Topology Change and Wormhole Contributions

A topology-changing saddle can contribute only after the theory declares which cobordisms are summed, what boundary Hilbert spaces they connect, and how the amplitude composes. Lorentzian causal restrictions, Euclidean wormhole saddles, and baby-universe descriptions answer different questions. A stationary metric on a nontrivial manifold is therefore not sufficient evidence that topology change belongs to the quantum theory.

Required background. Gravitational Actions and Boundary Terms supplies boundary data and gluing actions; Flat-Spacetime Quantum Field Theory supplies functional-integral composition and connected correlators.

Helpful background. Einstein Equations supplies classical saddle conditions; Quantum Information supplies Hilbert-space factorization and mixed-state tests; and Quantum Gravity supplies programme-wide evidence criteria.

Cobordisms and composition

Let a cobordism $\mathcal M:\Sigma_{\mathrm{in}}\to\Sigma_{\mathrm{out}}$ have boundary $\partial\mathcal M=\bar\Sigma_{\mathrm{in}}\sqcup\Sigma_{\mathrm{out}}$. A formal amplitude is

$$K_{\mathcal M}[h_{\mathrm{out}},h_{\mathrm{in}}] =\int_{\substack{g|_{\Sigma_{\mathrm{in}}}=h_{\mathrm{in}}\\ g|_{\Sigma_{\mathrm{out}}}=h_{\mathrm{out}}}} \mathcal Dg\,e^{iS_{\mathcal M}[g]}.$$

If two cobordisms are glued along $\Sigma$, consistency asks for

$$K_{\mathcal M_2\circ\mathcal M_1} =\int\mathcal Dh_\Sigma\, K_{\mathcal M_2}[h_{\mathrm{out}},h_\Sigma] K_{\mathcal M_1}[h_\Sigma,h_{\mathrm{in}}],$$

with the correct gauge quotient, boundary measure, and corner factors. A topology sum adds weights, automorphism factors, and a rule for which manifolds are allowed. Without those data, “sum over geometries” does not define a linear map between state spaces.

Lorentzian causal obstruction

Smooth Lorentzian topology change is constrained by causality. In a globally hyperbolic spacetime, all Cauchy hypersurfaces have the same topology: each inextendible timelike trajectory meets each such hypersurface once, allowing one to transport a spatial slice to another. A smooth compact time-oriented Lorentzian cobordism with spacelike initial and final boundaries is more tightly constrained: under Geroch’s causal hypotheses, different boundary topologies require closed timelike curves or failure of the hypotheses Geroch 1967, pp. 782–786. Arbitrary spacelike submanifolds need not be Cauchy surfaces, so changing the topology of a drawn slice alone is not a counterexample.

This theorem does not forbid every quantum topology-changing history. A proposal can allow degenerate metrics, singular geometries, causal discontinuity, boundaries, or Euclidean/complex saddles. Each escape changes a hypothesis and creates a new consistency test. For example, permitting degenerate metrics requires a measure near degeneracy and a rule for propagating fields through it. Invoking “quantum fluctuations” without specifying that rule does not answer the theorem.

Wormholes and baby universes

A Euclidean wormhole connects regions or boundaries through a common saddle. In an effective description, small wormholes can generate bilocal terms. Introducing auxiliary parameters can rewrite those terms as local couplings whose values vary between sectors, the origin of the baby-universe or $\alpha$-parameter language. Coleman developed this mechanism and its coupling-distribution interpretation in Coleman 1988, pp. 870–879.

The calculation is sensitive to the conformal contour, negative modes, wormhole measure, dilute-gas approximation, and ultraviolet completion. A semiclassical saddle with action $S_{\mathrm{wh}}$ contributes schematically $e^{-S_{\mathrm{wh}}}$ only after those determinants and integration cycles are fixed. Summing arbitrarily many disconnected components may exponentiate, diverge, or invalidate the dilute approximation.

