It is known that for a free particle in $\mathbb{R}$, the $C^*$ algebra of the observables is given by the Heisenberg algebra i.e. generated by $p,q$ such that $[q,p]= i$ ($h=1$). For technical reasons (problems in defining a norm) it is convenient to consider the Weyl $C^*$ algebra i.e. generated by $e^{i\alpha q}, e^{i\beta p}$ where $\alpha, \beta \in \mathbb{R}$ with the relation $e^{i\alpha q} e^{i\beta p} = e^{i\beta p}e^{i\alpha q}e^{-i \alpha \beta} $.

What is the $C^*$ algebra of observables for a particle in a ring i.e. $\mathbb{S}^1$? or a ring of radius $R$ ( i.e. $R$$\mathbb{S}^1$) I have read that it is generated by $e^{i n \theta}, e^{i \beta p_\theta}$ where $n\in \mathbb{Z}$ and $\beta \in \mathbb{R}$ but I can't find the commutation relations. Also I would like to understand how we obtain such an algebra of observables.


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