A Canonical and Quantum Formulation of the Self-Dual Yang-Mills System: Algebras and Hierarchies

Ling-Lie Chau；Itaru Yamanaka

Chinese Journal of Physics

30卷6期（1992 / 12 / 01）

885 - 898

英文

We construct a canonical formulation of the Self-Dual Yang-Mills (SDYM) system formulated in the gauge-invariant group-valued J-fields and derive their Hamiltonian and quadratic algebras of the fundamental Dirac brackets. We also show that the quadratic algebras satisfy Jacobi identities and their structure matrices satisfy modified Yang-Baxter equations. From these quadratic algebras, we construct Kac-Moody-like and Virasoro-like algebras. We also discuss their related symmetries, involutive conserved quantities, and hierarchies of nonlinear and linear equations.We then quantize the quadratic algebras of the self-dual Yang-Mills system and obtain a system of four-dimensional quantum exchange algebras and a system of four-dimensional quantum modified Yang-Baxter equations. We are hopeful that such quantum formulation of the self-dual Yang-Mills system will give an explicit example of how to obtain nonperturbative results for four-dimensional quantum field theories.