  We investigate nonlocal field theories, a subject that has attracted some
renewed interest in connection with nonlocal gravity models. We study, in
particular, scalar theories of interacting delocalized fields, the
delocalization being specified by nonlocal integral kernels. We distinguish
between strictly nonlocal and quasi-local (compact support) kernels and impose
conditions on them to insure UV finiteness and unitarity of amplitudes. We
study the classical initial value problem for the partial integro-differential
equations of motion in detail. We give rigorous proofs of the existence but
accompanying loss of uniqueness of solutions due to the presence of future, as
well as past, "delays," a manifestation of acausality. In the quantum theory we
derive a generalization of the Bogoliubov causality condition equation for
amplitudes, which explicitly exhibits the corrections due to nonlocality. One
finds that, remarkably, for quasi-local kernels all acausal effects are
confined within the compact support regions. We briefly discuss the extension
to other types of fields and prospects of such theories.
