Background-Field Quantization, Gauge Fixing, and Ghosts

The background-field method makes graviton loops calculable without turning gauge-fixed metric components into observables. A complete one-loop definition contains the metric Hessian, gauge-fixing operator, Faddeev–Popov ghosts, measure/Jacobian choices, and background Ward identity. A matter determinant with no internal metric line remains a different sector.

Required background. Applying EFT Power Counting to Gravity fixes the loop order; The 1PI Effective Action and Mean-Field Equations fixes the Hessian; and Changes of Variables and Regulated Jacobians fixes measure dependence.

Helpful background. Covariant Free-Photon Quantization and Propagator supplies the gauge-fixing analogy, while Regulated Jacobians and Measure Variation supplies the anomaly caution.

Background split and gauge complex

Use

$$g_{\mu\nu}=\bar g_{\mu\nu}+\frac{2}{M_{\mathrm{Pl}}}h_{\mu\nu}.$$

At linear order a quantum diffeomorphism acts as $\delta_\epsilon h_{\mu\nu}=\bar\nabla_\mu\epsilon_\nu+\bar\nabla_\nu\epsilon_\mu$ after absorbing the split normalization into $\epsilon$. Choose the covariant de Donder functional

$$F_\mu[h]=\bar\nabla^\nu h_{\mu\nu} -\frac12\bar\nabla_\mu h$$

and a gauge parameter $\alpha$. The quadratic functional has the form

$$S^{(2)}_{\mathrm{tot}} =\frac12\int\sqrt{-\bar g}\, h^{\mu\nu}\mathcal H_{\mu\nu}{}^{\rho\sigma}(\alpha)h_{\rho\sigma} +\int\sqrt{-\bar g}\,\bar c^\mu\mathcal M_\mu{}^\nu c_\nu.$$

With the site curvature convention, variation of $F_\mu$ gives

$$\mathcal M_\mu{}^\nu =\delta_\mu{}^\nu\bar\Box+\bar R_\mu{}^\nu.$$

The curvature sign is fixed directly rather than imported from DeWitt. Indeed, $\bar\nabla^\rho\bar\nabla_\mu\epsilon_\rho -\bar\nabla_\mu\bar\nabla^\rho\epsilon_\rho =\bar R_{\mu\nu}\epsilon^\nu$ follows from $[\bar\nabla_\rho,\bar\nabla_\mu]V^\rho =\bar R_{\sigma\mu}V^\sigma$ in the site convention. The remaining terms give $\bar\Box\epsilon_\mu$, establishing the plus sign in $\mathcal M=\bar\Box+\bar R$.

The anticommuting vector ghosts contribute with the opposite determinant power from the bosonic Hessian:

$$\Gamma_{\mathrm{1\,loop}}^{g+\mathrm{gh}} =\frac{i}{2}\operatorname{Tr}\log\mathcal H -i\operatorname{Tr}\log\mathcal M +\Gamma_{\mathrm{measure}}.$$

This is the defining separation from a matter-only $\frac{i}{2}\operatorname{Tr}\log P_{\mathrm m}$.

The structure map places ghosts on the metric-loop branch rather than the matter branch.

Graviton and ghost determinants form one gauge-consistent metric-loop sector; matter determinants are classified separately even though all renormalize the same local curvature basis. The map is schematic and not to scale.

First application: an Einstein background

Let $\bar R_{\mu\nu}=\Lambda\bar g_{\mu\nu}$ in four dimensions. Define the Lichnerowicz operator

$$(\Delta_L^{(2)}h)_{\mu\nu} =-\bar\Box h_{\mu\nu} -2\bar R_{\mu\rho\nu\sigma}h^{\rho\sigma} +2\bar R_{(\mu}{}^\rho h_{\nu)\rho}.$$

