Conformal-Factor Problem, Integration Contours, and Resurgent-Completion Proposals

The conformal-factor problem is a failure of the naive real Euclidean integration cycle: rapidly varying Weyl rescalings make the Euclidean Einstein action arbitrarily negative, so their Boltzmann weight grows instead of decaying. This is not, by itself, a propagating Lorentzian ghost, and it does not say that every semiclassical gravitational calculation is meaningless. It says that a gravitational amplitude needs more input than the formal symbol $\int \mathcal Dg\,e^{-I_E[g]/\hbar}$. One must declare a complex cycle, its boundary conditions, and the continuation that selects it.

This page first solves one unstable conformal mode exactly. That calculation exposes the phase choice hidden in the phrase “rotate the conformal factor.” It then separates three increasingly strong claims: a convergent one-loop Gaussian, a globally specified gravitational integration cycle, and a resurgent nonperturbative completion. Only the first is supplied by the elementary rotation.

Required background. Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions distinguishes boundary, contour, and topology data for the amplitude whose contour is at issue. Complex Saddles, Lefschetz Thimbles, and Integration Cycles develops relative homology, orientations, intersection numbers, and the general Stokes geometry used here.

Helpful background. Stationary Phase, Coalescing Saddles, and Stokes Geometry explains how steepest-descent bases change. Complex Saddles, Quasi-Zero Modes, and Hidden Phases gives a controlled field-theory example of ambiguity cancellation.

Compactly supported conformal rescalings make the real cycle divergent

Take $D>2$ and use the Euclidean-signature convention

$$I_E[g]=-\frac{1}{16\pi G} \int_M d^D x\,\sqrt g\,(R-2\Lambda)+I_{\partial M}.$$

Separate a Weyl factor from a reference metric,

$$g_{\mu\nu}=e^{2\sigma}\bar g_{\mu\nu}.$$

To prove unboundedness without changing the prescribed boundary geometry, choose $\sigma$ smooth and compactly supported in the interior of $M$. Then $g$ and $\bar g$ agree in a neighborhood of the boundary, so the Gibbons–Hawking–York term and any local boundary counterterms are unchanged. More generally,

$$\sqrt h\,K =\sqrt{\bar h}\,e^{(D-2)\sigma} \left[\bar K+(D-1)\bar n^\mu\partial_\mu\sigma\right],$$

and the normal-derivative term cancels the boundary term produced when the Einstein–Hilbert Laplacian is integrated by parts. Compact support makes that cancellation automatic for the diagnostic below. The scalar curvature transforms as

$$R[g]=e^{-2\sigma} \left[ \bar R-2(D-1)\bar\nabla^2\sigma -(D-1)(D-2)(\bar\nabla\sigma)^2 \right].$$

After integrating the Laplacian term over $M$, with no support at the boundary, the action is

$$I_E[e^{2\sigma}\bar g] =-\frac{1}{16\pi G}\int_M d^D x\,\sqrt{\bar g}\, \left[ e^{(D-2)\sigma} \left(\bar R+(D-1)(D-2)(\bar\nabla\sigma)^2\right) -2\Lambda e^{D\sigma} \right]+I_{\partial M}[\bar g].$$

The derivative term therefore has the wrong sign:

$$I_E[\sigma]\supset -\frac{(D-1)(D-2)}{16\pi G} \int_M d^D x\,\sqrt{\bar g}\, e^{(D-2)\sigma}(\bar\nabla\sigma)^2.$$

Choose an oscillatory $\sigma$ of fixed small amplitude and wave number $k$ inside that compact region. The negative term grows as $-k^2$, whereas the curvature and cosmological-potential terms have no compensating $k^2$ growth. Hence $I_E\to-\infty$ as $k\to\infty$, while the boundary data stay fixed, and the real-metric weight $e^{-I_E/\hbar}$ diverges. This is the conformal-factor problem behind the one-loop prescription of Gibbons, Hawking, and Perry 1978, pp. 141–150.

