A further pathology of the Coulomb gauge in non-abelian Yang-Mills theories

Abstract

In the first part the vacuum structure of SU(2) Yang-Mills theories in the Coulomb gauge is discussed. It is proved that the only transverse pure gauge field A μ( x) = U −1 ϖ μ U with U( x) r→∞ → const. is the trivial one A μ ( x) ≡ 0; the features of other possible vacua with U(x) r→∞ → U(θ, ¢p) are studied. In the second part, regular Euclidean configurations that connect a vacuum state at x 4 = −∞ to another at x 4 = +∞ are discussed. It is proved, always working in the Coulomb gauge, that the perturbative vacuum A μ ( x) ≡ 0 cannot tunnel into any other one and that regular configurations with non-vanishing Pontryagin number q cannot affect such a vacuum. Moreover, strong arguments are given to show that many-instanton configurations (| q|⩾2) cannot be expressed at all in the Coulomb gauge, that is by a regular field A μ satisfying the transversality condition ϖ i A i ( x, x 4) = 0.