I have a discussion witrh a colleague regarding the ADM formalism and foliation independence. My argument goes as follows: once I have chosen one (arbitrary) foliation the theory has two constraints H and D. H ~ 0 guarantuess that physical states do not change in coordinate-time, so physics essentially stays the same along the direction perpendicular to the foliation. Because the initially chosen foliation was arbitrary, the Hilbert space does not depend on the initial foliation and it does not change as the foliation changes in time (nothing changes in time as H ~ 0). Therefore the theory (Wheeler-deWitt, LQG) is foliation-independent. My argument says that this is similar to the relativistic particle with h = p²-m² ~ 0 which guarantuees that the theory is reparametrization-invariant (parameterization of the world line of the particle). A specific foliation (= a specific parameterization) does not appear in the final theory, therefore the Hilbert space and the whole theory do not depend on it. His argument is different: H ~ 0 does not change the foliation at all. Reparameterization changes only the time coordinate t' = f(t) for each 3-space slice, but not the slice itself. Looking at the theory with (g=metric and A = wave functional A[g]) each foliation creates a different sequence of (g,A). These different sequences will lead to different paths in superspace. My argument only involves symmetries along one specific path. Therefore one specific (g,A) can be contained in different paths which describe different physics. The physics is not only in the state but in the paths which may be different, even if one specific state = slice are identical. He says that my argument with the relativistic particle fails as there is nothing in this picture which corresponds to the foliation. I am sure that my argument is correct (or at least that my conclusion regarding foliation independence is correct even if the argument itself is flawed). Can anybody explain this from a different point of view?