Heat equation and Schr\"{o}dinger equation with translation invariance on the infinite-dimensional vector space $\mathbb R^\infty$

The standard Laplacian $-\triangle_{\mathbb R^n}$ in $L^2(\mathbb R^n)$ is self-adjoint and translation invariant on the finite-dimensional linear space $\mathbb R^n$. In this paper, we define a translation invariant operator $-\triangle_{\mathbb R^\infty}$ on $\mathbb R^\infty$ as a non-negative self-adjoint operator in some non-separable Hilbert space $L^2(\mathbb R^\infty)$. The set $L^2(\mathbb R^\infty)$ is a translation invariant subset of the set $CM(\mathbb R^\infty)$ of all complex measures on the product measurable space $\mathbb R^\infty$. Furthermore, we show that for any $f\in L^2(\mathbb R^n)$ and any $u\in L^2(\mathbb R^\infty)$, the separations of variables $e^{\triangle_{\mathbb R^\infty}t}(f\otimes u) =(e^{\triangle_{\mathbb R^n}t}f)\otimes (e^{\triangle_{\mathbb R^\infty}t}u) \ (t\in [0,+\infty))$ and $e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}(f\otimes u) =(e^{\sqrt{-1}\triangle_{\mathbb R^n}t}f)\otimes (e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}u) \ (t\in (-\infty,+\infty))$ hold. This clearly shows that $-\triangle_{\mathbb R^\infty}$ is an analog of $-\triangle_{\mathbb R^n}$. The starting point for the discussion in this paper is to naturally introduce a translation invariant structure of Hilbert space into $CM(\mathbb R^\infty)$. $L^2(\mathbb R^\infty)$ is a closed linear subspace of $CM(\mathbb R^\infty)$. The inner product of $L^2(\mathbb R^\infty)$ is defined as that of $CM(\mathbb R^\infty)$. For a manifold, H\"{o}rmander defined an inner product that does not depend on a particular measure. In fact, the way we introduce the inner product into $CM(\mathbb R^\infty)$ is a generalization of his. Not only is a statistical manifold on $\mathbb R^\infty$ a submanifold of $CM(\mathbb R^\infty)$, but the real inner product $\mathrm{Re}(\langle \cdot, \cdot \rangle_{CM(\mathbb R^\infty)})$ induces Fisher information metric.