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We construct an effective theory for the SM with extra dimensions. We startfrom a theory governed by the extra-dimensional groups ISO$(1,3+n)$ and $G_{\rmSM}({\cal M}^{4+n})=SU_C(3,{\cal M}^{4+n})\times SU_L(2,{\cal M}^{4+n})\timesU_Y(1,{\cal M}^{4+n})$, characterized by an unknown energy scale $\Lambda$ andvalid at energies far below this scale. Assuming that the extra dimensions aremuch larger than the distance scale of this theory, we construct an effectivetheory with symmetry groups ISO(1,3), $G_{\rm SM}({\cal M}^{4})$. The theoriesare connected by a canonical transformation that hides ISO$(1,3+n)$, $G_{\rmSM}({\cal M}^{4+n})$ into ISO(1,3), $G_{\rm SM}({\cal M}^{4})$; KK fieldsreceive mass. Using a set of orthogonal functions$\{f^{(\underline{0})},f^{(\underline{m})}\}$, generated by the Casimirinvariant $\bar{P}^2$ associated with the translations subgroup$T(n)\subset$ISO$(n)$, we expand the degrees of freedom of ISO$(1,3+n)$,$G_{\rm SM}({\cal M}^{4+n})$ in general Fourier series, whose coefficients arethe degrees of freedom of ISO(1,3), $G({\cal M}^{4})$. These functions, whichcorrespond to the projection on $\{|\bar{x}\big>\}$ of the discrete basis$\{|0\big>,|p^{(\underline{m})}\big>\}$ of $\bar{P}^2$, are central to definethe effective theory. Components along the ground state$f^{(\underline{0})}=\big <\bar x|0\big>$ do not receive mass at thecompactification scale, so they are the SM fields; components along excitedstates $f^{(\underline{m})}=\big<\bar x|p^{(\underline{m})}\big>$ get mass atthis scale, so they are KK excitations. For any direction$|p^{(\underline{m})}\neq0\big>$ there are a massive gauge field and apseudo-Goldstone boson. We stress resemblances of this mass-generatingmechanism with the Englert-Higgs mechanism and discuss physical implications.We include a full catalog of Lagrangian terms that can be used to calculateFeynman rules.

The Standard Model in extra dimensions and its Kaluza-Klein effective Lagrangian