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arxiv: 2606.01296 · v2 · pith:YRDPRGULnew · submitted 2026-05-31 · ✦ hep-th

Notes on Wasserstein distance and wormholes

Ville Keranen This is my paper

Pith reviewed 2026-06-28 16:45 UTC · model grok-4.3

classification ✦ hep-th
keywords Boltzmann-Wasserstein distancewormholesSchwinger-Keldyshblack hole horizonsthermal correlation functionsBTZ black holesT bar T deformation
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The pith

The Boltzmann-Wasserstein distance between two quantum theories reduces to a squared horizon-area difference for small entropy shifts and equals a thermal two-point function when Hamiltonians differ by an operator V.

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

The paper defines the Boltzmann-Wasserstein distance as the optimal W2 transport cost between the Boltzmann-weighted energy spectra of two quantum theories. This optimum is realized by the comonotone partition function C_max that pairs states according to their energy rank. For semiclassical theories that differ by a small entropy shift the normalized distance simplifies to a direct function of the difference in horizon areas evaluated at equal energy. When the Hamiltonians differ by an operator V the distance is given by a long-time average of the real-time thermal two-point function of V, with a four-point function appearing at next order if the one-point function vanishes. A classical Schwinger-Keldysh wormhole saddle built from two Euclidean caps and Lorentzian interpolating segments reproduces both the saddle-point equations and the on-shell value of C_max.

Core claim

The paper establishes that the dominant wormhole is the one realizing the comonotone partition function C_max, which pairs energy eigenstates by rank order, and that this quantity is exactly reproduced by a classical Schwinger-Keldysh saddle consisting of two Euclidean caps sharing a horizon joined by Lorentzian segments that adiabatically connect the two theories. The on-shell action of this saddle matches both the saddle conditions and the value of C_max, while a purely Euclidean interpolation is exponentially suppressed. The saddle captures only the rearrangement of the spectrum; perturbative expansions in the difference operator V additionally retain the variance of its matrix elements.

What carries the argument

The comonotone partition function C_max, which pairs states by rank in the energy basis and is realized as the dominant wormhole in the doubled Hilbert space.

If this is right

  • For semiclassical theories differing by a small entropy shift the normalized BW distance squared approximates (delta A / 4G)^2 / 8 with areas evaluated at equal energy.
  • When Hamiltonians differ by an operator V the BW distance equals the long-time average of the real-time thermal two-point function of V.
  • If the thermal one-point function of V vanishes, a four-point function representation appears at the next order in the expansion.
  • The gravity saddle computes only the spectral rearrangement while perturbative representations retain the variance of the matrix elements of V.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a geometric diagnostic for how close two gravitational theories are when their spectra are only mildly rearranged.
  • The same wormhole saddle may be used to compare theories whose spectra are known only approximately, such as those obtained from T bar T deformations.
  • Because the Lorentzian segments are required for the leading contribution, any purely Euclidean path-integral approach to the distance would miss the dominant term.

Load-bearing premise

That the comonotone pairing by energy rank gives the dominant wormhole contribution and that the constructed Schwinger-Keldysh saddle exactly reproduces both its saddle conditions and its on-shell value.

What would settle it

Numerical computation of the optimal W2 transport distance between the Boltzmann-weighted spectra of two explicit models, such as BTZ black holes with different cosmological constants, and direct comparison against the predicted horizon-area formula at equal energy.

Figures

Figures reproduced from arXiv: 2606.01296 by Ville Keranen.

Figure 1
Figure 1. Figure 1: Wormhole interpretation of the cross-term [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of a localised perturbation on the Bernstein criterion. Left: spectral densities [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Schwinger–Keldysh wormhole geometry. The blue and red Euclidean half-discs [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The TT¯ comonotone geometry. Left: the Euclidean disc split into two caps with different boundary radii — Cap 1 (H1, blue) at Rmax → ∞ and Cap 2 (H2, red) at the TT¯ cutoff rc, sharing the horizon r+. The Lorentzian transitions (green) implement the cutoff interpolation. Right: the radial profile R(τ ) of the cutoff surface around the thermal circle. For the TT¯ map, T0(E1) = 1 − √ 1 − 4µE1 2µ , T′ 0 (E1) … view at source ↗
Figure 5
Figure 5. Figure 5: Normalised BW distance W˜ 2 (β) as a function of inverse temperature β. (a) Two BTZ black holes with unequal AdS radii (including the vacuum contribution). The distance approaches 1 at both high and low temperature; theories with different central charge are far apart at all scales. The minimum near the Hawking–Page transition βHP = 2π is deepest for nearby theories. (b) T T¯ deformation. The distance drop… view at source ↗
Figure 6
Figure 6. Figure 6: The area comparator. The two theories are compared at the common saddle energy [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
read the original abstract