Why this is not immediately a Lorentzian ghost. The trace of a metric perturbation is not an independent physical scalar before the diffeomorphism constraints, gauge condition, ghosts, Jacobians, and boundary modes are treated together. The wrong sign above diagnoses an off-shell integration direction of the naive Euclidean field-space integral, not a negative-norm particle in the Lorentzian graviton spectrum. Gauge fixing can change how scalar and longitudinal variables mix, but it does not automatically supply a convergent cycle. Indeed, modern quadratic contour prescriptions can depend on the auxiliary DeWitt metric used on perturbation space Liu, Marolf, and Santos 2024, §§ 2–4. That dependence is one reason to state the contour prescription rather than treating it as invisible notation.

A regulated Gaussian fixes what contour rotation means

To isolate the unstable direction without misidentifying it as a physical determinant eigenmode, first restrict the bare Einstein action to a one-dimensional conformal slice. Let $\bar g$ be an Einstein saddle,

$$\bar R=\frac{2D\Lambda}{D-2},$$

and choose a normalized scalar mode $u(x)$ obeying the fixed-boundary condition,

$$-\bar\nabla^2u=\kappa^2u, \qquad \int_M d^D x\,\sqrt{\bar g}\,u^2=1, \qquad u|_{\partial M}=0,$$

Writing $\sigma(x)=q\,u(x)$ and expanding the restricted bare action gives

$$I_E[e^{2qu}\bar g] =I_E[\bar g]-\frac{\lambda_{\mathrm{EH}}}{2}q^2+O(q^3), \qquad \lambda_{\mathrm{EH}} =\frac{(D-1)(D-2)\kappa^2-2D\Lambda}{8\pi G}.$$

For sufficiently large $\kappa$, $\lambda_{\mathrm{EH}}>0$. The boundary condition keeps the prescribed boundary metric fixed while $q$ is complexified. This result is exact at quadratic order on the bare conformal slice. A covariant gauge-fixing term can also scale as $\kappa^2$, and diagonalizing the full scalar–longitudinal sector need not preserve the form $\sigma=qu$. Ghosts, Jacobians, zero modes, and the measure must therefore be included before this Gaussian is interpreted as a factor in a gravitational one-loop determinant. In compact-Einstein-space examples, such full accounting can cancel phases that an isolated mode would appear to contribute Shi and Turiaci 2025, §§ 2–3.

For the purpose of exposing the contour data, set $\lambda=\lambda_{\mathrm{EH}}>0$ and study the finite-dimensional normal form

$$Z_q(\mathcal C)=\int_{\mathcal C}dq\, \exp\!\left(\frac{\lambda q^2}{2\hbar}\right).$$

Scope of the calculation. The object is the restricted conformal Gaussian, not yet the full one-loop amplitude. The convention is the Euclidean weight $e^{-I_E/\hbar}$ with real $\hbar,\lambda>0$. The domain is a finite-mode Gaussian neighborhood of one fixed saddle. The logical status is an exact analytic result within that normal form, independently checked by direct parametrization and by continuation from a stable Gaussian. It samples $q=O(\sqrt{\hbar/\lambda})$, where omitted interactions must be parametrically small. Its uncertainty is structural rather than numerical: the quadratic approximation does not determine the nonlinear contour tails, the coupled gauge-fixed determinant, or the topology sum.

The endpoints and orientations are essential:

Oriented cycle	Parametrization	Weight on the parameter line	Exact result	What convergence establishes
$\mathcal C_{\mathbb R}:-\infty\to+\infty$	$q=y$, $y:-\infty\to+\infty$	$e^{+\lambda y^2/(2\hbar)}$	divergent	The original real Euclidean Gaussian is not defined.
$\mathcal C_{\uparrow}:-i\infty\to+i\infty$	$q=iy$, $y:-\infty\to+\infty$	$e^{-\lambda y^2/(2\hbar)}$	$+i\sqrt{2\pi\hbar/\lambda}$	A locally convergent continuation with upward orientation.
$\mathcal C_{\downarrow}:+i\infty\to-i\infty$	$q=-iy$, $y:-\infty\to+\infty$	$e^{-\lambda y^2/(2\hbar)}$	$-i\sqrt{2\pi\hbar/\lambda}$	A locally convergent continuation with downward orientation.

The last two rows are the same geometric imaginary axis with opposite orientation. They are not two independent thimbles of this pure Gaussian. Their opposite phases record two ways of analytically continuing a divergent expression.

