We calculate the Gelfand functionsE(f,g;a) for quantized field φ in lattice space,a being the lattice constant. In the limita → 0 the functionals take on two different forms depending upon the “potential”F[φ] of the lattice Hamiltonian (coupling between different lattice sites not included). IfF[φ] is of a short-range type (see text for definition) the limit functional is Gaussian. The corresponding representation of CCR is reducible and its realization apparently non-unique unlessF[φ] is quadratic. The most natural realization is to represent the field as a linear combination of Fock fields whose masses are given by the excitation energies of the lattice Hamiltonian. IfF[φ] is of a long-range type, the limit functional takes the more general form once studied byAraki.