Covariant quantization of gauge theories with Lagrange multipliers

TL;DR

This work establishes a covariant path-integral framework that demonstrates the quantum equivalence between first-order and second-order formulations for Yang–Mills theory and gravity, including matter couplings. Central to the analysis are structural identities that relate Green's functions of auxiliary (first-order) fields to composite-field Green's functions of the second-order theories, with explicit verification at the integrand level in YM and at one-loop order in gravity. The Faddeev–Senjanović determinant plays a crucial role in covariant quantization, canceling tadpole-like contributions and ensuring equivalence across finite-temperature and regularization schemes; for gravity, a manifestly covariant determinant is identified and interpreted as the Senjanović determinant. To address issues arising from LM fields, the authors develop a modified LM formalism that enforces field-redefinition invariance by introducing ghost sectors, yielding a theory whose perturbative expansion is truncated to one loop while preserving unitarity and covariance; the formalism and BRST structure are shown to commute with FP quantization. Overall, the work provides a covariant, quantum-consistent route to relate first- and second-order gauge theories and offers a robust framework for incorporating LM fields in gravity and gauge theories while preserving equivalence with conventional formulations.

Abstract

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-Senjanović procedure. The Senjanović 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.

Figures (22)

• Figure 1: Pinched diagrams. Diagrams (a) and (b) respectively represent the contributions to the left-hand side of the structural identities \ref{['eq:YM:SI1']} and \ref{['eq:YM:SI2']} at one-loop order.
• Figure 2: Extended Feynman rule for the interaction vertex with a composite field $\mathbb{H}_{\mu \nu}^{a}$, which is represented by a double spring line.
• Figure 3: Pinched diagrams of Fig. \ref{['fig:HH1p']} represented by extended Feynman rules. The symmetry factors are the same of those of the original pinched diagrams.
• Figure 4: Green's functions (at one-loop order) of the FOYM formalism equivalent to the pinched diagrams in Fig. \ref{['fig:pinchedYM']}.
• Figure 5: In diagram (a), we have the one-loop contribution to the mixed propagator $\langle 0| TH^\lambda_{\mu\nu} \phi^{\pi\tau}|0\rangle$. In diagram (b), we have the one-loop contribution to the propagator $\left \langle 0|TH_{\mu \nu}^{\lambda} H_{\pi \tau}^{\rho}|0\right\rangle$. The momenta satisfy the condition $q=p+k$.