To see the continuation directly, begin with the convergent parent Gaussian

$$G(a)=\int_{-\infty}^{\infty}dq\, e^{-a q^2/(2\hbar)} =\sqrt{2\pi\hbar}\,a^{-1/2}, \qquad \operatorname{Re}a>0.$$

Write $a=\lambda e^{i\theta}$ and rotate $q=e^{-i\theta/2}y$ so that $a q^2=\lambda y^2$. Continuing $a$ from $+\lambda$ to $-\lambda$ through the upper half-plane, $0\leq\theta\leq\pi$, ends on $q=-iy$ and gives $G=-i\sqrt{2\pi\hbar/\lambda}$. Continuing below the origin, $0\geq\theta\geq-\pi$, ends on $q=iy$ and gives the opposite phase. The two continuations encircle the branch point of $a^{-1/2}$ on different sides.

This calculation sharpens the usual instruction:

• Convergence fixes the allowed asymptotic sectors for $q$.
• It does not choose clockwise versus counterclockwise continuation.
• A Lorentzian $i\epsilon$ prescription, a boundary-state definition, or an independently declared relative cycle must make that choice.
• With infinitely many modes, the product of these phases is itself regulator dependent and must be combined with the remaining fluctuation and ghost determinants.

Nonlinear terms can create several critical points and genuinely different relative-homology classes. At that stage a choice can change not only an overall Gaussian phase but also which exponentially weighted saddles contribute. The prerequisite page’s signed quartic Stokes jump develops that general geometry; repeating its figure here would obscure the gravitational input that remains to be supplied.

From one mode to a gauge-fixed metric integral

A full prescription must say what is being complexified and what is held fixed:

Input	Required declaration
Boundary problem	Fixed real sources or state data, allowed bulk complexification, and the boundary action that makes the variation well posed.
Regulator	Finite mode, lattice, or other truncation and the intended regulator-removal limit.
Gauge treatment	Gauge slice, Faddeev–Popov factors, stabilizers, zero modes, and boundary conditions.
Parent cycle	Real Lorentzian contour with a stated $i\epsilon$, a convergent Euclidean domain, or a specified analytic continuation.
Singular loci	Poles, branch sets, degenerate metrics, lapse zeroes, and allowed endpoints.
Output	A conditional relative cycle, oriented thimble coefficients, and determinant phases.
Stop rule	If no parent cycle is given, regulator removal is unstable, or the relative problem changes without a transport rule, no global gravitational completion has been defined.

Minisuperspace makes the missing information visible without pretending to solve the functional integral. After integrating other variables, a wavefunction may reduce schematically to

$$\Psi(a_f)=\int_{\mathcal C_N}dN\,\mu(N) \exp\!\left[\frac{i}{\hbar}S_{\mathrm{red}}(N;a_f)\right],$$

where the lapse $N$ imposes the Hamiltonian constraint. The contour must specify its endpoints and how it avoids $N=0$, where a configuration-space integrand can have an essential singularity. Starting from reduced phase space, Banihashemi and Jacobson found that integrating momenta before the lapse selects a contour passing below the origin in their construction Banihashemi and Jacobson 2025, §§ II–III. This is a concrete derivation in a specified framework, not a universal instruction for every gravitational amplitude.

The cosmology literature also supplies an instructive contrast. A positive half-line Lorentzian lapse contour in Feldbrugge, Lehners, and Turok selects one saddle combination, while the contour and state definition used by Díaz Dorronsoro, Halliwell, Hartle, Hertog, and Janssen yield a real no-boundary combination Feldbrugge, Lehners, and Turok 2017, §§ II–IV Díaz Dorronsoro et al. 2017, §§ I–III. Picard–Lefschetz theory correctly analyzes each declared integral; it cannot decide which initial cycle represents the desired quantum state. Earlier contour analyses reached the same conceptual conclusion: convergence and semiclassical consistency constrain a no-boundary contour but can leave physical freedom Halliwell and Hartle 1990, pp. 1815–1834.

