Gravitational Path Integrals and Boundary Conditions

A gravitational path integral is not specified by the symbol $\int\mathcal Dg\,e^{iS[g]}$ alone. One must say which metrics and topologies are included, which diffeomorphisms are gauge, what boundary data are held fixed, how zero modes and determinants are treated, and which integration cycle makes an oscillatory or unbounded integral meaningful. This chapter builds that full specification before using semiclassical saddles, cosmological boundary proposals, or wormholes.

Helpful background. Flat-Spacetime Quantum Field Theory supplies gauge-fixed functional integrals and perturbation theory; Einstein Equations supplies the saddle equations; Quantum Information supplies amplitudes, states, and factorization tests; Quantum Gravity supplies the evidence and scale distinctions; and Gravitational Actions and Boundary Terms supplies the well-posed variational problems used here.

What the integral must define

For a fixed manifold $\mathcal M$ and boundary data $B$, a schematic Lorentzian amplitude is

$$Z[B]=\frac{1}{\operatorname{Vol}(\mathrm{Diff}_0)} \int_{\mathcal C(B)}\mathcal Dg\,\mu[g]\,e^{iS[g]}.$$

Each symbol carries physical content. $\mathrm{Diff}_0$ denotes transformations treated as gauge rather than boundary symmetries; $\mu[g]$ is the field-space measure before gauge fixing; $S$ contains boundary and corner terms appropriate to $B$; and $\mathcal C(B)$ is the original integration cycle. Gauge fixing replaces the formal division by orbit volume with a gauge condition and its Faddeev–Popov determinant; it must not count that quotient twice. A sum over topology, if present, is a further definition rather than an automatic consequence of integrating over metrics.

Perturbatively, gauge fixing can be developed around a background and checked through BRST identities. DeWitt’s covariant construction makes explicit the gauge operator and determinant DeWitt 1967, pp. 1203–1213. Nonperturbatively, however, the quotient by diffeomorphisms can have stabilizers, Gribov-like ambiguities, and inequivalent global sectors. A formal local determinant does not settle those questions.

Six steps through the chapter

The chapter follows the order in which a defensible calculation must make choices.

1. Configuration Space and Gauge Volume identifies the metrics, topology, quotient, zero modes, and boundary data.
2. Measures, Determinants, and BRST Structure derives the perturbative Faddeev–Popov determinant, ghosts, and collective-coordinate factors.
3. The Conformal-Factor Problem shows why the Euclidean Einstein action is unbounded along conformal directions and compares contour prescriptions.
4. No-Boundary and Tunneling Proposals compares cosmological wave functions only after fixing their endpoint and contour data.
5. Topology Change and Wormhole Contributions asks when additional cobordisms are compatible with the Hilbert space, composition law, and factorization target.
6. Picard–Lefschetz Theory and Lorentzian Contours determines which complex saddles are connected to an original Lorentzian cycle and validates the method with an executable finite-dimensional benchmark.

Readers interested in perturbative field theory can take the first two pages. Readers interested in cosmology should read the first, third, fourth, and fifth. The last page is a conceptual and mathematical bridge, not a replacement for the dedicated black-hole information or string and holographic treatments.

Lorentzian origin, Euclidean tools

The Lorentzian weight $e^{iS}$ is oscillatory. A Wick rotation can turn a nongravitational Gaussian into a decaying Euclidean integral, but no general global map sends every Lorentzian spacetime and boundary problem to a real positive-definite metric. Worse, under $g_{\mu\nu}=e^{2\phi}\bar g_{\mu\nu}$, the Euclidean Einstein action contains a conformal kinetic term with the wrong sign. Gibbons, Hawking, and Perry derive this unbounded direction and propose a conformal contour rotation in Gibbons, Hawking, and Perry 1978, pp. 143–148.

Picard–Lefschetz theory instead complexifies the variables and deforms a declared original cycle into steepest-descent thimbles. A saddle contributes only when its upward cycle has nonzero intersection number with the original contour. Intersection numbers can jump at a Stokes transition even though the exact integral remains continuous. This distinguishes finding a stationary point from proving that it contributes.

