Nonlocality I: Nonlocality, singularity, and elastic scattering

In this paper we first express the quantum nonlocality of AB-effect type in the form of singularity using the method of path integrals. By considering the gauge fix term in path integrals as usually done, the condition that AB effect happens is automatically satisfied. And then one can reach the conclusion that all the scattering processes are accompanied by nonlocal effect. The formulae are extended to field theory that includes the scalar potential along a rational routine. It turns out that the degree of freedom of nonlocality in quantum fields is just the degree of the ghosts. Furthermore, it is found that renormalization method is relative to this type of nonlocal effect