Boundary conditions cannot be appended after the saddle analysis. In a one-loop de Sitter minisuperspace calculation, changing the metric-fluctuation boundary data changes the consistent boundary action and can shift the corrected lapse saddles Ailiga, Mallik, and Narain 2025. Likewise, defining a thermal quantity from a real Lorentzian contour can evade the real-Euclidean conformal divergence while still recovering selected Euclidean black-hole saddles semiclassically Marolf 2022, §§ 2–5. These are observable- and boundary-condition-dependent constructions, not interchangeable definitions of a single universal “gravitational path integral.”

Stokes transport changes the basis, not automatically the amplitude

Let $\zeta$ denote boundary sources or couplings, and let $\Gamma(\zeta)$ be a convergent relative cycle transported through a nonsingular parameter region. Away from a Stokes wall it can be expanded in downward thimbles,

$$\Gamma(\zeta)=\sum_s N_s(\zeta)\,\mathcal J_s(\zeta), \qquad N_s\in\mathbb Z,$$

with orientations included in $\mathcal J_s$. A necessary candidate condition for saddles $s$ and $t$ to lie on a Stokes wall is phase alignment,

$$\operatorname{Im}\!\left(\frac{I_s-I_t}{\hbar}\right)=0;$$

the gradient-flow connectivity and real-part ordering determine whether a jump actually occurs.

This relative-homology formulation and its orientation data are reviewed in Witten 2011, §§ 3.1–3.2.

Suppose crossing the wall changes the basis by

$$\mathcal J_s^{+}=\mathcal J_s^{-}+m_{st}\mathcal J_t^{-}, \qquad \mathcal J_t^{+}=\mathcal J_t^{-}, \qquad m_{st}\in\mathbb Z.$$

For the same transported cycle $\Gamma$, the coefficients must transform inversely:

$$N_s^{+}=N_s^{-}, \qquad N_t^{+}=N_t^{-}-m_{st}N_s^{-}.$$

Thus an individual saddle contribution and its coefficient can jump while the exact integral remains continuous. Holding the coefficients fixed while changing the thimble basis silently changes the integration cycle. If endpoints, convergence sectors, or singular loci change, even the relative-homology problem can change; continuity then requires a separate physical and analytic argument rather than this algebra alone.

The adversarial test is therefore:

1. Vary a source across a candidate Stokes wall from both sides.
2. Verify an actual connecting flow, not phase alignment alone.
3. Compute the thimble-basis jump and the inverse coefficient jump.
4. Check the lateral limits of the complete amplitude under the declared transport.
5. Repeat with a homologically inequivalent parent cycle.

If the fifth step changes an exponentially small contribution while leaving the perturbative series about the leading saddle unchanged, perturbation theory has not selected a unique amplitude. The strongest surviving statement is then “a consistent completion of the declared cycle,” not “the unique completion of quantum gravity.”

A Borel singularity exposes the missing completion datum

Let the formal loop expansion about saddle $s$ be

$$\Phi_s(\hbar)\sim\sum_{k=0}^{\infty}a_{s,k}\hbar^k, \qquad \widehat\Phi_s(t)=\sum_{k=0}^{\infty}\frac{a_{s,k}}{k!}t^k.$$

When the Borel transform can be analytically continued with suitable growth, a directional Borel sum is

$$\mathcal S_{\theta}\Phi_s(\hbar) =\frac{1}{\hbar} \int_{0}^{e^{i\theta}\infty}dt\, e^{-t/\hbar}\widehat\Phi_s(t).$$

A singularity on that ray forces lateral sums $\mathcal S_{\theta^\pm}$. Large-order relations can then connect the coefficients near $s$ to action differences and Stokes constants associated with other sectors. A transseries organizes a candidate answer as

$$Z(\hbar;\boldsymbol\sigma) =\sum_s \sigma_s\,\hbar^{b_s}e^{-I_s/\hbar} \mathcal S_{\theta}\Phi_s(\hbar),$$

where $b_s$ includes determinant and zero-mode power counting, with logarithmic or multi-instanton sectors added when required. The generally complex parameters $\sigma_s$ are not the same objects as the integer thimble coefficients $N_s$, although a specified finite-dimensional integral can relate them. Topology weights are further independent data.

