  We construct an effective theory for the SM with extra dimensions. We start
from a theory governed by the extra-dimensional groups ISO$(1,3+n)$ and $G_{\rm
SM}({\cal M}^{4+n})=SU_C(3,{\cal M}^{4+n})\times SU_L(2,{\cal M}^{4+n})\times
U_Y(1,{\cal M}^{4+n})$, characterized by an unknown energy scale $\Lambda$ and
valid at energies far below this scale. Assuming that the extra dimensions are
much larger than the distance scale of this theory, we construct an effective
theory with symmetry groups ISO(1,3), $G_{\rm SM}({\cal M}^{4})$. The theories
are connected by a canonical transformation that hides ISO$(1,3+n)$, $G_{\rm
SM}({\cal M}^{4+n})$ into ISO(1,3), $G_{\rm SM}({\cal M}^{4})$; KK fields
receive mass. Using a set of orthogonal functions
$\{f^{(\underline{0})},f^{(\underline{m})}\}$, generated by the Casimir
invariant $\bar{P}^2$ 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 are
the degrees of freedom of ISO(1,3), $G({\cal M}^{4})$. These functions, which
correspond to the projection on $\{|\bar{x}\big>\}$ of the discrete basis
$\{|0\big>,|p^{(\underline{m})}\big>\}$ of $\bar{P}^2$, are central to define
the effective theory. Components along the ground state
$f^{(\underline{0})}=\big <\bar x|0\big>$ do not receive mass at the
compactification scale, so they are the SM fields; components along excited
states $f^{(\underline{m})}=\big<\bar x|p^{(\underline{m})}\big>$ get mass at
this scale, so they are KK excitations. For any direction
$|p^{(\underline{m})}\neq0\big>$ there are a massive gauge field and a
pseudo-Goldstone boson. We stress resemblances of this mass-generating
mechanism with the Englert-Higgs mechanism and discuss physical implications.
We include a full catalog of Lagrangian terms that can be used to calculate
Feynman rules.
