Abstract
We study the effect of a momentum ([ital k]) lattice as a regulator of quantum field theory. An an example, we compute the vacuum polarization in noncompact (linearized) QED from [ital k]-lattice perturbation theory to one-loop order and study the continuum limit. The amplitude has a finite part plus logarithmically, linearly, and quadratically divergent terms. The amplitude violates gauge invariance (Ward identity) and Lorentz (Euclidean) invariance and is nonlocal. For example, the linear term [similar to][Lambda][vert bar][ital k][vert bar] is nonlocal. Renormalization requires nonlocal counterterms, which is not inconsistent because the original action on the [ital k] lattice already has a nonlocality. We explicitly give the counterterms, which render the amplitude Lorentz and gauge invariant to recover the standard result.
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