Jonathan Dimock: Ultraviolet Stability for Quantum Electrodynamics in d=3

We report on results for quantum electrodynamics on a finite volume Euclidean spacetime in dimension $d=3$. The theory is formulated as a functional integral on a fine toroidal lattice involving both fermion fields and abelian gauge fields. The main result is that, after renormalization, the partition function is bounded uniformly in the lattice spacing. This is a first step toward the construction of the model. The result is obtained by renormalization group analysis pioneered by Balaban. A single renormalization group transformation involves block averaging, a split into large and small field regions, and an identification of effective actions in the small field regions via cluster expansions. This leads to flow equations for the parameters of the theory. Renormalization is accomplished by fine-tuning the initial conditions for these equations. Large field regions need no renormalization, but are shown to give a tiny contribution.

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