We develop the Boltzmann-Wasserstein (BW) distance, a temperature-dependent metric on the space of quantum theories, defined as the optimal $W_2$ distance between Boltzmann-weighted energy spectra. Computing it is an optimisation over wormholes: each unitary identification of the two energy bases defines a coupling of the two boundaries in the doubled Hilbert space, and the optimum - the comonotone partition function $C_{\max}$, which pairs states by rank - is the dominant wormhole connecting the two theories. For semiclassical theories differing by a small entropy shift, the normalised BW distance collapses to a squared horizon-area comparator, $\tilde{\mathcal{W}}^2 \approx (\delta A/4G)^2/8$, with the two areas evaluated at equal energy. When the Hamiltonians differ by an operator $V$, the BW distance equals a long-time average of the real-time thermal two-point function of $V$; when the thermal one-point function of $V$ vanishes - for instance for $V$ odd under an unbroken discrete global symmetry - a four-point representation appears at the next order. On the gravity side we construct the classical saddle that computes $C_{\max}$: a Schwinger-Keldysh wormhole built from two Euclidean caps sharing a single horizon, joined by Lorentzian segments that adiabatically interpolate between the two theories. Its on-shell action reproduces the spectral saddle of $C_{\max}$ - both the saddle-point conditions and the on-shell value - and the Lorentzian segments are essential: a purely Euclidean interpolation is exponentially suppressed. The saddle captures only the rearrangement of the spectrum; the perturbative representations retain in addition the variance of the matrix elements of $V$, invisible to the classical geometry. We work out two examples - two BTZ black holes with different cosmological constants and a $T\bar{T}$ deformation of BTZ.

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.

Referee Report

2 major / 2 minor

Summary. The paper defines the Boltzmann-Wasserstein (BW) distance as the optimal W_2 transport distance between Boltzmann-weighted energy spectra of two quantum theories. It identifies the optimum with the comonotone partition function C_max realized geometrically as a wormhole in the doubled Hilbert space, constructs an explicit Schwinger-Keldysh saddle (two Euclidean caps sharing a horizon joined by Lorentzian adiabatic segments) whose on-shell action is claimed to reproduce both the saddle-point equations and the value of C_max, derives that for small entropy shifts the normalized distance reduces to a squared horizon-area difference evaluated at equal energy, and shows that when the Hamiltonians differ by an operator V the distance equals a long-time average of the thermal two-point function of V (or a four-point function when the one-point vanishes). Two BTZ examples are worked out.

Significance. If the central identification between the SK wormhole saddle and the comonotone optimum holds without circularity, the work supplies a concrete gravitational realization of an optimal-transport distance on the space of theories, together with falsifiable reductions to area comparisons and thermal correlators. The explicit construction of the saddle and the demonstration that purely Euclidean interpolation is exponentially suppressed are positive features.

