Heat equation and Schrödinger equation with translation invariance on the infinite-dimensional vector space ${\mathbb{R}}^{\mathrm{\infty }}$

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