For a transverse-traceless fluctuation, direct variation of $G_{\mu\nu}+\Lambda g_{\mu\nu}$ gives

$$\delta(G_{\mu\nu}+\Lambda g_{\mu\nu}) =\frac12(\Delta_L^{(2)}-2\Lambda)h_{\mu\nu}^{\mathrm{TT}}.$$

Thus the physical spin-two Hessian contains $\Delta_L^{(2)}-2\Lambda$. The trace, longitudinal, and ghost blocks depend on $\alpha$, but their principal symbols are minimal Laplace type at $\alpha=1$. On the Einstein background the ghost operator reduces to

$$\mathcal M_\mu{}^\nu =\delta_\mu{}^\nu\bar\Box+\Lambda\delta_\mu{}^\nu.$$

Two convention checks are immediate. In the flat limit, $\Delta_L^{(2)}\to-\Box$ and the TT equation becomes $\Box h_{\mu\nu}^{\mathrm{TT}}=0$, while the ghost operator becomes $\Box\delta_\mu{}^\nu$. On a maximally symmetric four-dimensional background, $\bar R_{\mu\rho\nu\sigma}h^{\rho\sigma}_{\mathrm{TT}} =-\Lambda h_{\mu\nu}^{\mathrm{TT}}/3$, so the equation reduces to $(\Box-2\Lambda/3)h_{\mu\nu}^{\mathrm{TT}}=0$. These round trips verify both signs despite the different Riemann convention used in DeWitt’s source.

These formulas supply three checks: the TT zero modes solve the linearized Einstein equation; the ghost operator is exactly $\delta F/\delta\epsilon$; and the combined principal-symbol count removes gauge directions. The original background-field construction and covariant gravitational Ward identities are developed in DeWitt 1967, §§2–5.

Background diffeomorphism invariance acts simultaneously on $\bar g$, $h$, and ghosts. The resulting identity may be written schematically as

$$\bar\nabla_\mu \frac{\delta\Gamma}{\delta\bar g_{\mu\nu}} +\frac{\delta\Gamma}{\delta h_{\rho\sigma}} \mathcal L^\nu h_{\rho\sigma} +\text{ghost terms}=0.$$

At vanishing mean fluctuation and ghost fields it reduces to covariant conservation of the background effective equation. BRST invariance supplies the corresponding quantum gauge identity; the Faddeev–Popov construction is the determinant representation of this gauge orbit Faddeev and Popov 1967, pp. 29–30.

Gauge and parametrization test

Change $\alpha$, or replace the linear split by an exponential metric parametrization. The off-shell Hessian and local coefficients in $\Gamma[\bar g]$ can change. A matched on-shell amplitude, asymptotic charge, or properly transformed relational observable must not. Failure of that comparison can signal omitted ghosts, a Jacobian, an incomplete counterterm basis, or use of an off-shell metric component as the output.

An Einstein background simplifies the Hessian but does not make its off-shell determinant gauge independent. Zero and negative modes also require a declared contour and collective-coordinate treatment rather than being silently included in $\operatorname{Tr}\log$.

Domain and failure conditions

The chapter comparison table licenses a metric loop only after split, gauge functional, parameter, ghost boundary conditions, measure, regulator, and observable are stated. Nonperturbative quantum geometry is not inferred from this expansion, and matter-only heat-kernel calculations remain in Chapter 8.

The failure map’s gauge branch is tested by varying $\alpha$ and the field parametrization while holding the on-shell matching conditions fixed.

Off-shell effective actions may reorganize across gauges and parametrizations; the declared invariant observable is the quantity required to agree. The map is schematic and not to scale.

References

• DeWitt, B. S. “Quantum Theory of Gravity. II. The Manifestly Covariant Theory.” Physical Review 162, 1195–1239 (1967). doi:10.1103/PhysRev.162.1195
• Faddeev, L. D., and V. N. Popov. “Feynman Diagrams for the Yang–Mills Field.” Physics Letters B 25, 29–30 (1967). doi:10.1016/0370-2693(67)90067-6