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In every book/reference on Kaluza-Klein (KK) dimensional reduction, one uses that fluctuations $\delta\Phi(x,y)$ can be expanded as follows $$\delta\Phi(x,y)= \sum_n\delta\Phi_n(x)\,h_n(y)$$ where $\{x,y\}$ are coordinates for a geometry $\mathbb{R}^{1,3}\times X$ and $h_n(y)$ are eigenfunctions of some mass operator $$ M^2 h_n=m_n^2\,h_n\ .$$ For the modes $h_0$ whose mass is zero, in many cases this means they correspond to harmonic forms with respect to a suitable Laplacian. In order to achieve this, a "gauge fixing" conditions is required on the fluctuations which is somehow analogous to Lorentz gauge of electrodynamics.

Now, choosing this gauge is great: it simplifies calculations and makes the particle interpretation of these fluctuations manifest. On the other hand, like for any choice of gauge, the physics should not depend on it. It seems very hard though, if not impossible, to perform the dimensional reduction and define the effective Lagrangian - obtained after integrating on the compact space - outside of this expansion.

Is there any direct way of seeing that the KK physics does not depend on picking Fourier modes?

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