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What kind of things can go wrong if we try to do quantum mechanics on a nonseparable Hilbert space? I have heard that usual mathematical manipulations that we take for granted will no longer hold. What I am looking for is either a general high-level argument explaining why things would go wrong or some specific examples where it is so.

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1 Answer 1

I'm normally loath to just quote Wikipedia, but it does have some relevant things to say:

The article on separabel spaces tells us that

every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.

(cf densely defined operator) and that

many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into algorithms for use in numerical analysis [...]. A famous example of a theorem of this sort is the Hahn–Banach theorem.

Furthermore, the article on Hilbert spaces contains the following:

A Hilbert space is separable if and only if it admits a countable orthonormal basis.

In case of field theory, it states:

Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined).

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Probably compactness, metrizablility, and Hahn-Banach are most important. To understand those pretty much requires at least a masters in math though... –  daaxix Apr 11 '13 at 7:17
This answer doesn't really address what is wrong with nonseparable spaces. –  Sudip Paul Apr 12 '13 at 0:34
@SudipPaul: quote 1 implies that in general, you can no longer define operators on just a countable subset; quote 2 implies that Hahn-Banach becomes non-constructive: Zorn's lemma will tell you that there is a linear extension, but you have no way to actually get it; quote 3 implies that there's no (countable!) Hilbert basis, which is often assumed in QM when you treat infinite spaces like finite ones; quote 4 implies that restriction to separable spaces doesn't have a known down-side as far as physics go –  Christoph Apr 12 '13 at 6:14
Again, none of these quotes address what is wrong with doing physics on a nonseparable space. Is there some physical result that holds only for the separable case? The reason I asked this question because in LQG you generically get nonseparable Hilbert spaces and this fact is sometime used to criticize LQG. I wanted to know why that should be a problem. –  Sudip Paul Apr 14 '13 at 4:38
According to Arnold Neumaier's answer of physics.stackexchange.com/q/29740, "Due to superselection sectors, the Hilbert space of QED is already nonseparable" –  jjcale May 30 '13 at 16:08

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