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The most general case would of course be to take the fields to be of the form $\Phi ( x , y ) = \sum _n \phi _n (x,y) $. By taking the field to be of the form, $\Phi ( x , y ) = \sum _n f _n (y) \phi _n (x) $ and just carrying out the integral over $y$, we are considering an effective field theory of the full theory. As you can clearly see the effect of the ...


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The integrals essentially follow from the fact that the Kaluza-Klein scalar is expanded in a Fourier series in terms of an orthonormal basis. This can be understood if we write down $f(y)$ explicitly: $$f_n(y)=\exp(iny/R),$$ where n can take on values between $-\infty$ and $\infty$, and $R$ is the radius of the compactified dimension. The orthonormality ...


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Isn't the first integral equation just the orthonormality assumption that the basis functions $f_n$ must possess when trying to expand an arbitrary function using them, just like the Fourier basis $f_n(y)=e^{iny}$, etc? I am not sure where the second equation comes from (maybe you could give a reference), but assuming the same exponential basis one finds: ...



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