It would be extremely useful to know whether a particular low energy effective theory might have come from a compactification of a higher dimensional space. Here, this problem is approached from the ground up by considering theories with multiple interacting massive gravitons. It is actually very difficult to construct discrete gravitational dimensions which have a local continuum limit. In fact, any model with only nearest neighbor interactions is doomed. If we could find a non-linear extension for the Fierz-Pauli Lagrangian for a graviton of mass $m_g,$ which does not break down until the scale ${\Lambda }_2=\sqrt{m_g{M}_{\mathrm{Pl}}},$ this could be used to construct a large class of models whose continuum limit is local in the extra dimension. But this is shown to be impossible: a theory with a single graviton must break down by ${\Lambda }_3{=(m}_g^2{M}_{\mathrm{Pl}}{)}^{1/3}.$ Next, we look at how the discretization prescribed by the truncation of the Kaluza-Klein tower of an honest extra dimension raises the scale of strong coupling. It dictates an intricate set of interactions among various fields which conspire to soften the strongest scattering amplitudes and allow for a local continuum limit, at least at the tree level. A number of candidate symmetries associated with locality in the discretized dimension are also discussed.

• Received 9 April 2003

DOI:https://doi.org/10.1103/PhysRevD.68.024029