York time in JT gravity

Onkar Parrikar; Sunil Kumar Sake

REVIEW 2 major objections 3 minor 3 cited by

York time in JT gravity

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In JT gravity, the Hartle-Hawking wavefunction satisfies a Schrödinger equation in York time, generated by a manifestly Hermitian York Hamiltonian.

desk verdict Real new result for York time in JT gravity, but the printed York Hamiltonian and WDW equation are only the large-length parts; the exact generator has an extra Hermitian dilation term. read the letter →

arxiv 2505.19231 v2 pith:DQ4RXACB submitted 2025-05-25 hep-th

Onkar Parrikar , Sunil Kumar Sake This is my paper

classification hep-th

keywords YorktimeJTgravityHartle-HawkingwavefunctionWheeler-DeWittequationlengthbasisunitaryevolutionquantumAdS2

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how a notion of time can be internal to a quantum theory of gravity rather than supplied by an external boundary clock. Working in Jackiw-Teitelboim (JT) gravity on two-dimensional anti-de Sitter space, it foliates the spacetime by slices of constant trace of the extrinsic curvature $k$, a geometric quantity known as York time, and computes the Hartle-Hawking wavefunction with fixed renormalized length on such a slice. Using both canonical quantization and the JT path integral, the paper shows this wavefunction satisfies a Schrödinger equation in $k$, with a York Hamiltonian that is manifestly Hermitian after the length basis is given a $k$-dependent normalization. The paper interprets the $k$-dependence as a unitary rotation of the fixed-length basis states, not as physical time evolution of the dual boundary state. A correct result would provide an explicit, operator-ordering-free bulk clock Hamiltonian in a solvable quantum gravity model, recasting the Wheeler-DeWitt constraint as unitary evolution.

What carries the argument

The central object is the York-time slice basis $|\ell,k\rangle$ of JT gravity: states labeled by renormalized length $\ell$ on a slice whose trace of the extrinsic curvature is fixed to $k$. The argument is carried by the large-$\ell$ identity $\partial_k \tilde d_\infty \simeq \frac{k\ell}{1+k^2}\partial_\ell \tilde d_\infty$ (eq. 4.76), which converts the $k$-derivative of the Bessel-function wavefunction into a length derivative and turns the Wheeler-DeWitt constraint into a first-order Schrödinger equation. The prefactor $(1+k^2)^{1/4}$ that defines the normalized states removes the $k$-dependent norm of the naive basis, turning the non-Hermitian $\ell P$ term into the symmetric combination $\frac12(\ell P+P\ell)$ and yielding the manifestly Hermitian $H_{\mathrm{York}}$. The same Hamiltonian is recovered from the zero-mode Wheeler-DeWitt constraint including corner terms, checked to leading order in the large-$\ell$ and small-$\delta k$ expansion.

What would settle it

Evaluate the exact expression (4.71) for $\langle \ell,k|E\rangle$ at finite $\ell$ and compare the numerical $k$-derivative with the action of $H_{\mathrm{York}}$ from (4.84); any discrepancy beyond $O(1/\ell)$ would show the Schrödinger form is only approximate. A direct alternative is to compute the $O(\ell^{-1})$ correction to (4.76) and check whether it introduces non-Hermitian terms into the Hamiltonian.

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Extended reading notes

Core claim

The central claim is that the generalized Hartle-Hawking wavefunction $\langle \ell,k|E\rangle$, the overlap between an energy eigenstate of the boundary Hamiltonian and a bulk state of fixed renormalized length $\ell$ on a slice of constant extrinsic curvature $k$, obeys $-i\,\partial_k \langle \ell,k|E\rangle = \langle \ell,k|H_{\mathrm{York}}(k)|E\rangle$. With the normalized states $|\ell,k\rangle_{\!\sim}=(1+k^2)^{1/4}|\ell,k\rangle$, the York Hamiltonian is $H_{\mathrm{York}}(k)=\frac12\frac{k}{1+k^2}(\ell P+P\ell)+\frac{2q}{\sqrt{1+k^2}}$, where $P=-i\partial_\ell$ and $q$ is the boundary dilaton value. This equation is equivalent to the Wheeler-DeWitt constraint, the zero-mode Hamiltonian constraint of canonical quantum gravity, and the paper shows that the same Hamiltonian emerges from canonical quantization and from the path integral. The $k$-dependence is then interpreted as a unitary transformation of the fixed-length basis states rather than as evolution of the physical state in the dual boundary theory.

