Configuration Space and Gauge Volume

Before assigning a measure or finding saddles, a gravitational path integral must identify its configuration space. The integration is over metrics satisfying declared signature, regularity, topology, and boundary data, modulo only those diffeomorphisms regarded as gauge. Diffeomorphisms that act at a boundary can instead be physical symmetries with charges. Confusing these classes either overcounts gauge copies or divides away real states.

Required background. Gravitational Actions and Boundary Terms supplies well-posed metric boundary data and surface actions; Flat-Spacetime Quantum Field Theory supplies gauge-fixed functional integrals and determinants.

Helpful background. Einstein Equations supplies the saddle equations; Quantum Information supplies state and composition concepts; and Quantum Gravity supplies regime and evidence distinctions.

Metrics, boundaries, and equivalence classes

Fix a differentiable manifold $\mathcal M$, a metric signature, and boundary data $B$. A Lorentzian configuration space may be written

$$\mathcal Q_B=\operatorname{Met}_{-+++}(\mathcal M;B)/\mathrm{Diff}_0(\mathcal M;B).$$

The numerator still needs a regularity class, nondegeneracy condition, time orientation, and any causal restrictions. The denominator consists of diffeomorphisms connected to the identity that preserve the chosen boundary data and are declared gauge. Large diffeomorphisms can label additional sectors, while transformations nontrivial at an asymptotic boundary may generate energy, momentum, or angular momentum rather than redundancy.

Boundary data depend on the amplitude. Fixing the induced metric on a non-null boundary calls for the Einstein–Hilbert action plus the Gibbons–Hawking–York term. Fixing conjugate data requires a Legendre-transformed action. Null boundaries and corners have additional terms. Gibbons and Hawking explain how boundary terms make the metric variational problem and semiclassical partition function well defined in Gibbons and Hawking 1977, pp. 2753–2756.

Gauge slices and stabilizers

Choose a gauge condition $F_\mu[g]=0$. Under an infinitesimal diffeomorphism generated by $\xi^\mu$,

$$\delta_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu, \qquad M_\mu{}^\nu=\frac{\delta F_\mu[g^\xi]}{\delta\xi_\nu}\bigg|_{\xi=0}.$$

Formally inserting

$$1=\Delta_{\mathrm{FP}}[g]\int\mathcal D\xi\, \delta[F(g^\xi)]$$

turns division by gauge volume into an integral over a gauge slice. For an ordinary real delta-function change of variables the Jacobian is $\lvert\det M\rvert$; perturbative ghost formulas use an oriented $\det M$ in a chart where its sign is fixed. The identity assumes that the slice intersects each relevant orbit once and transversely. An allowed Killing vector is a zero mode of the gauge action at a symmetric metric; its stabilizer volume must be treated separately from the determinant. A collective coordinate instead parametrizes a genuine family of distinct saddles. Multiple intersections are the functional analogue of Gribov copies.

DeWitt’s background-covariant formulation derives the local gauge operator and its determinant around a fixed saddle DeWitt 1967, pp. 1203–1213. It provides the perturbative quotient, not a proof that one gauge chart covers the full space of metrics.

Fixed topology is a choice

Metrics on different manifolds are not points in one ordinary linear space. A proposed topology sum has the schematic form

$$Z[B]=\sum_{[\mathcal M]}w(\mathcal M) \int_{\mathcal Q_B(\mathcal M)}\mathcal D[g]\,e^{iS[g]},$$

where the allowed cobordisms, weights $w(\mathcal M)$, automorphism factors, and boundary identifications must be supplied. There is no canonical measure on all four-manifold topologies. Even at fixed topology, degenerate metrics can connect components only after enlarging the original configuration space, which changes the theory.

The same care applies to Euclideanization. Replacing Lorentzian metrics by positive-definite metrics changes the real slice; it is not automatically a coordinate change on $\mathcal Q_B$. A complex contour can connect them only when the original cycle, singularities, and convergence sectors are controlled.

Independent checks

A configuration-space definition should pass four tests. The gauge-fixed amplitude must be independent of small changes in $F_\mu$ for gauge-invariant boundary observables; zero modes must be counted separately; gluing two regions must integrate over precisely the shared boundary data; and the semiclassical stationary point must satisfy both the bulk Einstein equation and the declared boundary conditions. Failure of any test identifies a missing quotient, charge, Jacobian, or boundary term.

This page does not choose a string or holographic microscopic completion, and it does not sum over topology by default. It defines the equivalence classes needed before those additional proposals can be evaluated.

Exercises

Let $I=(\operatorname{Vol}\mathbb R)^{-1}\int dx\,dy\,e^{-x^2/2}$ with gauge action $y\mapsto y+\alpha$. Use the gauge condition $F=y=0$ to evaluate the quotient integral and identify the Faddeev–Popov determinant. What changes for $F=2y$?

Solution

For $F=y$, $\partial F(y+\alpha)/\partial\alpha=1$, so inserting $\int d\alpha\,\delta(y+\alpha)=1$ removes the infinite orbit volume and leaves $I=\int dx\,e^{-x^2/2}=\sqrt{2\pi}$. For $F=2y$, the delta function contributes $\delta(2y)=\delta(y)/2$, while the determinant is $\lvert\partial(2y+2\alpha)/\partial\alpha\rvert=2$. The factors cancel and give the same quotient. Omitting the determinant would make the answer depend on a harmless rescaling of the gauge condition, exposing an inconsistent gauge-volume treatment.

References

• DeWitt, Bryce S. “Quantum Theory of Gravity. II. The Manifestly Covariant Theory.” Physical Review 162 (1967): 1195–1239. DOI.
• Gibbons, G. W., and S. W. Hawking. “Action Integrals and Partition Functions in Quantum Gravity.” Physical Review D 15 (1977): 2752–2756. DOI.