An exact toy model shows both the power and the limitation of Stokes cancellation. For real $g>0$, consider

$$\Phi(g)=\sum_{n=0}^{\infty}n!\,g^{n+1}, \qquad \widehat\Phi(t)=\frac{1}{1-t}.$$

The Borel transform has a pole on the positive integration ray. If $\mathcal S_+$ passes above the pole and $\mathcal S_-$ below it, the residue theorem gives

$$\mathcal S_+\Phi-\mathcal S_-\Phi =2\pi i\,e^{-1/g}.$$

Now define two lateral transseries representations

$$Z_+=\mathcal S_+\Phi+\sigma_+e^{-1/g}, \qquad Z_-=\mathcal S_-\Phi+\sigma_-e^{-1/g}.$$

Requiring $Z_+=Z_-$ fixes the jump relation

$$\sigma_+-\sigma_-=-2\pi i,$$

but it does not fix the common baseline value of the two parameters. Ambiguity cancellation is therefore a consistency condition, not a boundary condition.

A gravitational completion must provide all of the following:

• Sector data: the saddles or other nonperturbative sectors, including which topology classes are even in the problem;
• large-order data: Borel singularity locations, local behavior, and Stokes constants in a stated regulator and renormalization scheme;
• ambiguity cancellation: a demonstration that lateral Borel ambiguities cancel against the appropriate nonperturbative sectors;
• initial transseries parameters: values fixed by the physical cycle or boundary state, not guessed from the perturbative coefficients; and
• global consistency: compatible continuation across source space, singular loci, gauge choices, boundary conditions, and regulator removal.

The remaining logical gap can be proved without any gravitational complication. If $\operatorname{Re}(A/\hbar)>0$, then

$$Z_{\kappa}(\hbar)=Z_0(\hbar)+\kappa e^{-A/\hbar}$$

has the same power-series asymptotic expansion as $Z_0$ for every constant $\kappa$, because $e^{-A/\hbar}$ is smaller than every power of $\hbar$ as $\hbar\to0$ in that sector. No finite or all-orders list of perturbative coefficients can determine $\kappa$ without additional analytic or boundary information.

Resurgent ambiguity cancellation has been demonstrated in controlled quantum-cosmology integrals: Honda, Matsui, Okabayashi, and Terada relate the Borel ambiguity around tunneling saddles to no-boundary saddle contributions in a minisuperspace model Honda et al. 2024, §§ II–III. That is evidence that the mechanism can operate in a gravitational reduction. It is not evidence that the full higher-dimensional metric integral, its measure, topology policy, and physical cycle have thereby been constructed.

Evidence, controls, and limits

Evidence cutoff: 29 August 2026. The levels below should not be collapsed into one claim.

Statement	Status at the stated cutoff	Decisive control or missing ingredient
The real Euclidean Einstein action is unbounded along short-wavelength Weyl rescalings for $D>2$.	Exact structural result within the displayed action and fixed-boundary deformation.	Direct Weyl transformation and high-wave-number limit.
An imaginary one-mode contour makes the unstable Gaussian convergent.	Exact finite-dimensional calculation.	Endpoints, orientation, determinant phase, and analytic-continuation side must be declared.
A declared minisuperspace or one-loop gravitational cycle can be decomposed into thimbles.	Established method for specified regulated integrals.	The parent cycle, singularities, boundary data, gauge fixing, and regulator are inputs.
Resurgent Stokes ambiguities can cancel between gravitational saddle sectors.	Demonstrated in selected minisuperspace models.	Extension to the full metric field space and all relevant sectors remains unproved.
A unique nonperturbative contour-and-resurgence definition of higher-dimensional Einstein gravity follows from perturbation theory.	Not established.	A physical global cycle, measure, topology policy, regulator removal, and uniqueness theorem are missing.

Adding familiar restrictions does not by itself repair the real Euclidean integral. Explicit 2025 families show that imposing gravitational constraints on a chosen foliation, or fixing scalar curvature in the studied boundary classes, can still leave the Einstein action unbounded below Horowitz, Marolf, and Santos 2025.

