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The fundamental QFT is formulated in a separable Hilbert space. But mostly approaches in condensed matter physics, e. g. thermal field dynamics, use a non-separable Hilbert space. It looks like it is not easy to find how a theory with separable Hilbert space can act like a basement for non-separable Hilbert space of condensed matter physics. Is it really (open) problem or not? if yes, whether it is possible to hope that the decoherence theory can shed light?

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    $\begingroup$ Could you please give me some reference where nonseparable Hilbert spaces are mentioned and used in condensed matter? I suspect that it is due to the attempt to avoid the algebraic approach and describe all unitarily inequivalent representations in an overall Hilbert space, thus nonseparable. $\endgroup$ – Valter Moretti Aug 19 '16 at 19:04
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    $\begingroup$ Yes, exactly, nonseparable Hilbert space used to describe all unitarily inequivalent representations in an overall Hilbert space (see, for example, chapter 2 in Umezawa, Matsumoto, Taethiki "Thermo field dynamics and condensed states" book). Do you want to say that unitarily inquivalent representations exists and are important in a fundamental QFT despite of using separable Hilbert space? $\endgroup$ – warlock Aug 19 '16 at 19:18
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    $\begingroup$ That is the same situation as in QFT, as soon as you drop Poincare' invariace. I mean in curved spacetime. There you may have uncountably many inequivalent representations of the same algebra of observables just by varying some continuous parameter (curvatures). If you put all these reps in orthogonal sectors in a common Hilbert space, it must be nonseparable. Each sector is however separable. $\endgroup$ – Valter Moretti Aug 19 '16 at 19:23
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    $\begingroup$ However I do not see any cogent reason to use that overall space. The algebraic approach is easier to handle. Each single Hilbert space sector is constructed by means of the GNS construction... $\endgroup$ – Valter Moretti Aug 19 '16 at 19:39
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    $\begingroup$ But if we have plain spacetime, there is only one sector? If so, I can not understand how it is possible to describe, for example, ferromagnetic solid body "from scratch", or any other phenomenon with Bose–Einstein condensate. $\endgroup$ – warlock Aug 19 '16 at 19:39
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That is the same situation as in QFT, as soon as you drop Poincaré invariance. I mean in curved spacetime. There you may have uncountably many inequivalent representations of the same algebra of observables just by varying some continuous parameter (curvatures). If you put all these reps in orthogonal sectors in a common Hilbert space, it must be non.separable. Each sector is however separable.

However I do not see any cogent reason to use that overall space. The algebraic approach is easier to handle. Each single Hilbert-space sector is constructed by means of the GNS construction.

In flat spacetime it is not necessary to introduce uncountably many orthogonal sectors making the overall Hilbert space non-separable. In condensed matter I suspect that orthogonality and thus non.separability arises only taking the thermodynamical limit. So these sectors cannot co-exist simultaneously in our universe.

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