A class of interacting classical random fields is constructed using deformed *-algebras of creation and annihilation operators. The fields constructed are classical random field versions of "Lie fields". A vacuum vector is used to construct linear forms over the algebras, which are conjectured to be states over the algebras. Assuming this conjecture is true, the fields constructed are "quantum random fields" in the sense that they have Poincare invariant vacua with a fluctuation scale determined by Planck's constant. A nonlocal particle interpretation of the formalism is shown to be the same as a particle interpretation of a quantum field theory.