The parity-preserving $U_A\left(1\right)\times U_a\left(1\right)$ massive QED${}_3$ is ultraviolet finiteness – exhibits vanishing $\beta$-functions, associated to the gauge coupling constants (electric and pseudochiral charges) and the Chern–Simons mass parameter, and all the anomalous dimensions of the fields – as well as is parity and gauge anomaly free at all orders in perturbation theory. The proof is independent of any regularization scheme and it is based on the quantum action principle in combination with general theorems of perturbative quantum field theory by adopting the Becchi–Rouet–Stora (BRS) algebraic renormalization method in the framework of Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) subtraction scheme.

保持奇偶校验 ${ü}_{\mathrm{一种}}（\mathrm{1个}）\times {ü}_{\mathrm{一种}}（\mathrm{1个}）$ 大规模QED${}_3$ 是紫外线的限制–消失了 $\beta$函数，与量具耦合常数（带电和拟手性电荷）和Chern-Simons质量参数以及所有场的异常尺寸有关-以及扰动理论中所有阶次的奇偶校验和量具异常都没有。证明与任何正则化方案无关，它基于量子作用原理并结合微扰量子场论的一般定理，在Bogoliubov-Parasiuk-Hepp框架内采用Becchi-Rouet-Stora（BRS）代数重整化方法–Zimmermann（BPHZ）减法方案。