Constructing orthonormal bases in sub-Hilbert spaces of $L^2(\mathbb R)$ defined for particular spatial regions?

Question

I've explored the entanglement of modes by expanding the ground-state solution of a many-body problem as an infinite sum of Slater determinants of one-particle Hermite functions. The one-particle basis of Hermite functions is however defined over all space.

I'm intrigued to see how certain correlation measures relate to geometric features of the system but I would need to expand the ground state in terms of a basis that is defined over a particular spatial region for a particular sub-Hilbert space.

For instance, the 2d radial Hermite functions could be piecewise-defined and divided into two spatial regions; a circle with radius a, for the spatial region $(0,a)$ and a "donut" over the region $(a,\infty)$. This however seems rather hand-wavy and as far as I can tell doesn't seem to give rise to an orthonormal basis for a particular sub-Hilbert space. This should at least serve as an illustration of what I am trying to do.

My question is the following; does anyone have any suggestions as to how I can partition the Hilbert space such that the sub-Hilbert spaces are defined over certain spatial regions which can be easily intuited? and particularly what sort of orthonormal bases would span these Hilbert spaces?

I realize this may not actually be possible given; Can we always find a Hilbert space corresponding to a region of spacetime? (or at least very challenging!). I'm not well versed in QFT but many thanks in advance for any help or suggestions.

Answer 1

The $1$-particle Hilbert space corresponding to a region $\Omega$ of space is $L^2(\Omega)$ and you have : $$L^2(\Omega) \oplus L^2(\Omega^c) = L^2(\mathbb R^d)$$

Now, for a $1$-particle Hilbert space $\mathfrak h$, the $N$-particle space associated with it is $S_\nu \mathfrak h^{\otimes n}$ where $S_\nu$ is the symmetrization (resp. anti-symmetrization) operator for bosons (resp. fermions) and the full Fock space is : $$\mathcal F_\nu(\mathfrak h) = \bigoplus_{n=0}^{+\infty}S_\nu \mathfrak h^{\otimes n}$$ If is easy to see that we have : $$S_\nu(\mathfrak h_1 \oplus \mathfrak h_2)^{\otimes n} = \bigoplus_{k=0}^n(S_\nu\mathfrak h_1^{\otimes k})\otimes (S_\nu\mathfrak h_2^{\otimes (n-k)})$$ and therefore: $$\mathcal F_\nu(\mathfrak h_1 \oplus \mathfrak h_2) = \mathcal F_\nu(\mathfrak h_1) \otimes \mathcal F_\nu(\mathfrak h_2)$$

When considering a many-body system over a partition of space $(\Omega,\Omega^c)$, you need the full Fock-space for the Hilbert space to split as a nice tensor product.