Covariant Gauge Fixing and Ghost Fields

The quadratic Einstein operator cannot be inverted until diffeomorphism redundancy is fixed. A covariant gauge adds an invertible kinetic operator but also changes the functional measure by a Faddeev–Popov determinant. Anticommuting ghost fields represent that determinant. They never label external physical gravitons; inside loops they cancel the gauge-dependent tensor modes required by a covariant calculation.

Required background. Flat-Spacetime Quantum Field Theory supplies functional measures and BRST quantization. Weak-Field Gravity supplies linearized diffeomorphisms. Gravitational Actions and Boundary Terms supplies the action and variational boundary conditions.

Helpful background. Einstein Equations supplies covariant dynamics, while Quantum Information supplies the physical-state and unitarity language.

Harmonic gauge and its determinant

About Minkowski space, use the canonical normalization $g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$, $\kappa^2=32\pi G$, and define

$$F_\mu[h]=\partial^\nu h_{\mu\nu}-\frac12\partial_\mu h.$$

At linear order, $\delta_\xi h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$, so

$$\delta_\xi F_\mu=\Box\xi_\mu.$$

The gauge slice $F_\mu=0$ is implemented by

$$\mathcal L_{\mathrm{gf}}=-\frac{1}{\alpha}F_\mu F^\mu,$$

with gauge parameter $\alpha$. This normalization makes $\alpha=1$ the simple harmonic propagator gauge: adding $-F^2$ to the canonical Fierz–Pauli density cancels its mixed-divergence terms, leaving

$$\mathcal L_2+\mathcal L_{\mathrm{gf}} =-\frac12\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu} +\frac14\partial_\lambda h\partial^\lambda h.$$

The same normalization is explicit in Donnelly and Giddings 2016, Appendix B, Eqs. (B3), (B9)–(B11), PDF. A convention with $-F^2/(2\alpha)$ is also possible, but its simple-propagator value is $\alpha=1/2$. The limit $\alpha\to0$ enforces the gauge slice sharply; finite $\alpha$ implements a Gaussian average over it. The change of variables from gauge orbits to the slice produces

$$\Delta_{\mathrm{FP}}[h]=\det M_\mu{}^\nu[h], \qquad M_\mu{}^\nu=\frac{\delta F_\mu[h^\xi]}{\delta\xi_\nu}\bigg|_{\xi=0}.$$

At strictly linear order, $M_\mu{}^\nu=\Box\delta_\mu{}^\nu$ is independent of $h$, so its determinant changes only the normalization and free ghosts decouple. In nonlinear gravity, the diffeomorphism of $h_{\mu\nu}$ contains terms proportional to $\kappa h\partial\xi$ and $\kappa\xi\partial h$. The determinant then depends on $h$ and cannot be discarded.

Ghost action and Feynman role

Introduce Grassmann-odd vector fields $c^\mu$ and $\bar c_\mu$. The determinant is represented by the ghost density

$$\mathcal L_{\mathrm{gh}}=-\bar c^\mu M_\mu{}^\nu[h]c_\nu.$$

Its free part is $-\bar c^\mu\Box c_\mu$ and its interaction begins with one graviton, two ghosts, and derivatives. A closed ghost loop carries a minus sign because the fields anticommute. Faddeev and Popov’s determinant construction explains why these auxiliary fields are forced by the gauge-orbit measure rather than added as new particles Faddeev and Popov 1967, pp. 29–30.

Ghosts have propagators but are not asymptotic states. The physical external space contains the two transverse-traceless graviton helicities. Including ghost cuts or external ghosts in a formal gauge-fixed identity can be useful, but a measurable amplitude must project onto the BRST cohomology.

At tree level with only external conserved matter sources, ghosts often do not appear explicitly. At loop level, graviton diagrams include longitudinal and trace components whose gauge dependence is canceled by ghost diagrams. Omitting them generally breaks the Ward identities and changes the ultraviolet divergence.

