Abstract
Given a number field K with a Hecke character chi, for each place v we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of chi. These theories appear in the study of p-adic string theory and p-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of chi. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places v, the field theory two-point functions corresponding to the Green's functions satisfy an adelic product formula, which is equivalent to the global functional equation for Zeta integrals. In particular, this points out a role of Tate's thesis in adelic physics.