Time ordering, anomalies, and chiral gauge theories

For chiral gauge theories, singularity of the time-ordered products results in some ambiguity spoiling the Tomonaga-Schwinger equation; this is explicitly demonstrated in two- and four-dimensional Abelian theories. The ambiguity can be eliminated by the requirement of gauge invariance and, thus, nonpolynomially modifies the conventional quantization of chiral gauge theories. A correctly quantized theory should suffer no gauge anomalies, neither the perturbative ''triangle'' nor the nonperturbative Witten anomalies