Load-bearing premise

The calculation relies on the York slice having large renormalized length $\ell$, so that the derivative relation $\partial_k \tilde d_\infty \simeq \frac{k\ell}{1+k^2}\partial_\ell \tilde d_\infty$ is valid; the paper does not quantify the finite-length corrections, and the Wheeler-DeWitt check is likewise only at leading order in $\delta k$ and large $\ell$.

Editorial extensions

If this is right

The Wheeler-DeWitt constraint of JT gravity can be rewritten as a unitary Schrödinger equation in York time, with the operator-ordering ambiguity fixed by the normalized length basis.
The York Hamiltonian is proportional to the squeezing operator $\ell P+P\ell$, so York time evolution acts as a unitary rotation of the length basis rather than a change of the physical boundary state.
The canonical and path-integral derivations give the same generalized Hartle-Hawking wavefunction, and gluing two such wavefunctions along the York boundary reproduces the JT partition function.
Because the normalized length states have a $k$-independent norm, York-time evolution is norm-preserving, making the bulk clock a genuine unitary generator.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to compute the next-order correction in $1/\ell$ to the derivative identity (4.76); if finite-length terms break the Hermiticity of $H_{\mathrm{York}}$, the Schrödinger form would be a large-length emergence rather than an exact statement.
If the squeezing form of $H_{\mathrm{York}}$ survives in the dual matrix model, York evolution might act as a Clifford or squeezing unitary on the discrete basis of length/chord states in the dual theory, a microscopic proposal the paper raises but does not establish.
The paper's reading of York time as a basis rotation suggests that, in higher-dimensional generalizations, interior time evolution toward a black hole singularity could be re-expressed as a unitary change of slicing basis rather than as evolution of the quantum state.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a genuine and mostly solid calculation, with one underplayed approximation. The k≠0 Hartle-Hawking wavefunction, the two independent derivations, and the identification of a Hermitian generator of York time evolution are all real. But eq. (4.84) is not the exact York Hamiltonian; it is the large-ℓ part. From the d̃∞ in (4.71), ∂k d̃∞/∂ℓ d̃∞ = kℓ/(1+k²) − 2k/(1+k²)^{3/2}. The paper keeps only the first term, so the exact generator contains an extra −2k/(1+k²)^{3/2} P before symmetrization, and after the (1+k²)^{1/4} rescaling the Hermitian form is ½ k/(1+k²)(ℓP+Pℓ) − 2k/(1+k²)^{3/2} P + 2q/√(1+k²). That is still manifestly Hermitian, so the qualitative claim survives. The extra term also changes the unitary map from k=0: ℓ gets an affine shift as well as a rescaling. The WDW constraint (4.79) misses the same term; Appendix C.2 checks only at O(δk)^0 and large ℓ, so it cannot detect this.

What the paper does well: the wavefunction is computed twice, canonically and via the path integral, with agreement after the stated state-normalization identification; the saddle point matches the classical on-shell action; the gluing reproduces the JT thermal partition function; the k=0 limit reduces to known results. The operator-ordering discussion is honest, and the basis-rotation interpretation is clean. The citation of prior work looks accurate, and the nonzero-k result is new relative to Yang, Harlow–Jafferis, and Iliesiu et al.