major comments (2)
  1. [Schwinger-Keldysh wormhole saddle construction (around the description of the two Euclidean caps and Lorentzian segments] The central claim that the constructed SK wormhole saddle exactly reproduces both the saddle-point conditions and the on-shell value of C_max (the comonotone optimum) is load-bearing for all subsequent results, including the area-comparator formula and the two-point-function representation. The manuscript must show explicitly that the variational equations obtained from varying the gravity action coincide with the optimality condition for rank-ordered pairing without presupposing the result; any mismatch or additional contribution from the Lorentzian segments would invalidate the equality between the geometric on-shell value and the spectral optimum.
  2. [Semiclassical limit and area comparator] The reduction ilde{\mathcal{W}}^2 ≈ (\delta A/4G)^2/8 for semiclassical theories differing by a small entropy shift is stated to hold when the areas are evaluated at equal energy. The derivation of this leading-order expression, the precise definition of the normalization, and the regime of validity (including the size of higher-order corrections in the entropy shift) must be supplied with the intermediate steps shown.
minor comments (2)
  1. [Introduction/definition of BW distance] Notation for the normalized distance \tilde{\mathcal{W}} and the precise relation between the partition function C_max and the W_2 cost should be introduced with an explicit equation early in the text.
  2. [Comparison of Euclidean vs. SK saddles] The statement that the Lorentzian segments are essential because a purely Euclidean interpolation is exponentially suppressed should be accompanied by a quantitative estimate of the suppression factor.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [Schwinger-Keldysh wormhole saddle construction (around the description of the two Euclidean caps and Lorentzian segments] The central claim that the constructed SK wormhole saddle exactly reproduces both the saddle-point conditions and the on-shell value of C_max (the comonotone optimum) is load-bearing for all subsequent results, including the area-comparator formula and the two-point-function representation. The manuscript must show explicitly that the variational equations obtained from varying the gravity action coincide with the optimality condition for rank-ordered pairing without presupposing the result; any mismatch or additional contribution from the Lorentzian segments would invalidate the equality between the geometric on-shell value and the spectral optimum.

    Authors: We agree that an explicit demonstration without circularity is required. In the revised manuscript we will insert a dedicated calculation that varies the on-shell gravity action of the SK wormhole (two Euclidean caps joined by Lorentzian segments) and shows that the resulting saddle-point equations are identical to the optimality conditions for comonotone rank-ordered pairing. We will also verify that the Lorentzian segments contribute only through the adiabatic matching and do not alter the on-shell value or introduce extraneous terms. revision: yes

  2. Referee: [Semiclassical limit and area comparator] The reduction ilde{\mathcal{W}}^2 ≈ (\delta A/4G)^2/8 for semiclassical theories differing by a small entropy shift is stated to hold when the areas are evaluated at equal energy. The derivation of this leading-order expression, the precise definition of the normalization, and the regime of validity (including the size of higher-order corrections in the entropy shift) must be supplied with the intermediate steps shown.

    Authors: We acknowledge that the semiclassical reduction was stated without full intermediate steps. In the revision we will expand the relevant section to derive the normalized distance ilde{\mathcal{W}} from the definition of C_max, perform the small-\delta S expansion at fixed energy, obtain the leading (\delta A/4G)^2 term, define the normalization factor explicitly, and bound the higher-order corrections O((\delta S/S)^2). The regime \delta S o 0 with rac{ ext{area difference}}{ ext{total area}} o 0 will be stated clearly. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines the BW distance as the optimal W_2 distance between Boltzmann-weighted energy spectra and identifies the comonotone partition function C_max (pairing by rank) as the mathematical optimum, which is then interpreted as the dominant wormhole. It constructs an explicit Schwinger-Keldysh saddle (two Euclidean caps sharing a horizon joined by Lorentzian adiabatic segments) and states that its on-shell action reproduces both the saddle-point conditions and value of this C_max. The semiclassical area comparator and two-point function representations are derived from this equivalence rather than presupposed. No self-citation load-bearing steps, fitted inputs renamed as predictions, or definitional reductions (e.g., geometry defined to match C_max by fiat) appear; the central claim is an independent gravitational computation of a spectral optimum.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Central claims rest on the new definition of BW distance via optimal transport and the assumption that the comonotone coupling dominates; no free parameters fitted to data are mentioned.

axioms (2)
  • domain assumption Optimal transport theory applies directly to Boltzmann-weighted energy spectra of quantum theories
    Used to define the BW distance as the W2 distance between spectra.
  • ad hoc to paper The comonotone coupling (pairing by rank) achieves the optimum C_max
    Identified as the dominant wormhole without independent justification in the abstract.
invented entities (2)
  • Boltzmann-Wasserstein distance no independent evidence
    purpose: Metric on the space of quantum theories
    Newly defined via optimal transport of spectra.
  • Schwinger-Keldysh wormhole saddle no independent evidence
    purpose: Classical geometry computing C_max
    Constructed to reproduce the spectral saddle; Lorentzian segments claimed essential.

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discussion (0)

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Reference graph

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