Criteria restricting allowable complex metrics can remove pathological saddles, but even their application to quantum gravity remains a proposed physical condition and does not by itself furnish the full cycle Witten 2022, pp. 245–280. For selected supersymmetric AdS$_5$ black holes with two angular momenta, the Kontsevich–Segal–Witten and microscopic-index boundary conditions agree analytically, while numerical bulk tests support the corresponding full-metric equivalence Benetti Genolini, Janssen, and Murthy 2026, §§ 2.4 and 3.2–3.5. A complementary rotating-black-hole study finds criterion violations where the corresponding statistical index description breaks down Krishna and Larsen 2026. These are nontrivial, model-specific diagnostics; neither result proves a global integration cycle or fixes its transseries parameters.

Likewise, convergence does not guarantee unitarity. The Osterwalder–Schrader discussion starts from well-defined Euclidean correlators and imposes reflection positivity and other hypotheses to reconstruct a Lorentzian theory. A complex gravitational contour generally has a complex weight, so those properties require separate checks; they do not follow from steepest descent.

For any proposed calculation, record the observable, boundary data, parent cycle, regulator, gauge and measure, saddle set, loop order, Borel direction, Stokes prescription, topology policy, and limiting procedure. Then test at least the Gaussian phase, constraint equations, boundary variation, complex conjugation or reality condition, Stokes transport, gauge-parameter dependence, and a known semiclassical or canonical limit. A failure should lower the claim rather than be absorbed into an unspecified “contour choice.”

The next page, Saddles, Negative Modes, and Steepest-Descent Cycles, analyzes saddle-specific physical negative modes. Those are distinct from the universal short-wavelength conformal instability isolated here. For a cosmological application of lapse contours, continue to No-Boundary and Tunneling Wavefunction Proposals.

Common pitfalls

“The wrong Euclidean sign proves a propagating ghost.” It does not. A propagating ghost is a statement about the constrained Lorentzian spectrum and its norm or residue. The conformal-factor problem is a statement about convergence of a naive Euclidean integration direction.

“Rotate $q\to iq$” completely defines the integral. The map omits endpoints, orientation, the side of analytic continuation, and the phase of the measure. The one-mode table shows two opposite answers unless those data are supplied.

“The contributing saddles are intrinsic.” They are intrinsic only after the parent relative cycle is fixed. Different state or boundary prescriptions can have different intersection numbers with the upward cycles.

“Phase alignment forces a Stokes jump.” It only identifies a candidate wall. A relevant connecting flow or nonzero incidence is also needed.

“A Stokes jump makes the exact amplitude discontinuous.” A thimble basis can jump while a transported cycle remains continuous because its coefficient vector jumps inversely. Genuine singularities, changing endpoints, or a deliberately chosen branch cut require a separate analysis.

“All perturbative coefficients determine the nonperturbative answer.” They may encode action differences and Stokes data under resurgence assumptions, but exponentially small homogeneous freedom remains until a cycle, boundary condition, or other global condition fixes the transseries parameters.

Exercises

1. Recover the wrong-sign kinetic term

Starting from the displayed Weyl transformation of $R[g]$, integrate the $\bar\nabla^2\sigma$ term by parts for compactly supported $\sigma$ and derive the coefficient of $(\bar\nabla\sigma)^2$ in $I_E$. Explain why a cosmological term cannot control the short-wavelength instability.

Solution

Let $f=e^{(D-2)\sigma}$. Compact support removes the boundary term, so

$$\int\sqrt{\bar g}\,f\,\bar\nabla^2\sigma =-\int\sqrt{\bar g}\,\bar\nabla_\mu f\,\bar\nabla^\mu\sigma =-(D-2)\int\sqrt{\bar g}\,f(\bar\nabla\sigma)^2.$$

The Laplacian term in $\sqrt gR$ therefore contributes

$$+2(D-1)(D-2)f(\bar\nabla\sigma)^2,$$

while the explicit gradient term contributes

$$-(D-1)(D-2)f(\bar\nabla\sigma)^2.$$

Their sum is $+(D-1)(D-2)f(\bar\nabla\sigma)^2$ inside $\int\sqrt gR$. The overall minus sign in $I_E$ makes its coefficient negative. For a mode of amplitude $\epsilon$ and wave number $k$, this contribution scales as $-\epsilon^2k^2$. The cosmological term depends on $e^{D\sigma}$ but has no derivatives, so at fixed small $\epsilon$ it scales as $k^0$ and cannot compete as $k\to\infty$.

