Convex decomposition of the gauge field and background-field quantization

The 1PI (one-particle-irreducible) two-point function of a pure Yang-Mills gauge theory is computed. The background field method is employed in a slightly altered form that makes use of the convexity of the space of gauge fields. It is shown how this avoids the singularity of the matrix ∂2S/∂V∂V thereby allowing the calculation of the Gaussian integral in the generating functional without having to fix the gauge. It is also shown how, when it comes to actually calculating the 1PI two-point function by a perturbative method, a singularity in a particular term, M(0), of the total matrix M(0) necessitates the introduction of a gauge-fixing term. The 1PI two-point function is shown to be identical to that of the conventional background-field method except for the presence of a new parameter, t, introduced by the convex decomposition of the gauge field. (author)