  By averaging over an ensemble of field configurations, a classical field
theory can display many of the characteristics of quantum field theory,
including Lorentz invariance, a loop expansion, and renormalization effects.
There is additional freedom in how the ensemble is chosen. When the field mode
amplitudes have a Gaussian distribution, and the mode phases are randomly
distributed, we review the known differences between the classical and quantum
theories. When the mode amplitudes are fixed, or have a nongaussian
distribution, the quartic and higher correlations among the free fields are
modified, seemingly in a nonlocal way. We show how this in turn affects the
perturbative expansion. We focus on $\lambda\phi^4$ theory in 1+1 dimensions
and use lattice simulations to augment our study. We give examples of how these
nonlocal correlations induce behavior more similar to quantum field theory, at
both weak and strong coupling.
