# Why do quantum observables form an associative algebra in some contexts?

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In elementary quantum mechanics, we learn that quantum observables are self-adjoint operators that act on the Hilbert space of states.

However, in more advanced context, we talk of local operators, which are supposed to be quantum observables supported locally, forms an algebra.

First, there is the Weyl algebra appearing in deformation quantization. It is the universal enveloping algebra of the Heisenberg Lie algebra, and obtained by deforming the the space of classical observables.

Second, in CFT, we assume operator product expansion. By multiplying two fields, we obtain another.

I would be grateful to anybody who could clear things out for me.

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