Consistency Conditions for the First-Order Formulation of Yang-Mills Theory

We examine the self-consistency of the first-order formulation of the Yang-Mills theory. By comparing the generating functional $Z$ before and after integrating out the additional field $F^a_{\mu\nu}$, we derive a set of structural identities that must be satisfied by the Green's functions at all orders. These identities, which hold in any dimension, are distinct from the usual Ward identities and are necessary for the internal consistency of the first-order formalism. They relate the Green's functions involving the fields $F^a_{\mu\nu}$, to Green's functions in the second-order formulation which contain the gluon strength tensor $f^a_{\mu\nu}$. In particular, such identities may provide a simple physical interpretation of the additional field $F^a_{\mu\nu}$.