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

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.

• Are you referring to an $N$-particle model in which the Hilbert space consists of completely symmetric/antisymmetric functions of the particles' coordinates (a wavefunction), and is that symmetry/antisymmetry the main obstacle? The model can be reformulated in terms of local creation/annihilation operators, and then the symmetry/antisymmetry is encoded in the operator algebra instead. Part 2 of my answer to another question describes that formulation in some detail. It doesn't answer your question directly, but it gives a conceptual foundation. Jul 8 at 19:36
• Sorry, maybe I should have specified: the model I am referring to is called the N-Harmonium which is a model of N particles in a harmonic trap with harmonic inter-particle interactions. This model can encompass fermionic or bosonic particles and has an analytic ground-state solution, I am looking at fermionic constituents in this case. I have been able to construct the anti-symmetric wavefunctions in the particles' coordinates, but thanks anyhow! Jul 9 at 7:46
• That's consistent with what I thought you were asking, but maybe my request-for-clarification wasn't clear enough. The formulation using an (anti)symmetric wavefunction tends to obscure the very thing that I think you're asking about (if I understand the question correctly), namely how the Hilbert space can be partitioned into mutually orthogonal subspaces corresponding to different regions of space. The local creation/annihilation operator formalism makes that feature more obvious, because it highlights the fact that observables are tied to regions of space, not to particles. Jul 9 at 17:03

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.