I'm reading about the uncertainty principle formulated in the language of $ C^*$ algebras. It states that for any state $\omega: A \rightarrow \mathbb{C}$ on a $C^*$ algebra and for any self-adjoint $a, b \in A$, we have $$ \sigma_{\omega}(a)^{2} \sigma_{\omega}(b)^{2} \geq \frac{1}{4} \omega(i[a, b])^{2} $$ So, what about the particle on a ring (one can read about it on wiki)? LHS vanishes here. In contrast to this formulation, in usual QM we have condition for domains of operators. Should this principle be reformulated because of this case?
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phase operatpr(it is position op but on circle its just phase bounded in finite interval and periodic) is messy, so instead build your alebra from exponential of phase ops – physshyp Nov 03 '20 at 15:57
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see also https://physics.stackexchange.com/a/338057/36194 – ZeroTheHero Nov 03 '20 at 17:58
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thanks for helpful comments! – nabzdyczony Nov 03 '20 at 18:07
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There is a beautiful paper by Levy-Lebond on this: Who is afraid of nonhermitian operators? It's one of my favourites. – Philip Nov 03 '20 at 19:33
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I'm voting to close this since I feel the linked question is essentially a duplicate: Why doesn't the uncertainty principle contradict the existence of definite-angular momentum states? – Philip Nov 03 '20 at 19:36