Imaginary-time normalization for a coupled rotor chain ($\mathcal{S}=\int d\tau dx[I{\dot{\varphi }}^2/(2{\hslash }^2)+J{\varphi }_x^2/2]$)

When imaginary time is divided by $\hslash$ and ranges from zero to $\beta$, integrating rotor momenta gives $I{\dot{\varphi }}^2/(2{\hslash }^2)$ but leaves the potential stiffness term $J({\partial }_x\varphi {)}^2/2$. Factoring $1/{\hslash }^2$ out of both terms requires spatial coefficient ${\hslash }^{2}J$ inside the bracket. Setting $\hslash =1$ hides this distinction, but restoring units must preserve it.