Notes on Wasserstein distance and wormholes

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.

v2: Two errors corrected: the area comparator is evaluated at equal energy, not equal temperature; vanishing one-point functions at finite volume do not follow from conformal invariance (now an explicit assumption). Added: resummed crossover form of the comparator, shifted-saddle TTbar expansion, first-order relation of adiabatic evolution to the comonotone unitary. Main results unchanged