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 $\mathbb R^\infty$
https://www.researchgate.net/publication/371006617

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

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