Authors: Brian Slovick
In the conventional approach of renormalization, divergent loop integrals are regulated and combined with counterterms to satisfy a set of renormalization conditions. While successful, the process of regularization is tedious and must be applied judiciously to obtain gauge-invariant results. In this Letter, I show that by recasting the renormalization conditions as the initial conditions of momentum-space differential equations for the loop amplitudes, the need for regularization disappears because the process of differentiating under the loop integrals renders them finite. I apply this approach to successfully renormalize scalar $\phi^4$ theory and QED to one-loop order without requiring regularization or counterterms. Beyond considerable technical simplifications, the ability to perform renormalization without introducing a regulator or counterterms can lead to a more fundamental description of quantum field theory free of ultraviolet divergences.
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