2. Determine both Gaussian continuation phases

For $G(a)=\sqrt{2\pi\hbar}\,a^{-1/2}$ with $a>0$, continue $a$ to $-\lambda$ once above and once below the origin. Derive the corresponding oriented $q$ contours and verify the two exact phases by direct integration.

Solution

Set $a=\lambda e^{i\theta}$ and $q=e^{-i\theta/2}y$. For the upper continuation $\theta:0\to\pi$, the endpoint is $q=-iy$. As $y$ runs from $-\infty$ to $+\infty$, the contour runs from $+i\infty$ to $-i\infty$ and $dq=-i\,dy$. Hence

$$G_{\mathrm{upper}} =-i\int_{-\infty}^{\infty}dy\, e^{-\lambda y^2/(2\hbar)} =-i\sqrt{\frac{2\pi\hbar}{\lambda}}.$$

For the lower continuation $\theta:0\to-\pi$, one has $q=iy$, the contour runs from $-i\infty$ to $+i\infty$, and $dq=i\,dy$. Therefore

$$G_{\mathrm{lower}} =+i\sqrt{\frac{2\pi\hbar}{\lambda}}.$$

These are the two boundary values of the square root around its branch point. Convergence alone does not choose between them.

3. Preserve a cycle across a Stokes jump

Suppose $\mathcal J_1^+=\mathcal J_1^-+2\mathcal J_2^-$ and $\mathcal J_2^+=\mathcal J_2^-$. A cycle has coefficients $(N_1^-,N_2^-)=(3,-1)$ below the wall. Find $(N_1^+,N_2^+)$ above the wall and verify that the geometric cycle is unchanged.

Solution

The inverse coefficient transformation gives

$$N_1^+=N_1^-=3, \qquad N_2^+=N_2^- -2N_1^-=-1-6=-7.$$

Then

$$3\mathcal J_1^+ -7\mathcal J_2^+ =3(\mathcal J_1^-+2\mathcal J_2^-)-7\mathcal J_2^- =3\mathcal J_1^- -\mathcal J_2^-,$$

which is the original cycle. Keeping $(3,-1)$ on both sides would instead add $6\mathcal J_2^-$ and change the amplitude.

4. Cancel a Borel ambiguity without claiming uniqueness

For the toy series $\Phi(g)=\sum_{n\geq0}n!g^{n+1}$, compute the discontinuity between contours passing above and below the pole of $\widehat\Phi(t)=1/(1-t)$. Determine the required jump in the coefficient of $e^{-1/g}$ and identify what remains free.

Solution

The closed contour formed by the upper ray and the reversed lower ray encloses the pole at $t=1$. With the orientations used in the text,

$$\mathcal S_+\Phi-\mathcal S_-\Phi =2\pi i\,e^{-1/g}.$$

For $Z_\pm=\mathcal S_\pm\Phi+\sigma_\pm e^{-1/g}$, equality of the two lateral representations requires

$$2\pi i+\sigma_+-\sigma_-=0,$$

or $\sigma_+-\sigma_-=-2\pi i$. Adding the same constant $c$ to both $\sigma_+$ and $\sigma_-$ preserves this relation, so the Stokes cancellation does not determine the common baseline. A parent integration cycle or another global condition must fix it.

5. Prove perturbative nonuniqueness

Let $A>0$ and define $Z_\kappa(\hbar)=Z_0(\hbar)+\kappa e^{-A/\hbar}$ for real $\hbar\to0^+$. Show that every $Z_\kappa$ has the same asymptotic power series as $Z_0$, and state what additional datum can distinguish them.

Solution

For every nonnegative integer $N$,

$$\lim_{\hbar\to0^+}\frac{e^{-A/\hbar}}{\hbar^N}=0.$$

Set $x=A/\hbar$; the ratio is $A^{-N}x^Ne^{-x}$, which tends to zero as $x\to\infty$. Thus $\kappa e^{-A/\hbar}$ is beyond all orders in $\hbar$ and changes none of the asymptotic power-series coefficients. A boundary condition, parent integration cycle, analyticity or growth condition, exact quantization condition, or equivalent global datum is needed to fix $\kappa$.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

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