= Solution
For smooth phase differences, $-J\cos(\phi_{n+1}-\phi_n)=-J+J(\phi_{n+1}-\phi_n)^2/2+\cdots$. After integrating the angular momenta in the same inverse-energy time coordinate $\tau$, the <coupled quantum rotor chain> has the fluctuation action
$$
\boxed{\mathcal S=\int_0^\beta d\tau\int dx\left[\frac{I}{2\hbar^2}(\partial_\tau\phi)^2+\frac J2(\partial_x\phi)^2\right].}
$$
Unit spacing replaces the lattice sum and difference by their continuum limits. The constant energy $-J$ per bond contributes an overall factor $e^{\beta JN}$, suppressed when describing fluctuations. The phase is compact; the Gaussian action describes its smooth sector.
There is an <imaginary-time normalization for a coupled rotor chain> issue in the printed formula. Factoring out $1/\hbar^2$ from the action above places $\hbar^2J$, not $J$, in its spatial bracket. The printed bracket is therefore correct in $\hbar=1$ units, or with its spatial coefficient redefined as $J_{\rm bracket}=\hbar^2J_{\rm Hamiltonian}$. With the same dimensional $J$ as in the Hamiltonian, it lacks that factor.
In physical real time the wave equation is $I\partial_t^2\phi=J\partial_x^2\phi$. The <harmonic phase mode of a quantum rotor chain> consequently has
$$
\boxed{\omega(q)=\sqrt{J/I}\,|q|,\qquad E(q)=\hbar\sqrt{J/I}\,|q|.}
$$
The lattice expression is $\omega(q)=2\sqrt{J/I}|\sin(q/2)|$. The mode is a gapless phase wave analogous to sound in a <superfluid>, with rotor angular momentum conjugate to its phase. In one dimension this harmonic regime has strong long-distance phase fluctuations rather than true continuous-symmetry long-range order; compact <phase slips> can change the infrared phase outside the smooth-phase approximation.
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