Cosmological proposals sharpen the issue. The no-boundary and tunneling states can solve the same Wheeler–DeWitt equation while selecting different branches, lapse domains, endpoints, or regularity conditions. Lehners reviews the corresponding minisuperspace saddles and perturbation debate in Lehners 2023, §§3.2, 3.6, and 4.1–4.2. A saddle weight is not automatically a normalized probability measure over universes.

Results and open questions as of 2026-08-20

Semiclassical expansions are powerful once their defining data are fixed, but a unique nonperturbative gravitational measure and contour are not settled. The main positions make substantively different starting assumptions.

Formulation or proposal	Controlled result or evidence	Scope, adverse check, or limitation
Background-field gauge-fixed integral	Reproduces perturbative graviton amplitudes and loop determinants order by order	Local construction depends on background, regulator, and boundary conditions; it is not a global quotient
Real Euclidean metrics with conformal rotation	Gives familiar instanton expansions after a prescribed deformation	Euclidean action is unbounded on the original conformal direction; the rotated cycle is extra data
Lorentzian Picard–Lefschetz contour	Thimbles and intersection numbers define finite-dimensional oscillatory integrals rigorously	Infinite-dimensional metric space, singular lapse endpoints, and Stokes structure remain model-dependent
No-boundary or tunneling boundary condition	Produces explicit minisuperspace wave functions and saddle weights	Contour, perturbation branch, normalization, and extension beyond truncation remain disputed
Sum over topologies and wormholes	Connected saddles are controlled in selected low-dimensional or semiclassical settings	Compatibility with a fixed Hilbert space, factorization, and four-dimensional ultraviolet completion is unresolved

The adverse check is simple and general: changing only the contour while leaving the stationary points fixed can change which saddles contribute. Therefore an argument that lists classical solutions but never states the original cycle has not defined an amplitude. Feldbrugge, Lehners, and Turok exhibit this sensitivity for Lorentzian minisuperspace lapse integrals Feldbrugge, Lehners, and Turok 2017, §§II–IV.

The conclusions could change after a regulator-independent construction controls an infinite-dimensional Lorentzian cycle, gauge quotient, and continuum limit, or after new calculations settle whether competing cosmological contours give stable perturbations and a consistent composition law beyond minisuperspace.

Check your preparation

Evaluate the ordinary integral $I=\int_{-\infty}^{\infty}dx\,e^{iax^2/2}$ for $a\gt0$ by rotating to the ray $x=e^{i\pi/4}y$. Which orientation and Jacobian data must be retained, and what goes wrong for the opposite ray?

Solution

On $x=e^{i\pi/4}y$, $dx=e^{i\pi/4}dy$ and $iax^2/2=-ay^2/2$, so $I=e^{i\pi/4}\sqrt{2\pi/a}$ with the ray oriented from negative to positive $y$. Rotating to $x=e^{-i\pi/4}y$ gives an exponentially growing factor rather than a convergent Gaussian for this sign of $iS$. Reversing the orientation would add a minus sign. Thus the critical point $x=0$ is insufficient: the original cycle, allowed deformation, decay sector, Jacobian, and orientation determine the answer.

References

• DeWitt, Bryce S. “Quantum Theory of Gravity. II. The Manifestly Covariant Theory.” Physical Review 162 (1967): 1195–1239. DOI.
• Feldbrugge, Job, Jean-Luc Lehners, and Neil Turok. “Lorentzian Quantum Cosmology.” Physical Review D 95 (2017): 103508. DOI.
• Gibbons, G. W., S. W. Hawking, and M. J. Perry. “Path Integrals and the Indefiniteness of the Gravitational Action.” Nuclear Physics B 138 (1978): 141–150. DOI.
• Lehners, Jean-Luc. “Review of the No-Boundary Wave Function.” Physics Reports 1022 (2023): 1–82. arXiv:2303.08802.