Is the angular degree of freedom on a circle considered a gauge degree?

plasmon
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Is it okay to classify the angular degree of freedom of a point on a circle as a gauge degree

of freedom?

(i). One can add 2Pi to angle without changing the position of point.

(ii). U(1) symmetry exp (i theta)
 
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Typically this corresponds to global translational invariance; in order to be gauge trf. it should be a local symmetry.
 

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