 Solution

Lift the circle to its real-valued angular covering coordinate. The free classical equation is $\ddot{\theta }=0$. For each winding number $p\in \mathbb{Z}$, the endpoint lift is $\theta +2\pi p$, andPositive and negative windings are distinct classical paths with the same physical endpoints. Reducing the angle modulo $2\pi$ returns their trajectories on the ring.