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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.

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  • $\begingroup$ 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. $\endgroup$ Jul 8 at 19:36
  • $\begingroup$ 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! $\endgroup$ Jul 9 at 7:46
  • $\begingroup$ 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. $\endgroup$ Jul 9 at 17:03
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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.

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  • $\begingroup$ This is definitely the correct way to formalize the problem, thanks for spelling it out. I'm still not entirely sure how exactly this will work in practice, i.e. what basis to choose for a region of space and its complement but I have a better idea as to how to approach the problem now. $\endgroup$ Jul 9 at 13:39

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