Finite-size effects in the phi(4) field and lattice theory above the uppercritical dimension

We demonstrate that the standard O(n) symmetric phi(4) field theory does no t correctly describe the leading finite-size effects near the critical poin t of spin systems with periodic boundary conditions on a d-dimensional latt ice with d > 4. We show that these finite-size effects require a descriptio n in terms of a lattice Hamiltonian. For n --> infinity and n = 1, explicit results are given for the susceptibility and for the Binder cumulant. They imply that these quantities do not have the universal properties predicted previously and that recent analyses of Monte Carlo results for the five-di mensional Ising model are not conclusive.