Domain of sum of momentum operators

Question

This is a repost from MathStackExchange (https://math.stackexchange.com/q/4840786/) where however no solution has been found so far.

Given the tensor product of Hilbert spaces $\otimes_{i \in \mathcal{Z}} (\mathcal{H}_i, \psi_i)$ (here $\mathcal{Z}$ is the set of integer numbers, $\mathcal{H}_i = \mathcal{L}^2(\mathcal{R}, dx)$ and $\psi_i \in \mathcal{H}_i$), I would like to understand the following things:

1. What is the domain for the sum of two self-adjoint operators $\hat{p}_1+\hat{p}_2$ acting on the Hilbert space $\mathcal{H}_1\otimes\mathcal{H}_2$, where each $\hat{p} = -i \frac{d}{dx}$ is the momentum operator?

2. I would like also to understand what is the domain of existence for the operator defined as limit $\lim_{n \to \infty}\sum_{i = -n}^{n} \hat{p}_i$, and if it is still self-adjoint with dense domain. Any help is really appreciated.

P.S. Here I assume the Von Neumann definition of infinite tensor products of Hilbert spaces, but I am not sure if there are different definitions not equivalent to this one.

Answer 1

Trivially, all operators $$(A_k): \mathit H_k\to \ H_k$$ act on factor $k$ only and act as identity on the complement, so the embedding functor to its tensor form is

$$(A_k) \to 1_1 \otimes \dots\ 1_{k-1} \otimes A_k \otimes \ 1_{k+1}\dots $$

This is modelled for use of multidimensional integrals as the norm functional for each factor.

$$\int_{X_1,\dots, X_n} f_1(x_1) f_2(x_2) \dots \partial_{x_1} g_1(x_1) g_2(x_2) \dots d^nx =\int f_1(x) g'(x_1) dx_1 * \prod_{k=2}^n f_k(x_k)g_k(x) dx_2\dots dx_n $$

It follows that $$\partial_{x_k} f(x_1,\dots\, x_n)=0 \quad \text{iff the k^{th} factor is the constant function}$$ that is, by the philosophy of tensor products, the factor $$f_k(x_k)=1,$$ the product does not depend on $x_k$ or $f_k(x_k)=0$, the product belongs to the subspace without that variable.

The difficulty to construct infinite tensor products is the endless series of formation of equivalence classes yielding a transfinite limit of finite products or of infinite products yielding finite expectations.

For finite tensor products, work begins with the step, to show, that the Hilbert space of function over $\mathbb R^n$ has bases of products of a single basis in $\mathit L^2(\mathbb R)$ by their tensor product in all variables, that reduce to ordinary products under the multidimensional integral as the constituting bilinear form.