Baby-universe language can be formulated as a Hilbert space carrying operators that change the number of connected components. Conditioning on an eigenstate of commuting baby-universe operators may restore factorization for parent-universe observables, whereas averaging over sectors produces correlations resembling an ensemble. Marolf and Maxfield give a modern Hilbert-space account and distinguish state conditioning from ensemble averaging in Marolf and Maxfield 2020, §§2–4.

The factorization diagnostic

For two disconnected boundaries $B_1$ and $B_2$, a fixed quantum theory with independently prepared systems ordinarily obeys

$$Z[B_1\sqcup B_2]=Z[B_1]Z[B_2].$$

A connected wormhole saddle adds $Z_{\mathrm{conn}}[B_1,B_2]$. That may indicate an averaged theory, a shared superselection sector, a failure of the assumed tensor factorization, or an incomplete saddle sum. It cannot be interpreted before the boundary Hilbert space and preparation protocol are declared.

This issue is handed off in detail to black-hole information and to string and holographic constructions when a specific duality or microscopic realization is used. Here the general requirement is narrower: a topology-changing contribution must coexist with the theory’s composition law, causal and contour choices, and boundary state space.

Results and open questions as of 2026-08-20

Topology-changing mechanisms are controlled in selected low-dimensional, semiclassical, and dual models; the definition of a unique four-dimensional sum over topology remains unresolved.

Position or formulation	Evidence or controlled result	Scope, adverse check, or limitation
Fixed-topology Lorentzian quantization	Preserves a declared causal state space and standard composition	Excludes topology change by definition and may miss nonperturbative sectors
Lorentzian cobordisms with enlarged histories	Can include topology change after relaxing degeneracy or causality hypotheses	Must control singular or causally discontinuous propagation and the measure
Euclidean wormhole saddle sum	Produces explicit connected contributions in selected semiclassical models	Conformal contour, negative modes, dilute approximation, and ultraviolet completion are unsettled
Baby-universe Hilbert space or sector conditioning	Explains correlations and possible factorization within fixed sectors	Physical sector selection and relation to a unique four-dimensional theory remain open

The adverse test is factorization on disconnected boundaries. If a proposed wormhole term gives a connected answer, one must show whether the observable is ensemble averaged, conditioned on a baby-universe state, or genuinely nonfactorizing. Relabeling the same calculation as “topology fluctuation” does not resolve the conflict.

The conclusions could change after a four-dimensional construction defines a convergent topology sum with gauge, contour, and gluing data, or after an independent result proves that a proposed baby-universe sector restores—or necessarily violates—factorization for a specified boundary Hilbert space beyond a solvable model.

Exercises

Suppose a saddle calculation gives $Z[B_1\sqcup B_2]=Z_1Z_2+\epsilon$ with $\epsilon\ne0$. Compute the connected normalized quantity and explain two logically different ways it could arise without immediately concluding that quantum mechanics fails.

Solution

The connected part is $Z_{\mathrm{conn}}=Z[B_1\sqcup B_2]-Z_1Z_2=\epsilon$, or $\epsilon/(Z_1Z_2)$ after normalization by the disconnected amplitude. One possibility is ensemble averaging: for random sector-dependent amplitudes $Z_\alpha[B]$, the average $\langle Z_\alpha[B_1]Z_\alpha[B_2]\rangle$ need not equal the product of averages. Another is a shared baby-universe state that correlates the boundary experiments; conditioning on a definite $\alpha$ sector may restore factorization. These interpretations make different preparation claims. A fixed, independently prepared tensor-product theory would instead require cancellation of $\epsilon$ after the complete saddle sum.

References

• Coleman, Sidney. “Black Holes as Red Herrings: Topological Fluctuations and the Loss of Quantum Coherence.” Nuclear Physics B 307 (1988): 867–882. DOI.
• Geroch, Robert. “Topology in General Relativity.” Journal of Mathematical Physics 8 (1967): 782–786. DOI.
• Marolf, Donald, and Henry Maxfield. “Transcending the Ensemble: Baby Universes, Spacetime Wormholes, and the Order and Disorder of Black Hole Information.” Journal of High Energy Physics 2020 (2020): 044. arXiv:2002.08950.