A finitistic quantum field theory, as a model for a unified theory of elementary particles is proposed, making the assumption that all spatial and temporal distances can only assume integer values of a fundamental length and time, for which we have chosen the Planck length and Planck time. To satisfy the condition of causality in its quantized version, the theory must be exactly nonrelativistic, because only then can the concept that points in space are separated by multiples of a fundamental length be formulated in an invariant way. The theory is formulated as a partial finite difference equation, invariant under translations and rotations in space, and translations in time, and can be expressed as a partial differential equation of infinite order. The theory is free of all divergencies, has a positive definite metric in Hilbert space and therefore no ghost states. Possessing a fundamental length, chosen to be equal the Planck length, the theory has an inbuilt cut-off at the Planck energy. For energies sufficiently below the Planck energy, the theory can be approximated by the previously described Planck aether model, which can be viewed as a superfluid consisting of Planck masses, leading to special relativity as a dynamic symmetry in the asymptotic limit of low energies.