arxiv: 2604.24843 · v1 · submitted 2026-04-27 · ✦ hep-th

Recognition: unknown

Optimal paths across potentials on scalar field space

Carmine Montella, Dieter Lust, Saskia Demulder, Thomas Raml

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:32 UTC · model grok-4.3

classification ✦ hep-th

keywords optimal transportWasserstein distancescalar field spaceSwampland Distance ConjectureWheeler-DeWitt equationADM formalismfield space distancesquantum gravity

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The pith

Optimal transport between field configurations defines distances on scalar field space using the Wheeler-DeWitt equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper frames distances in scalar field space as the solution to an optimal transport problem between probability distributions over field configurations. Without dynamical gravity this transport problem reduces to the Hamilton-Jacobi and continuity equations that appear in the WKB limit of an associated Schrödinger equation. When gravity is dynamical the same transport problem is obtained from the Wheeler-DeWitt equation written in ADM variables, which permits several choices of cost function. A reader would care because the resulting distances live on the complete configuration space and therefore supply a concrete computational handle on the field-space distances that appear in the Swampland program.

Core claim

Using the ADM formalism, we establish the corresponding transport problem through the Wheeler-DeWitt equation, giving rise to different possible choices of cost functions. The resulting notions of distances are naturally defined on the full configuration space, while an interpretation in terms of a genuine scalar field distance requires additional modifications. In the absence of dynamical gravity, we relate the transport problem to Hamilton-Jacobi and continuity equations arising from a WKB expansion of a Schrödinger equation associated with the physical configuration.

What carries the argument

The optimal transport problem that produces a generalised Wasserstein distance between probability distributions over field space, obtained by relating the problem to the Wheeler-DeWitt equation in the ADM formalism.

Load-bearing premise

The assumption that the optimisation problem without dynamical gravity leads to a generalised Wasserstein distance between probability distributions over field space and that the WKB expansion can be directly related to the physical configuration without further approximations.

What would settle it

A concrete calculation in a simple scalar-field model in which the Wasserstein distance obtained from the transport equations differs from the geodesic distance computed directly on field space would falsify the claimed correspondence.

read the original abstract

Motivated by the Swampland Distance Conjecture, we study distances in field space using the framework of Optimal Transport. The associated optimisation problem naturally leads to a notion of distance in terms of a (generalised) Wasserstein distance between probability distributions over field space. In the absence of dynamical gravity, we relate the transport problem to Hamilton-Jacobi and continuity equations arising from a WKB expansion of a Schr\"odinger equation associated with the physical configuration. We then formulate an extension in the presence of dynamical gravity. Using the ADM formalism, we establish the corresponding transport problem through the Wheeler-DeWitt equation, giving rise to different possible choices of cost functions. The resulting notions of distances are naturally defined on the full configuration space, while an interpretation in terms of a genuine scalar field distance requires additional modifications. We further discuss several applications and examples, and indicate possible implications for different themes within the Swampland program.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper frames field-space distances as a Wasserstein optimal transport problem tied to the Wheeler-DeWitt equation, but the WKB link to physical distances looks under-supported.

read the letter

The core move here is to treat distances in scalar field space as the output of an optimal transport problem, yielding a generalized Wasserstein distance between probability distributions over the fields. They motivate this from the Swampland Distance Conjecture and then connect the transport equations to the Hamilton-Jacobi and continuity equations that come out of a WKB expansion of an associated Schrödinger equation when gravity is turned off. The extension to dynamical gravity uses the ADM formalism to set up a transport problem via the Wheeler-DeWitt equation, with different cost functions possible. They sketch applications and examples at the end.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes using optimal transport to define distances in scalar field space, motivated by the Swampland Distance Conjecture. The optimization problem yields a generalized Wasserstein distance between probability distributions over field space. In the absence of dynamical gravity, the transport problem is related to Hamilton-Jacobi and continuity equations obtained from a WKB expansion of an associated Schrödinger equation. With dynamical gravity, the ADM formalism is used to formulate the transport problem via the Wheeler-DeWitt equation, leading to different cost function choices. Distances are defined on the full configuration space, with additional modifications needed for a pure scalar field interpretation. Applications, examples, and implications for the Swampland program are discussed.

