Theory of massive and massless Yang-Mills fields

Abstract

Introducing the Lagrangian multiplier field χ^→(x), a canonical formalism for the Yang-Mills fields f^→_μ(x) with mass M≥0 is proposed within the framework of an indefinite-metric quantum field theory. The formalism for the massive f^→_μ has a well-defined zero-mass limit, and the reduction of the physical components of f^→_μ as M→0 is embodied in an elegant way. Using the field equation for χ^→ (x) and the path integral, we find that the "extra" factor in the amplitude due to the interaction of χ^→ (x) in the intermediate states is [det(1+(□+M²)⁻¹gf^→_μ×∂^μ)]^−½≡D_M^−½ for the massive f^→_μ, and that the extra factor is D_M≡0⁻¹ for the massless f^→_μ because of their different degrees of observable freedom. Thus, the resultant rules for the Feynman diagrams for M > 0 and M=0 are not smoothly connected. The theory is covariant, renormalizable, and unitary after the extra parts are removed from the amplitudes. The problems of unitarization and renormalizability are discussed.