Feynman Rules for the Yang-Mills Field: A Canonical Quantization Approach. I

Abstract

An attempt has been made to derive covariant Feynman rules for the massless Yang-Mills field, starting with canonical methods of quantization. In this paper we will summarize the techniques involved in such a program, along with a few preliminary results. Working in the radiation gauge (${\partial }_i{\overrightarrow{\mathrm{b}}}_i=0\mathrm{}$), we find that there is an infinity of noncovariant vertices. We obtain a noncovariant set of rules to describe them to any order. Working with the suggested set of rules, we first prove that all tree diagrams can be described by a covariant set of Feynman rules. Secondly, to order $g^2$, we find that the one-loop diagram can also be made covariant. However, apart from the usual three-vector and four-vector vertices, the covariant loop contains an extra vertex of vector-scalar-scalar type and the scalar loop occurs with a weight factor of -2 with respect to the vector loop.