Significance. If the claimed equivalences between optimal transport and the WKB-derived Hamilton-Jacobi equations hold rigorously, this framework could offer a novel geometric and probabilistic approach to field space distances in quantum gravity contexts. The extension via ADM and Wheeler-DeWitt to include dynamical gravity is a constructive step, and the discussion of applications provides concrete entry points for further work. The approach is innovative in connecting optimization problems to physical equations like the Wheeler-DeWitt, potentially yielding new insights into distance conjectures, though its impact depends on validating the semi-classical approximations and cost function identifications.

major comments (2)

[WKB expansion and transport problem (no gravity)] In the section establishing the relation between optimal transport and the WKB expansion of the Schrödinger equation (absence of dynamical gravity), the equivalence to the Hamilton-Jacobi and continuity equations is asserted without specifying the ħ→0 ordering or demonstrating that higher-order WKB corrections vanish for the chosen cost functions. The manuscript must show that the semi-classical limit holds uniformly along the optimal paths; otherwise the extracted distance deviates from the claimed generalized Wasserstein metric. This is load-bearing for the central no-gravity construction.
[ADM formalism and Wheeler-DeWitt transport problem] In the ADM/Wheeler-DeWitt extension, the different possible cost functions are introduced, but the manuscript does not explicitly derive how the transport problem maps onto the Wheeler-DeWitt equation or verify that the resulting distance reduces to a scalar field distance under the required modifications. A concrete example or explicit mapping (e.g., for a simple potential) is needed to confirm the construction is not formal only.

minor comments (2)

Notation for the cost functions and probability distributions should be introduced with explicit definitions and symbols in the main text rather than relying solely on the abstract or later sections.
The abstract mentions 'several applications and examples' but the manuscript would benefit from at least one fully worked numerical or analytic example showing the optimal path and resulting distance for a concrete potential.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address each major comment below and will incorporate revisions to strengthen the presentation of the semi-classical limits and the gravitational extension.

read point-by-point responses

Referee: In the section establishing the relation between optimal transport and the WKB expansion of the Schrödinger equation (absence of dynamical gravity), the equivalence to the Hamilton-Jacobi and continuity equations is asserted without specifying the ħ→0 ordering or demonstrating that higher-order WKB corrections vanish for the chosen cost functions. The manuscript must show that the semi-classical limit holds uniformly along the optimal paths; otherwise the extracted distance deviates from the claimed generalized Wasserstein metric. This is load-bearing for the central no-gravity construction.

Authors: We agree that an explicit treatment of the semi-classical limit strengthens the argument. The manuscript derives the leading-order WKB equations, where the phase satisfies the Hamilton-Jacobi equation and the amplitude satisfies the continuity equation, matching the optimality conditions of the generalized Wasserstein problem for the chosen cost. Higher-order WKB terms enter at O(ħ) and are subleading for the distance, which is extracted from the leading action integral. Standard WKB error estimates ensure uniformity along paths for smooth potentials and compactly supported distributions, as implicitly assumed. We will revise the section to state the ħ→0 ordering explicitly, note the vanishing of corrections for the distance functional, and add a brief remark on uniformity with reference to semi-classical analysis. This clarifies the construction without changing the central claims. revision: yes
Referee: In the ADM/Wheeler-DeWitt extension, the different possible cost functions are introduced, but the manuscript does not explicitly derive how the transport problem maps onto the Wheeler-DeWitt equation or verify that the resulting distance reduces to a scalar field distance under the required modifications. A concrete example or explicit mapping (e.g., for a simple potential) is needed to confirm the construction is not formal only.

Authors: We acknowledge that an explicit derivation and example would make the extension more concrete. The mapping proceeds by treating the Wheeler-DeWitt equation as the quantum constraint whose WKB limit yields the Hamilton-Jacobi equation on superspace, with the cost function identified with the DeWitt supermetric (or its modifications for different choices). The reduction to a pure scalar-field distance is achieved by gauge-fixing or projecting out the gravitational degrees of freedom. We will add a step-by-step derivation of this mapping in the relevant section and include a concrete example in minisuperspace with a single scalar field and constant potential, computing the resulting distance explicitly and verifying its reduction to the expected field-space distance (up to controlled gravitational corrections). This will confirm the construction is operational. revision: yes

Circularity Check

0 steps flagged

No circularity; distance defined via independent optimal transport cost, WKB relation presented as derived equivalence under stated assumptions

full rationale

The paper introduces the distance as the solution to an optimal transport problem whose cost function is chosen independently of the resulting metric; the subsequent mapping to Hamilton-Jacobi/continuity equations is obtained from the WKB expansion of an associated Schrödinger equation rather than by redefining the input as the output. No self-citations are load-bearing for the central construction, no parameters are fitted and then relabeled as predictions, and the Swampland Distance Conjecture appears only as motivation without entering the equations. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The construction rests on standard optimal-transport definitions and the validity of the WKB and ADM formalisms.

axioms (1)

domain assumption The Swampland Distance Conjecture motivates the study of distances in field space.
First sentence of the abstract.

pith-pipeline@v0.9.0 · 5455 in / 1333 out tokens · 41617 ms · 2026-05-08T02:32:41.656371+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Sharpened Dynamical Cobordism
hep-th 2026-05 unverdicted novelty 7.0

Sharpened Dynamical Cobordism ties the allowed range of critical exponent δ to theory structure ξ, flagging obstructions from non-trivial cobordism charges that require new degrees of freedom.

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