Theory of massive and massless Yang-Mills fields

Abstract

Introducing the Lagrangian multiplier field $\overrightarrow{\chi }\mathrm{}\left(x\right)\mathrm{}$, a canonical formalism for the Yang-Mills fields ${\overrightarrow{\mathrm{f}}}_{\mu }\mathrm{}\left(x\right)\mathrm{}$ with mass $M\ge 0$ is proposed within the framework of an indefinite-metric quantum field theory. The formalism for the massive ${\overrightarrow{\mathrm{f}}}_{\mu }$ has a well-defined zero-mass limit, and the reduction of the physical components of ${\overrightarrow{\mathrm{f}}}_{\mu }$ as $M\to 0$ is embodied in an elegant way. Using the field equation for $\overrightarrow{\chi }\mathrm{}\left(x\right)\mathrm{}$ and the path integral, we find that the "extra" factor in the amplitude due to the interaction of $\overrightarrow{\chi }\mathrm{}\left(x\right)\mathrm{}$ in the intermediate states is ${\left[\det \right(1+{(\square +M^2)}^{-1\mathrm{}}g{\overrightarrow{\mathrm{f}}}_{\mu }\times {\partial }^{\mu }\left)\right]}^{\frac{-1\mathrm{}}{2}}\equiv {D_M}^{\frac{-1\mathrm{}}{2}}$ for the massive ${\overrightarrow{\mathrm{f}}}_{\mu }$, and that the extra factor is ${D_{M=0\mathrm{}}}^{-1\mathrm{}}$ for the massless ${\overrightarrow{\mathrm{f}}}_{\mu }$ because of their different degrees of observable freedom. Thus, the resultant rules for the Feynman diagrams for $M>\mathrm{}0\mathrm{}$ and $M=0\mathrm{}$ 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.