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Symmetric Ghost Lagrange Densities for the Coupling of Gravity to Gauge Theories

David Prinz

TL;DR

This work tackles the challenge of quantizing gauge theories in curved spacetime by deriving symmetric ghost Lagrangian densities for the coupling of gravity to gauge theories. It advances Curci–Ferrari and Baulieu–Thierry-Mieg formalisms to (effective) Quantum General Relativity and covariant Yang–Mills theory by using a total gauge fixing boson and the total super-BRST operator $\mathcal{D} = D \circ \overline{D}$ to generate BRST/anti-BRST symmetric ghosts and gauge fixings. The authors introduce graviton-ghost and gauge-ghost homotopies with parameters $\varsigma$ and $\vartheta$ that interpolate between FP, symmetric, and opposed FP constructions, and demonstrate how the complete GR–YM Lagrangian is obtained as $\mathcal{D}Y$ with a double-homotopy in $(\varsigma,\vartheta)$. This framework provides a structured path toward understanding cancellation identities and potential renormalization schemes for perturbative quantum gravity within the BRST/BV formalism.

Abstract

We derive and present symmetric ghost Lagrange densities for the coupling of General Relativity to Yang--Mills theories. The graviton-ghost is constructed with respect to the linearized de Donder gauge fixing condition and the gauge ghost with respect to the covariant Lorenz gauge fixing condition. Both ghost Lagrange densities together with their accompanying gauge fixing Lagrange densities are obtained from the action of the diffeomorphism and gauge super-BRST differential -- which we define as the composition of the BRST differential with its anti-BRST differential -- on suitable gauge fixing bosons. In addition, we introduce a total gauge fixing boson and show that the complete symmetric ghost and gauge fixing Lagrange density can be generated thereof using the total super-BRST differential. In particular, we generalize two earlier approaches for flat-spacetime Yang--Mills theories to General Relativity and covariant Yang--Mills theories: The original approach by Curci and Ferrari (1976), using the Faddeev--Popov method on non-linear gauge fixings, and the modern approach by Baulieu and Thierry-Mieg (1982), using BRST and anti-BRST symmetries with gauge fixing bosons.

Symmetric Ghost Lagrange Densities for the Coupling of Gravity to Gauge Theories

TL;DR

This work tackles the challenge of quantizing gauge theories in curved spacetime by deriving symmetric ghost Lagrangian densities for the coupling of gravity to gauge theories. It advances Curci–Ferrari and Baulieu–Thierry-Mieg formalisms to (effective) Quantum General Relativity and covariant Yang–Mills theory by using a total gauge fixing boson and the total super-BRST operator to generate BRST/anti-BRST symmetric ghosts and gauge fixings. The authors introduce graviton-ghost and gauge-ghost homotopies with parameters and that interpolate between FP, symmetric, and opposed FP constructions, and demonstrate how the complete GR–YM Lagrangian is obtained as with a double-homotopy in . This framework provides a structured path toward understanding cancellation identities and potential renormalization schemes for perturbative quantum gravity within the BRST/BV formalism.

Abstract

We derive and present symmetric ghost Lagrange densities for the coupling of General Relativity to Yang--Mills theories. The graviton-ghost is constructed with respect to the linearized de Donder gauge fixing condition and the gauge ghost with respect to the covariant Lorenz gauge fixing condition. Both ghost Lagrange densities together with their accompanying gauge fixing Lagrange densities are obtained from the action of the diffeomorphism and gauge super-BRST differential -- which we define as the composition of the BRST differential with its anti-BRST differential -- on suitable gauge fixing bosons. In addition, we introduce a total gauge fixing boson and show that the complete symmetric ghost and gauge fixing Lagrange density can be generated thereof using the total super-BRST differential. In particular, we generalize two earlier approaches for flat-spacetime Yang--Mills theories to General Relativity and covariant Yang--Mills theories: The original approach by Curci and Ferrari (1976), using the Faddeev--Popov method on non-linear gauge fixings, and the modern approach by Baulieu and Thierry-Mieg (1982), using BRST and anti-BRST symmetries with gauge fixing bosons.
Paper Structure (6 sections, 6 theorems, 61 equations)

This paper contains 6 sections, 6 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. General setup
  3. The case of (effective) Quantum General Relativity
  4. The case of covariant Quantum Yang--Mills theory
  5. The total construction
  6. Conclusion

Key Result

Proposition 3.1

The symmetric gauge fixing and ghost Lagrange density for (effective) Quantum General Relativity reads with the linearized de Donder gauge fixing functional ${d D \mspace{-2mu}}^{(1)}_\mu \coloneq \eta^{\rho \sigma} \Gamma_{\mu \rho \sigma}$. It can be obtained from the gauge fixing boson via $\mathcal{P} F$, where $\mathcal{P}$ is the diffeomorphism super-BRST operator.

Theorems & Definitions (25)

  • Definition 2.1: Spacetime
  • Definition 2.2: Spacetime-matter bundle
  • Definition 2.3: Sheaf of particle fields
  • Definition 2.4: Diffeomorphism (anti-)BRST operator
  • Definition 2.5: Gauge (anti-)BRST operator
  • Definition 2.6: Total (anti-)BRST operator
  • Definition 2.7: Super-BRST operators
  • Remark 2.8
  • Definition 2.9: Ghost conjugation, anti-Hermitian auxiliary field
  • Remark 2.10
  • ...and 15 more