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While asking about operators on this site, many answers mentioned "C*-algebras" to be the fundamental mathematical element corresponding to an observable (in QFT and QM at least), and choosing a representation maps an element of this algebra to a linear operator (I hope I got it right until here).

Now my problem is that I can't think of an intuitive way why an element of an c*-algebra corresponds to an observable. I'll quickly explain what I mean by intuitivity here: My usual understanding in classical mechanics is was that "something" happened "somewhere" "somehow", and although this seems rather abstract, it was pretty easy to translate this to mathematical objects: "Somewhere" and "Somewhen" translate to the mathematical object of $R^4$, or more generally to a 4dimensional manifold. The "something" was just the mere existance of a particle, which doesn't need another mathematical object. In the same way a field which will yield a force can intuitively be thought of as a little arrow attached to each point in space-time,so it's plausible that the fields are tensors over the tangential space of the said manifold.

In all cases described, observables ultimately can be mapped to some kind of number. With quantum mechanics, things get a little bit rough, but the the probabillity-interpretation gives an intuitive reasoning why classical observables are replaced by operators: Operators can have eigenvalues (that correspond to numerical values of a classical observable) and the states they act on can be thought as sum of eigenvectors.

Now with an element of an c*-algebra - there are no states that those elements can act on a priori, as it by itself doesn't at first sight correspond to any number. So my question is: Is there any meaningful way to relate a c*-algebra (which is thought of as the mathematical object to describe what we call "observables" to intuitive human conceptions like "When", "where", "what" or "how (big)"?

EDIT: Since the question is too broad, I'd like to restrict to the cases of QFT: Is there a meaningful way to relate an element of a C*-Algebra to the intuitive concepts like "When", "where", what", or "how (big)", more specially to meaningful numbers, in the same way that Operators are connected to classical observables in QM via the operators eigenvalues?

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    $\begingroup$ What algebras? Lie algebras? Associative algebras? $C^{\ast}$-algebras? This post (v1) seems too broad. $\endgroup$ – Qmechanic Oct 8 '19 at 21:14
  • $\begingroup$ I think there is something called G-structure which encodes stuff at places. I am a bit out of my depth here but perhaps that is what you are looking for. $\endgroup$ – Emil Oct 8 '19 at 21:34
  • $\begingroup$ I edited the question to ask only for C*-Algebras, because this seems to be the biggest instance of the word "algebra" being used in the answers I read so far. $\endgroup$ – Quantumwhisp Oct 8 '19 at 22:08
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    $\begingroup$ C*-algebras are more general than a specific physical system -- you put in the "when/where/what/how" when you specify which C*-algebra you're working with. $\endgroup$ – knzhou Oct 8 '19 at 22:19
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    $\begingroup$ It's like how vector spaces are used a lot in quantum mechanics. How do we relate an abstract vector space to "when/where/what/how"? It depends on the situation. Surely the vector space by itself doesn't know about it. $\endgroup$ – knzhou Oct 8 '19 at 22:20

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