Axioms for Quantum Yang-Mills Theories -- 1. Euclidean Axioms
Min Chul Lee
TL;DR
This work proposes a complete set of Euclidean axioms for Schwinger functions in non-Abelian gauge theories, focusing on gauge-invariant observables and independence from the chosen construction path (lattice or stochastic quantization). It formalizes local gauge actions, co-located operator products, and a renormalization scheme with counterterms to ensure well-defined, gauge-invariant Schwinger functions, and verifies the framework through sanity checks in 2D, 3D, and 4D Abelian/Non-Abelian settings. The paper also rigorously derives the chiral anomaly within this axiomatic scheme and connects local Euclidean formulations to non-local formalisms like Wilson loops. Overall, it lays a rigorous continuum foundation for constructing quantum Yang–Mills theories and sets the stage for Minkowski-space reconstructions and AQFT integration.
Abstract
This paper extends the notion of Schwinger functions to quantum Yang-Mills theories and proposes the axioms they should satisfy. Two main features of this axiom scheme are that we assume existence of gauge-invariant co-located Schwinger functions and impose physical properties only on them. This is in accordance with the fundamental principle of gauge theories that only gauge-invariant quantities can be physical observables.