Gravitons, Gauge Fixing, and Quantum Amplitudes

Weak metric fluctuations can be quantized with the same disciplined sequence used in gauge theory: identify the physical helicities, fix the redundant variables, invert the quadratic action, derive vertices, and assemble gauge-independent amplitudes. The Einstein coupling has negative mass dimension, so loops generate an expanding tower of local operators; nevertheless, unitarity and low-energy power counting leave universal long-distance predictions. This chapter develops that controlled effective description without presenting perturbation theory as an ultraviolet completion.

Helpful background. Flat-Spacetime Quantum Field Theory supplies gauge-fixed path integrals and amplitudes; Einstein Equations supplies the nonlinear dynamics; Quantum Information supplies unitarity language; and Weak-Field Gravity supplies the classical perturbation expansion.

From a metric perturbation to an amplitude

About Minkowski space, write

$$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}, \qquad \kappa^2=32\pi G.$$

The symmetric tensor has ten configuration components; on shell, four gauge choices and four independent constraint equations reduce them to two radiative helicities. Covariant calculations retain all ten components and add gauge-fixing and Faddeev–Popov ghosts; BRST symmetry ensures that unphysical states cancel from gauge-invariant on-shell quantities.

The free propagator carries a massless pole. Einstein self-interactions contain two derivatives and arbitrary powers of $h$, so a vertex with $n$ gravitons scales schematically as $\kappa^{n-2}p^2$. Each loop increases the low-energy order by $\kappa^2E^2/(16\pi^2)\sim GE^2$ and allows higher-curvature counterterms. DeWitt’s covariant perturbative formulation established the gravitational Feynman rules and gauge treatment DeWitt 1967, §§2–7.

Six calculations in order

1. Linearized Gravity and Graviton Polarizations reduces the symmetric tensor to two transverse-traceless helicities.
2. Covariant Gauge Fixing and Ghost Fields derives the Faddeev–Popov operator and BRST cancellation.
3. The Graviton Propagator inverts the quadratic action and isolates the conserved-source numerator.
4. Gravitational Vertices and Power Counting organizes derivative order, coupling powers, loops, and counterterms.
5. Tree-Level Gravitational Amplitudes works through matter exchange, pole factorization, and the soft limit.
6. Loops, Unitarity Cuts, and Counterterms separates universal nonanalytic terms from ultraviolet-sensitive local coefficients.

Every stage uses the same checks: replace an external polarization by a pure-gauge tensor, verify factorization residues, confirm dimensions and crossing signs, and distinguish the $i0$ prescription from a physical boundary condition.

Results and open questions as of 2026-08-20

Perturbative quantum GR is exceptionally well controlled as a low-energy EFT. Its status as a fundamental high-energy theory is different.

Result or formulation	Evidence or calculation	Scope and limitation
Free graviton and tree amplitudes	Gauge-fixed action gives two physical helicities and unitary on-shell factorization	Requires weak fluctuations about a declared background; no ultraviolet conclusion follows
One-loop pure gravity	On-shell one-loop divergence cancels in four-dimensional pure gravity, while matter or off-shell terms require curvature-squared counterterms ’t Hooft and Veltman 1974, §§3–6	The cancellation is special and does not make Einstein gravity renormalizable
Two-loop pure gravity	A nonzero on-shell cubic-curvature divergence occurs Goroff and Sagnotti 1985, pp. 81–86	Establishes the need for a new EFT coefficient, not inconsistency of the low-energy theory
Unitarity-cut methods	Products of on-shell trees determine discontinuities and nonanalytic long-distance terms	Rational and local pieces require additional information; regulator and scheme enter local coefficients

The conservative position is that the amplitude expansion demonstrates a predictive low-energy quantum theory of gravitons and matter, while leaving ultraviolet completion unspecified. A more permissive position treats exceptional amplitude structure—factorization, soft behavior, and cancellations—as clues to deeper microscopic organization. These competing readings differ in extrapolation; neither changes the established two-loop counterterm result.

The ultraviolet completion remains unresolved: no perturbative cancellation displayed here establishes an all-order finite theory.

The conclusions could change if a new all-order theorem changes perturbative ultraviolet power counting, an independent amplitude calculation revises a cited divergence or unitarity result, a controlled nonperturbative completion fixes the EFT coefficients, or evidence after 2026-08-20 changes the interpretation.

Domain and approximation boundary

The Hilbert space here is the asymptotic Fock or infrared-dressed space of gravitons and matter about Minkowski spacetime. On-shell amplitudes are the principal observables. Covariant gauges and dimensional regularization are calculation choices; physical answers must be gauge independent. The expansion assumes $GE^2\ll1$, weak background curvature, and kinematics away from uncontrolled long-time or strong-field regimes.

The first omitted effects depend on the page: higher powers of $\kappa h$, additional loops, higher-derivative operators, or nonperturbative saddles. A low-energy pole or logarithm can be universal even when a polynomial in momenta is absorbed into an unknown local coefficient. This analytic/nonanalytic distinction will recur in the EFT chapter.

Review the chapter

Diagnostic. In four dimensions, a symmetric tensor has ten components. Explain why a massless graviton has two physical polarizations, and check the answer by counting the independent components of a transverse-traceless spatial wave travelling in the $z$ direction.

Solution

Covariantly, four gauge choices and four constraint equations reduce the ten on-shell tensor components to two. Canonically, $h_{ij}$ has six configuration variables and twelve phase-space dimensions; four first-class constraints remove eight phase-space dimensions—one constraint surface condition and one gauge direction each—leaving four phase-space dimensions, or two canonical polarizations. For a wave along $z$, choose transverse-traceless gauge: $h_{0\mu}=0$, $h_{z\mu}=0$, and $h^i{}_i=0$. The remaining spatial components satisfy $h_{yy}=-h_{xx}$ and $h_{xy}=h_{yx}$. Thus

$$h_+=h_{xx}=-h_{yy},\qquad h_\times=h_{xy}$$

are the two independent polarizations. If the graviton were massive, the gauge symmetry would be absent and the same count would not apply.

References

• ’t Hooft, Gerard, and Martinus Veltman. “One-Loop Divergencies in the Theory of Gravitation.” Annales de l’Institut Henri Poincaré A 20 (1974): 69–94. Open scan.
• DeWitt, Bryce S. “Quantum Theory of Gravity. II. The Manifestly Covariant Theory.” Physical Review 162 (1967): 1195–1239. DOI.
• Goroff, Marc H., and Augusto Sagnotti. “Quantum Gravity at Two Loops.” Physics Letters B 160 (1985): 81–86. DOI.