The Graviton Propagator

The graviton propagator is the inverse of the gauge-fixed quadratic Einstein operator, not a sum over ten physical tensor modes. In harmonic gauge its conserved-source numerator projects onto massless spin-two exchange, while momentum-dependent gauge terms vanish from on-shell amplitudes. The pole prescription specifies the Green function; it must not be confused with a new physical degree of freedom.

Required background. Flat-Spacetime Quantum Field Theory supplies propagators and pole prescriptions. Weak-Field Gravity supplies trace reversal and conserved sources. Gravitational Actions and Boundary Terms supplies the quadratic action.

Helpful background. Einstein Equations supplies the classical exchange limit, while Quantum Information supplies the physical-state interpretation.

Inverting the quadratic operator

Use $g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$ and harmonic gauge

$$F_\mu=\partial^\nu h_{\mu\nu}-\frac12\partial_\mu h=0.$$

With the gauge-fixing normalization $\mathcal L_{\mathrm{gf}}=-F_\mu F^\mu/\alpha$, set $\alpha=1$. Integrating by parts gives

$$\mathcal L_2+\mathcal L_{\mathrm{gf}} =\frac12 h^{\mu\nu}\Box \left(h_{\mu\nu}-\frac12\eta_{\mu\nu}h\right).$$

Thus the algebraic kinetic map sends a symmetric tensor to its trace reverse. Taking its trace gives $\bar h=-(D-2)h/2$. Its inverse in $D\gt2$ is consequently

$$X_{\mu\nu}\longmapsto X_{\mu\nu}-\frac{\eta_{\mu\nu}X}{D-2}.$$

This derives the trace coefficient instead of assuming a polarization sum. Use the Fourier convention

$$h(x)=\int\frac{d^Dk}{(2\pi)^D}\,e^{ik\cdot x}h(k),$$

so $\Box$ becomes $-k^2$. The momentum-space Feynman propagator is therefore

$$D_{\mu\nu,\rho\sigma}(k)= \frac{-i}{k^2-i0}\,P_{\mu\nu,\rho\sigma},$$

with the mostly-plus metric and

$$P_{\mu\nu,\rho\sigma} =\frac12\left( \eta_{\mu\rho}\eta_{\nu\sigma} +\eta_{\mu\sigma}\eta_{\nu\rho} -\frac{2}{D-2}\eta_{\mu\nu}\eta_{\rho\sigma} \right).$$

The sign of the displayed propagator follows from $k^2=-\omega^2+\mathbf k^2$ and the $e^{-i\omega t}$ convention. Authors using the opposite metric write an apparently different overall sign; normalized amplitudes agree after all vertex conventions are changed consistently. DeWitt derives the covariant inverse operator and its gauge structure DeWitt 1967, §§4–6.

Conserved-source projection

Let two symmetric sources obey

$$k_\mu T^{\mu\nu}=0, \qquad k_\rho T'^{\rho\sigma}=0.$$

Their numerator contraction is

$$T^{\mu\nu}P_{\mu\nu,\rho\sigma}T'^{\rho\sigma} =T^{\mu\nu}T'_{\mu\nu}-\frac{1}{D-2}TT'.$$

In four dimensions, the trace coefficient is $-1/2$. Terms containing an explicit $k_\mu$ can be added to the propagator in a general covariant gauge, but conservation kills them between the sources. This is the practical reason the on-shell exchange is gauge-parameter independent even though individual tensor components of the propagator are not.

Replacing a polarization by $k_\mu\zeta_\nu+k_\nu\zeta_\mu$ gives the same check: contraction with a conserved source vanishes. If it does not, either the source is incomplete, the external states are off shell, or a contact term needed by the Ward identity has been omitted.

For a conserved source, every longitudinal numerator term vanishes, so the on-shell amplitude is gauge independent. This statement is stronger than choosing harmonic gauge: changing the covariant gauge parameter changes off-shell Green functions but cannot change the pole residue between physical conserved sources.

The dimension dependence is physically informative. The trace subtraction $1/(D-2)$ follows from the massless constraint structure. Its singularity at $D=2$ warns that the displayed inversion cannot be interpreted as ordinary propagating spin-two exchange there. A massive spin-two propagator has a different trace coefficient, $1/(D-1)$, because it carries additional helicities; taking its mass to zero does not reproduce the conserved-source numerator continuously in flat space. Thus the massless numerator is fixed jointly by Lorentz covariance, source conservation, and gauge symmetry—not by symmetrizing two vector propagators alone.

Pole, prescription, and physical spectrum

The only propagating pole in the free massless theory is at $k^2=0$. The Feynman $-i0$ prescription implements time ordering. Retarded or advanced Green functions use different contour prescriptions and answer causal-response questions rather than scattering questions. Changing the prescription does not change the numerator’s helicity content.

The residue on conserved sources is positive on the two transverse-traceless helicities. Trace and longitudinal components appearing in the covariant tensor are not additional external particles. BRST symmetry and ghosts remove them from the physical state space. A genuine extra pole would instead signal additional dynamics—for example, higher-derivative terms treated nonperturbatively—and its residue and EFT domain would need separate analysis.

Dimensional regularization requires retaining the $D$-dependent coefficient $1/(D-2)$ until renormalization is complete. Replacing it prematurely by $1/2$ can alter evanescent terms when multiplied by ultraviolet poles. For a purely four-dimensional tree exchange, the $1/2$ value is appropriate.

Newtonian and massless checks

For slowly moving sources, $T^{00}\simeq m$ and spatial stresses are negligible. With the mostly-plus metric, $T\simeq-m$. The four-dimensional numerator becomes

$$T^{\mu\nu}T'_{\mu\nu}-\frac12TT' \simeq\frac12m_1m_2.$$

Combining the $+\kappa h_{\mu\nu}T^{\mu\nu}/2$ coupling with correctly normalized source matrix elements and the standard Born matching gives the momentum-space potential $-4\pi Gm_1m_2/\mathbf q^2$, and therefore

$$V(r)=-\frac{Gm_1m_2}{r}.$$

This checks the factor $\kappa^2=32\pi G$, the trace subtraction, and the attractive sign. In the radiative limit, contracting $P$ with transverse-traceless polarizations instead gives a sum over the two helicities. These independent static and on-shell checks probe different parts of the numerator.

The approximation assumes weak perturbations and a flat asymptotic region. At $GE^2\sim1$, near strong curvature, or after resumming higher-derivative operators into new poles, the free propagator is no longer the complete description. Local counterterms modify contact terms; they do not move the massless pole order by order in the EFT.

Exercises

Exercise. In $D=4$, contract the numerator with two nonrelativistic conserved sources for which only $T^{00}=m_1$ and $T'^{00}=m_2$ are appreciable. Verify the factor of one half and explain why omitting the trace term gives the wrong Newtonian normalization.

Solution

Raising both time indices leaves $T^{00}=m_1$, so $T^{\mu\nu}T'_{\mu\nu}=m_1m_2$. Each trace is $T=\eta_{00}T^{00}=-m_1$ and similarly for $T'$. Therefore

$$T^{\mu\nu}T'_{\mu\nu}-\frac12TT' =m_1m_2-\frac12m_1m_2=\frac12m_1m_2.$$

Without the trace subtraction the exchange kernel would be twice the value fixed by the Einstein source coupling and would not reproduce the Newtonian potential for the declared normalization of $\kappa$.

References

• DeWitt, Bryce S. “Quantum Theory of Gravity. II. The Manifestly Covariant Theory.” Physical Review 162 (1967): 1195–1239. DOI.