No-Boundary and Tunneling Proposals

No-boundary and tunneling proposals are boundary conditions for a cosmological wave function, not different Wheeler–DeWitt equations. They differ through regularity requirements, lapse or metric contours, and the WKB branches retained at large scale factor. A fair comparison must use the same minisuperspace variables, factor ordering, perturbations, and probability rule. Neither proposal is presently a unique normalized measure on realistic cosmological histories.

Required background. Gravitational Actions and Boundary Terms supplies the action and endpoint variational problem; Flat-Spacetime Quantum Field Theory supplies saddle expansions and WKB connection formulas.

Helpful background. Einstein Equations supplies the classical cosmological saddles; Quantum Information supplies state, normalization, and conditional-probability concepts; and Quantum Gravity supplies the regime and evidence boundaries.

One constraint, different boundary data

An explicit example fixes the signs. Take a closed FLRW metric $ds^2=-N^2dt^2+a^2d\Omega_3^2$, where $a\gt0$ is the physical radius of a unit three-sphere, $N\gt0$, and the only source is $\Lambda\gt0$. Write $H^2=\Lambda/3$. Including the endpoint term appropriate to fixed three-geometry gives the reduced action

$$S=\frac{3\pi}{4G}\int dt\,N \left[-\frac{a\dot a^2}{N^2}+a-H^2a^3\right].$$

Thus $p_a=-3\pi a\dot a/(2GN)$. Multiplying the Hamiltonian constraint by $-3\pi a/G$ gives $p_a^2+U(a)=0$, where

$$U(a)=\frac{9\pi^2}{4G^2}a^2(1-H^2a^2), \qquad \left[-\partial_a^2+U(a)\right]\Psi(a)=0.$$

The last equation uses the simple ordering $p_a^2\mapsto-\partial_a^2$; ordering changes lower-derivative terms and endpoint conditions. Since $[a]=L$ and $[G]=L^2$, both $U$ and $\partial_a^2$ have dimension $L^{-2}$. The region $0\lt a\lt H^{-1}$ is nonoscillatory, while $a\gt H^{-1}$ admits real $p_a$. In that oscillatory region,

$$\Psi \simeq C_+\,\lvert p_a\rvert^{-1/2}e^{+iS(a)} +C_-\,\lvert p_a\rvert^{-1/2}e^{-iS(a)}.$$

Choose $S'(a)=\lvert p_a\rvert$. The $e^{-iS}$ term then has negative canonical momentum and describes expansion for positive lapse; $e^{+iS}$ describes contraction. A boundary proposal selects a relation between $C_+$ and $C_-$ and a continuation through the nonoscillatory region. With a dynamical scalar or inhomogeneous perturbations included, it must also specify their state; the pure-$\Lambda$ truncation does not determine it.

No-boundary regularity

Hartle and Hawking proposed a wave function obtained from compact regular four-geometries with one specified boundary carrying the final three-geometry and matter data Hartle and Hawking 1983, §§II–IV, pp. 2961–2968. In a saddle approximation, the geometry closes off smoothly rather than meeting an additional initial boundary. Regularity fixes local endpoint behavior, while a contour through complex metrics or lapse values is still needed to define the integral.

For a simple de Sitter minisuperspace saddle, analytic continuation often yields a real superposition of expanding and contracting WKB branches. Decoherence can make those branches approximately autonomous, but the underlying real combination is not identical to imposing a single outgoing branch. The relative saddle weights depend on the action sign, contour, and which fluctuation determinant is included.

Tunneling outgoing modes

Vilenkin’s tunneling condition selects outgoing flux at singular boundaries of superspace and supplements it with regularity conditions where no classical flux leaves Vilenkin 1988, §§II–IV, pp. 889–894. In a one-dimensional WKB region this sets one of $C_\pm$ to zero according to the orientation convention. It does not mean tunneling through a fixed external-time potential; the “barrier” lies in configuration space and the sign of the gravitational kinetic term matters.

For the pure-$\Lambda$ example, the positive barrier integral is elementary:

$$I=\int_0^{H^{-1}}da\,\sqrt{U(a)} =\frac{3\pi}{2G}\int_0^{H^{-1}}da\,a\sqrt{1-H^2a^2} =\frac{3\pi}{2G\Lambda}.$$

The conventional leading no-boundary and tunneling prescriptions give weights proportional to $e^{+I}$ and $e^{-I}$, respectively, before fluctuation prefactors and normalization. Equivalently, the regular Euclidean hemisphere has action $S_E=-I$. This explains how opposite weights can arise from the same constraint. In scalar-potential models the analogous comparison also assumes that a nearly constant effective $\Lambda$ is justified Lehners 2023, §§2–3. These are leading semiclassical amplitudes, not probabilities over theories with different $\Lambda$. Probabilities require a current or decoherent-histories construction, a measure over histories, and a conditioning question.

