Self-adjoint Laplace operator with translation invariance on infinite-dimensional space

It is an introduction to the manuscript.

Self-adjoint Laplace operator with translation invariance on infinite-dimensional space ${\mathbb{R}}^{\mathrm{\infty }}$
https://www.researchgate.net/publication/371006617

We define the translation-invariant Laplacian $-{\mathrm{△}}_{{\mathbb{R}}^{\mathrm{\infty }}}$ on the product measurable space ${\mathbb{R}}^{\mathrm{\infty }}$ as a non-negative self-adjoint operator in some Hilbert space $L^2({\mathbb{R}}^{\mathrm{\infty }})$, which is a subset of the set $CM({\mathbb{R}}^{\mathrm{\infty }})$ of all complex measures on ${\mathbb{R}}^{\mathrm{\infty }}$. Furthermore, we show that for any $f\in L^2({\mathbb{R}}^n)$ and any $u\in L^2({\mathbb{R}}^{\mathrm{\infty }})$, $e^{\sqrt{-1}{\mathrm{△}}_{{\mathbb{R}}^{\mathrm{\infty }}}t}(f\otimes u)=(e^{\sqrt{-1}{\mathrm{△}}_{{\mathbb{R}}^n}t}f)\otimes (e^{\sqrt{-1}{\mathrm{△}}_{{\mathbb{R}}^{\mathrm{\infty }}}t}u)(t\in (-\mathrm{\infty },+\mathrm{\infty }))$
and $e^{{\mathrm{△}}_{{\mathbb{R}}^{\mathrm{\infty }}}t}(f\otimes u)=(e^{{\mathrm{△}}_{{\mathbb{R}}^n}t}f)\otimes (e^{{\mathrm{△}}_{{\mathbb{R}}^{\mathrm{\infty }}}t}u)(t\in [0,+\mathrm{\infty }))$ hold.

Schrodinger equation, heat equation, diffusion equation, Dirichlet form, Gibbs measure, Feynman measure, canonical commutation relation (CCR), infinite particle system.