= Solution
The <particle on a circle> has the complete orthonormal basis of <energy eigenstates> $\{|n\rangle:n\in\mathbb Z\}$. Its spectral representation gives
$$
\begin{aligned}
K(\theta_f,T;\theta_i,0)
&=\sum_{n\in\mathbb Z}\langle\theta_f|n\rangle
e^{-iE_nT/\hbar}\langle n|\theta_i\rangle\\
&=\boxed{\frac1{2\pi}\sum_{n\in\mathbb Z}
\exp\!\left[in(\theta_f-\theta_i)
-\frac{in^2\hbar T}{2mR^2}\right]}.
\end{aligned}
$$
The integer $n$ is the quantized <angular momentum> in units of $\hbar$.
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