Lorentzian contour dispute

The modern disagreement is partly about what data define “the” no-boundary state. A Lorentzian lapse integral with a specified positive real contour can be deformed into particular complex thimbles. Feldbrugge, Lehners, and Turok derived the background saddle weighting in Feldbrugge, Lehners, and Turok 2017, “Lorentzian Quantum Cosmology,” §§II–IV. Their separate perturbation calculation found inverse-Gaussian tensor-fluctuation weighting for that prescription Feldbrugge, Lehners, and Turok 2017, “No Smooth Beginning for Spacetime,” tensor-perturbation calculation. Halliwell, Hartle, and Hertog argued that no-boundary regularity specifies a different contour/state construction and defended stable perturbative predictions within that definition Halliwell, Hartle, and Hertog 2019, §§II–V.

These are competing specifications, not merely two evaluations of an uncontested real integral. One must compare lapse domain, endpoint regularity, singularities, thimble intersection numbers, perturbation contour, and final probability rule. Lehners surveys the alternative definitions and their consequences in Lehners 2023, §§3.6, 4.1–4.2, and Appendix C.

Results and open questions as of 2026-08-20

Both proposals are explicit in symmetry-reduced models, but extension to an untruncated gravitational state and a unique empirical probability measure remains unresolved.

Proposal or formulation	Controlled result or evidence	Scope, adverse check, or limitation
Compact regular no-boundary saddle	Smooth complex saddles and WKB branches can be calculated in homogeneous minisuperspace	Choice of contour and fluctuation state is additional; no full configuration-space measure is known
Tunneling outgoing condition	Superspace current selects an outgoing semiclassical branch in simple models	“Outgoing” depends on orientation and clock; regularity elsewhere and normalization remain necessary
Positive-real Lorentzian lapse contour	Picard–Lefschetz intersections fix saddles for a declared finite-dimensional integral	Result need not coincide with contours used in other no-boundary definitions
Decoherent-histories no-boundary state	Probabilities can be conditioned on coarse-grained classical histories	Requires a decoherence functional, coarse graining, and controlled perturbative state

An adverse check is to reverse the WKB current convention while keeping the phrase “outgoing.” The selected coefficient changes unless the normal to superspace boundary and expanding branch are defined explicitly. Likewise, replacing one lapse contour by another can cross a singularity or Stokes line and change intersection numbers even if the saddle equations are unchanged.

The conservative position is that these are calculable candidate boundary conditions in minisuperspace. The stronger position that one uniquely determines cosmological initial conditions is conjectural. Relativistic cosmology develops data inference, while string and holographic realizations and semiclassical thermodynamics remain in their dedicated treatments.

The conclusions could change after an untruncated contour construction controls gauge volume, perturbations, and composition, or after competing prescriptions yield a common normalized probability for a discriminating observable—or demonstrably incompatible predictions—under the same matter model and observational conditioning.

Exercises

In an oscillatory region take $\Psi=C_+e^{iS}+C_-e^{-iS}$ with slowly varying real $S$ and positive $S'$. Compute $J=(i/2)\Psi^*\overleftrightarrow{\partial_a}\Psi$ after averaging rapid interference. Which condition is “outgoing” if positive $J$ is declared outgoing, and why is that declaration not a normalized probability?

Solution

Neglecting derivatives of the slowly varying amplitudes and averaging the cross terms gives $J=-S'\lvert C_+\rvert^2+S'\lvert C_-\rvert^2$ for the displayed current convention. Positive current therefore selects $C_+=0$ and retains the $e^{-iS}$ branch. Reversing the superspace normal or current definition reverses that label, which is why “outgoing” must be tied to an expanding geometry explicitly. The current is a flux on superspace, not a positive-definite norm; its magnitude also depends on normalization and the measure. Conditional probabilities require a physical inner product or decoherence construction in addition to branch selection.

References

• Feldbrugge, Job, Jean-Luc Lehners, and Neil Turok. “Lorentzian Quantum Cosmology.” Physical Review D 95 (2017): 103508. DOI.
• Feldbrugge, Job, Jean-Luc Lehners, and Neil Turok. “No Smooth Beginning for Spacetime.” Physical Review Letters 119 (2017): 171301. DOI. Open manuscript.
• Halliwell, Jonathan J., James B. Hartle, and Thomas Hertog. “What Is the No-Boundary Wave Function?” Physical Review D 99 (2019): 043526. DOI.
• Hartle, James B., and Stephen W. Hawking. “Wave Function of the Universe.” Physical Review D 28 (1983): 2960–2975. DOI.
• Lehners, Jean-Luc. “Review of the No-Boundary Wave Function.” Physics Reports 1022 (2023): 1–82. arXiv:2303.08802.
• Vilenkin, Alexander. “Quantum Cosmology and the Initial State of the Universe.” Physical Review D 37 (1988): 888–897. DOI.