Canonical methods in non-Abelian gauge theories

The exploitation of unrenormalized canonical field equations and equal-time commutation relations is generally not fruitful because of the need for infinite renormalization. The asymptotically free non-Abelian gauge theories are much more benign in their ultraviolet behavior. Under certain smoothness assumptions, canonical methods applied to the A_μᵃ(x) A_νᵇ(0) operator-product expansion are shown to lead to information on the exact singularity structure of the theory. As a result, we show (1) that the unrenormalized gauge coupling constant g₀(K) as a function of the cutoff K has the behavior g₀(K)~(lnK)^-1/2, (2) how to renormalize →a^μ×→a^ν (where →a^μ is the unrenormalized gauge field), and (3) that the field-strength tensor ${{∂}}^{{μ}}{{{→}}{a}}^{{ν}}{-}{{∂}}^{{ν}}{{{→}}{a}}^{{μ}} + {g}₀{{{→}}{a}}^{{μ}}×{}{{{→}}{a}}^{{ν}}$ becomes proportional to ${{{→}}{F}}^{{μ}{ν}} = {{∂}}^{{μ}}{{{→}}{A}}^{{ν}} {-} {{∂}}^{{ν}}{{{→}}{A}}^{{μ}}$ upon renormalization. These conclusions agree with results obtained by the use of renormalization-group equations.