Quantization of the self-dual Yang-Mills system: Exchange algebras and local quantum group in four-dimensional quantum field theories

Abstract

We have constructed a quantum field theory for the self-dual Yang-Mills system in terms of the group-valued fields J^. They satisfy exchange algebras, of which the structure matrices R^ satisfy Yang-Baxter equations. We show that the fields J^ form noncommutative vector spaces of a local quantum group and their products at short distances have nontrivial critical exponents. We obtain the quantum Hamiltonian and equations of motion; identify the generators for their symmetries; and construct the affine-Lie-algebra currents, Virasoro-algebra fields, and hierarchies of linear and nonlinear systems.