Making sense of dirac delta of a function $\delta(\psi) = \int \mathscr{D}\varphi e^{-i\int\bar{\varphi}\psi}$

Question

I am currently trying to learn about the MSR Formalism, which rephrases SDE problems in terms of path integrals. This is in the context of turbulence for plasma physics; I know that path integrals do not really have a rigorous mathematical interpretation yet, aside from the Feynman-Kac formula, but I was wondering if there is a way to make rigorous sense of the following representation of the Dirac $\delta$-function of a function in a paper of Canet and this website:

$$\delta(\psi) \sim \int\mathscr{D}\varphi\ e^{-i\int_t\bar{\varphi}\psi}.\tag{1}$$

Here, $\psi$ is some scalar field of spacetime, and $\mathscr{D}$ is supposed to be a sort of "measure" on the space of these fields. The field $\varphi$ is called an auxiliary or response field. I recognize this is formally like the definition of the Dirac $\delta$ via its inverse Fourier transform, but I am having some difficulty parsing this. Any ideas on how to make sense of this equation would be greatly appreciated!

—Edit—

After some further research, I think what I need is the calculus of white noise processes!

Answer 1

Here are some references which seem appropriate from my pedestrian perspective.

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The formula

from Hida et al.'s White Noise: An Infinite Dimensional Calculus, p.322 seems to match fairly well with the formula in question.

Here

• $\mathcal{S}'(\mathbb{R})$ is the complex valued tempered distributions on $\mathbb{R}$ (i.e. the dual space of the space of complex valued functions on $\mathbb{R}$ all of whose derivatives exist and decay faster than any polynomial) endowed with the cylinder $\sigma$-algebra (= optimal $\sigma$-algebra that makes finitely many simultaneous evaluations measurable),
• $\mu$ is a certain fully supported probability measure like a "Gaussian" on $\mathcal{S}'(\mathbb{R})$,
• $:\bullet:_y$ is an operator like a "renormalization" w/r/t $y\in\mathbb{S}'(\mathbb{R})$,
• $\widetilde{\delta}_0$ is the delta white noise functional at $0\in\mathcal{S}'(\mathbb{R})$.

See also Kubo & Yokoi's "A Remark on the Space of Testing Random Variables in the White Noise Calculus" for the definition of the delta white noise functional, as well as Hida's "Analysis of Brownian Functionals" (it seems there are at least three distinct publications by Hida with this title; one of them is available at https://hdl.handle.net/11299/4378)

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In the section before the section that includes the above formula (along with a formula for Fourier transform), some heuristics are provided: Let $(\Omega,\mathbb{P})$ be a probability space (to be suppressed), $y:\Omega\to\mathbb{R}^k$ be a Gaussian random variable with mean $0$ and covariance operator $\text{Id}$ (see https://en.wikipedia.org/wiki/Covariance_operator). Then for fixed $x\in\mathbb{R}^k$, the expectation $\mathbb{E}_y(\exp(-i\, x\bullet y))$ of the random variable $\Omega\to\mathbb{C}$, $\omega\mapsto \exp(-i\, x\bullet y(\omega))$ ($\bullet$ is the standard inner product on $\mathbb{R}^k$) w/r/t $y$ is $\exp(-|x|^2/2)$. Then the renormalization of $y\mapsto \exp(-i\, x\bullet y)$ w/r/t $y$ is defined by

$$:\exp(-i\, x\bullet y):_y = \dfrac{\exp(-i\, x\bullet y)}{\mathbb{E}_y(\exp(-i\, x\bullet y))} = \exp(-i\, x\bullet y+|x|^2/2).$$

Then for $f\in L^1(\mathbb{R}^k,\text{leb}_{\mathbb{R}^k};\mathbb{C})$ the Fourier transform is:

\begin{align*} \widehat{f}:\mathbb{R}^k\to\mathbb{C},\,\, y\mapsto &\dfrac{1}{(2\pi)^{k/2}}\int_{\mathbb{R}^k} f(x)\,\exp(-i\, x\bullet y)\, d\text{leb}_{\mathbb{R}^k}(x) \\ &= \int_{\mathbb{R}^k} f(x)\, :\exp(-i\, x\bullet y):_y\, d\text{gau}_{\mathbb{R}^k}(x), \end{align*}

where $\text{leb}_{\mathbb{R}^k}$ and $\text{gau}_{\mathbb{R}^k}$ are the Lebesgue measure and the standard Gaussian measure on $\mathbb{R}^k$, respectively. In the infinite-dimensional setting the formula above is based on this (although the renormalization is more involved, and the formula for Fourier transform is still more subtle).