Abstract
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 ${F}_{\ensuremath{\mu}\ensuremath{\nu}}^{a}$, 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 ${F}_{\ensuremath{\mu}\ensuremath{\nu}}^{a}$, to Green's functions in the second-order formulation which contain the gluon strength tensor ${f}_{\ensuremath{\mu}\ensuremath{\nu}}^{a}$. In particular, such identities may provide a simple physical interpretation of the additional field ${F}_{\ensuremath{\mu}\ensuremath{\nu}}^{a}$.
Highlights
The first-order formulation of gauge theories has a simple form that involves only cubic interactions of the gauge fields, which are momentum-independent
We examine different kinds of identities, which are necessary for the consistency of the first-order formulation
Which enables us to integrate out the auxiliary field Faμν. The equality of these functional forms leads to a set of structural identities among the Green’s functions which must be satisfied to all orders, in any dimension. These show that in the first-order formalism, Green’s functions containing only external gluon fields are the same as those which occur in the second-order formulation
Summary
The first-order formulation of gauge theories has a simple form that involves only cubic interactions of the gauge fields, which are momentum-independent. Faμν → Faμν þ faμν − 2Jaμν ð1:3Þ which enables us to integrate out the auxiliary field Faμν The equality of these functional forms leads to a set of structural identities among the Green’s functions which must be satisfied to all orders, in any dimension. These show that in the first-order formalism, Green’s functions containing only external gluon fields are the same as those which occur in the second-order formulation. Some details of the relevant calculations are given in the Appendices
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