BRST organization

Introduce an auxiliary bosonic field $B_\mu$. To leading order, a nilpotent BRST differential acts as

$$s h_{\mu\nu}=\partial_\mu c_\nu+\partial_\nu c_\mu+O(\kappa), \qquad s\bar c_\mu=B_\mu, \qquad sB_\mu=0,$$

while $s c^\mu$ begins at nonlinear order because diffeomorphisms form a non-Abelian algebra. The gauge-fixing plus ghost density can be written as a BRST variation of a gauge-fixing fermion. Nilpotency, $s^2=0$, encodes closure of the gauge algebra and organizes cancellations among unphysical modes.

Physical states satisfy $Q_{\mathrm{BRST}}\lvert\psi\rangle=0$ modulo states of the form $Q_{\mathrm{BRST}}\lvert\chi\rangle$. A change of $\alpha$ changes the representative propagators and off-shell Green functions, but not properly defined on-shell amplitudes between cohomology classes. DeWitt’s gravitational gauge-fixing treatment gives the covariant operator structure that underlies this cancellation DeWitt 1967, §§4–7.

Gauge-parameter and boundary checks

Three checks should accompany a calculation. First, differentiate a physical on-shell amplitude with respect to $\alpha$; the answer must vanish after all graviton and ghost contributions are included. Second, replace any external graviton polarization by $k_\mu\zeta_\nu+k_\nu\zeta_\mu$; the amplitude must vanish. Third, verify that boundary conditions make $M$ invertible apart from declared residual or zero modes.

The last point matters. On compact spaces or backgrounds with Killing vectors, the Faddeev–Popov operator can have zero modes. They must be divided out or treated as collective coordinates rather than assigned an ordinary propagator. Nonperturbative gauge copies can intersect the same gauge condition more than once; the local Faddeev–Popov derivation does not solve that global problem.

This page uses a perturbative expansion about Minkowski space, a covariant harmonic family, and field-independent asymptotic gauge parameters. Curved backgrounds replace ordinary derivatives with background covariant derivatives and add curvature to $M$. The first omitted terms are nonlinear BRST transformations, higher vertices, renormalized composite operators, and possible anomalies; a genuine BRST anomaly would make gauge independence fail.

Two interpretations should be excluded. A minus sign from a closed ghost loop is not evidence for a negative-probability particle: ghosts are absent from the physical asymptotic spectrum and serve the gauge-fixed measure. Conversely, calling ghosts “unphysical” does not make them optional. An internal calculation that propagates covariant tensor components without the corresponding determinant counts gauge-orbit volume incorrectly. The test is not whether an individual graph looks physical, but whether the complete on-shell result satisfies the BRST Ward identities and is independent of the gauge parameter.

Ghost decoupling is also formulation specific. It occurs for the free linearized operator because the gauge transformation is then field independent. It does not extend to nonlinear Einstein gravity, even in harmonic gauge, and it need not look the same in a noncovariant gauge. Different gauges redistribute cancellations among constraints, instantaneous terms, and ghosts while preserving the same physical helicity amplitudes.

Exercises

Exercise. Starting from $F_\mu=\partial^\nu h_{\mu\nu}-\partial_\mu h/2$, derive its linearized gauge variation. Explain why free ghosts decouple but interacting gravitational ghosts do not.

Solution

The trace varies as $\delta h=2\partial\cdot\xi$. Therefore

$$\delta F_\mu =\partial^\nu(\partial_\mu\xi_\nu+\partial_\nu\xi_\mu) -\partial_\mu(\partial\cdot\xi) =\Box\xi_\mu.$$

Thus the free Faddeev–Popov operator is $M_\mu{}^\nu=\Box\delta_\mu{}^\nu$, independent of $h$, and its determinant is an overall normalization. Nonlinear diffeomorphisms make $M[h]$ field dependent. Expanding it generates ghost–graviton vertices, so ghost loops are required to cancel the unphysical components of internal graviton lines and restore gauge-parameter independence.

References

• DeWitt, Bryce S. “Quantum Theory of Gravity. II. The Manifestly Covariant Theory.” Physical Review 162 (1967): 1195–1239. DOI.
• Donnelly, William, and Steven B. Giddings. “Diffeomorphism-Invariant Observables and Their Nonlocal Algebra.” Physical Review D 93 (2016): 024030; erratum 94 (2016): 029903. DOI. Open PDF.
• Faddeev, Ludvig D., and Victor N. Popov. “Feynman Diagrams for the Yang–Mills Field.” Physics Letters B 25 (1967): 29–30. DOI.