Covariant quantization of gauge theories with Lagrange multipliers

We revisited the equivalence between the second- and first-order formulations of the Yang-Mills (YM) and gravity using the path integral formalism. We demonstrated that structural identities can be derived to relate Green's functions of auxiliary fields, computed in the first-order formulation, to Green's functions of composite fields in the second-order formulation. In YM theory, these identities can be verified at the integrand level of the loop integrals. For gravity, the path integral was obtained through the Faddeev-Senjanovi\'c procedure. The Senjanovi\'c determinant plays a key role in canceling tadpole-like contributions, which vanish in the dimensional regularization scheme but persist at finite temperature. Thus, the equivalence between the two formalisms is maintained at finite temperature. We also studied the coupling to matter. In YM theory, we investigated both minimal and non-minimal couplings. We derived first-order formulations, equivalent to the respective second-order formulations, by employing a procedure that introduces Lagrange multipliers (LM). This procedure can be easily generalized to gravity. We also considered an alternative gravity model, which is both renormalizable and unitary, that uses LM to restrict the loop expansion to one-loop order. However, this approach leads to a doubling of one-loop contributions due to the additional degrees of freedom associated with Ostrogradsky instabilities. To address this, we proposed a modified formalism that resolves these issues by requiring the path integral to be invariant under field redefinitions. This introduces ghost fields responsible for canceling the extra one-loop contributions arising from the LM fields, while also removing unphysical degrees of freedom. We also demonstrated that the modified formalism and the Faddeev-Popov procedure commute, indicating that the LM can be viewed as purely quantum fields.