Integrate out the conjugate momentum to obtain a direct Lagrangian for the length mode

Derive the Lagrangian for the wormhole length mode n(τ), or its fluctuation field η(θ), by analytically integrating out the conjugate momentum k(τ) from the path integral representation of the Heisenberg model, namely starting from S[n(τ),k(τ)] = ∫_0^β dτ (i k(τ) · ḋn(τ) − √(2 n(τ)) cos k(τ)) and obtaining a closed-form Lagrangian solely in terms of n(τ) (or η(θ)) that reproduces the exact generating functional for length correlators.

Background

The paper derives a path integral for the length mode n(τ) and its conjugate momentum k(τ) from the Heisenberg model, leading to an action S[n,k] = ∫ (i k ḋn − √(2n) cos k). A natural next step would be to integrate out k to obtain a Lagrangian depending only on the length field. The authors instead infer the effective action using generating functionals and show non-local features, but they indicate that a direct integration of k is not currently available.

A direct derivation would clarify whether the effective action is local or inherently non-local, and would provide a more explicit dynamical description of the length mode consistent with the exact correlators obtained in the model.