Theory of Abelian projection

Analytic methods for Abelian projection are developed. A number of results are obtained related to string tension measurements. It is proven that even without gauge fixing, Abelian projection yields string tensions of the underlying non-Abelian theory. Strong arguments are given for similar results in the case where gauge fixing is employed. The methods used emphasize that the projected theory is derived from the underlying non-Abelian theory rather than {ital vice versa}. In general, the choice of subgroup used for projection is not very important, and need not be Abelian. While gauge fixing is shown to be in principle unnecessary for the success of Abelian projection, it is computationally advantageous for the same reasons that improved operators, e.g., the use of fat links, are advantageous in Wilson loop measurements. Two other issues, Casimir scaling and the conflict between projection and critical universality, are also discussed. {copyright} {ital 1999} {ital The American Physical Society}