would have been without resorting to generating functions.