= Imaginary-time normalization for a coupled rotor chain
{title2=$\mathcal S=\int d\tau dx[I\dot\phi^2/(2\hbar^2)+J\phi_x^2/2]$}
When imaginary time is divided by $\hbar$ and ranges from zero to $\beta$, integrating rotor momenta gives $I\dot\phi^2/(2\hbar^2)$ but leaves the potential stiffness term $J(\partial_x\phi)^2/2$. Factoring $1/\hbar^2$ out of both terms requires spatial coefficient $\hbar^2J$ inside the bracket. Setting $\hbar=1$ hides this distinction, but restoring units must preserve it.
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