Ulla Mattfolk No I actually have higher dimensional Lagrangians and lower D. You can consider the lower dimensions in isolation of higher because the the higher can be cast as stereogeaphic projection (in which case they don't go to zero) or as zero if you neglect for the higher D effects which are in most situations minor. I only actually set them to zero to obtain plain ordinary GR not for the purpose of the standard model which requires higher dimensions. You don't have to consider standard model effects to consider GR as recovered in isolation. So you have to consider what you want in your Lagrangian. The full Lagrangian in 4D contains stereogeaphic projections from higher D to accomodate both GR and the standard model in one dynamic 4D Lagrangian. But the complete model is in 9D with some properties only recovered at higher subshell dimensions. Each shell handles different properties, so the 9D isn't excessive in terms of motions or degrees of freedom. It just accounts for the full range.