Soft spots, in proportion: the large-ℓ approximation is not flagged in (4.84) or (4.79) as an approximation, which matters for a paper presenting “the” York Hamiltonian. Also, the exact relation between path-integral and canonical length states is fixed by demanding the TFD norm reproduce the thermal partition function—reasonable, but a choice. Neither is load-bearing for the overall picture, but both need to be stated honestly. If the authors either restrict the claim to large ℓ or add the exact dilation term, the paper is in good shape.

This deserves a serious referee. I would send it to review and ask the referee to push for the corrected Hamiltonian or an explicit statement that (4.84) is the leading large-ℓ term. I would also bring it to reading group.

Referee Report

2 major / 3 minor

Summary. The paper studies York time (constant extrinsic-curvature slicing) in JT gravity. It defines a generalized Hartle-Hawking wavefunction with fixed extrinsic curvature k on a Cauchy slice, and computes it by two methods: canonical quantization (eq. 4.18) and the boundary-particle path integral (eq. 4.51), identifying a state-normalization relation (eq. 4.56). The central technical claim is that this wavefunction satisfies a Schrödinger equation in York time k, with a manifestly Hermitian York Hamiltonian given in eq. (4.84) after a k-dependent redefinition of the length basis. The paper further shows that the saddle point of the quantum wavefunction reproduces the classical on-shell action, and presents consistency checks involving gluing of wavefunctions in Appendix B.3. The interpretation is that the k-dependence is a unitary rotation of the fixed-length basis states, not physical time evolution of the boundary state.

Significance. If the main claim holds, the paper provides an explicit Hermitian generator for an internal notion of time in a solvable quantum-gravity model, which is valuable for the problem of time and for de Sitter holography. The paper has notable strengths: the wavefunction is derived twice and the two results agree after a stated normalization identification; the classical saddle point is correctly reproduced; and the overlap/bootstrap checks in Appendix B.3 are nontrivial. The main quantitative result, however, is not exact as printed: eq. (4.84) omits a Hermitian correction term. Since the correction preserves the qualitative claim—Hermiticity and the Schrödinger form survive—the central message is defensible after a local but important correction.

major comments (2)

[4.3 (Eqs. 4.75, 4.76, 4.84)] The York Hamiltonian is not exact as printed. From d̃∞ = (1/(ϕ_B√(1+k²))) exp(ℓ√(1+k²)/2), the exact ratio is ∂_k d̃∞ / ∂_ℓ d̃∞ = kℓ/(1+k²) − 2k/(1+k²)^{3/2}. Substituting into eq. (4.75) and symmetrizing gives H_exact = (1/2) k/(1+k²) (ℓP + Pℓ) − 2k/(1+k²)^{3/2} P + 2q/√(1+k²), not eq. (4.84). The missing cP term is not a harmless subleading correction for the unitary map in eq. (4.85): it produces the affine shift ℓ → (ℓ + ln(1+k²))/√(1+k²), whereas the Hamiltonian in eq. (4.84) alone would generate only a scaling. The qualitative claim survives because the omitted term is manifestly Hermitian, but eqs. (4.78), (4.79), (4.84), and (4.85) must be corrected or explicitly labeled as a large-ℓ approximation.
[Appendix C.2 (after Eq. C.46)] The verification of the Wheeler-DeWitt equation is explicitly performed only at O(δk)^0 and at large ℓ. This is acknowledged in the manuscript, but it means the check cannot detect the missing −2k/(1+k²)^{3/2}P term discussed above. The main text presents eq. (4.79) as the WDW equation without carrying the same caveat. Please either verify the corrected WDW equation with the full k-dependence or clearly restrict the claim in the main text to the regime where the approximation is controlled.

minor comments (3)

[4.3 (Eq. 4.75)] The approximate sign in eq. (4.75) is used only once, but the subsequent equalities in eqs. (4.77)–(4.84) are written as exact statements; please propagate the approximation status explicitly or replace with the exact expression.
[4.3 (Eq. 4.84 and surrounding text)] The notation for the normalized Hamiltonian is inconsistent: the text refers to it as "eHYork" (presumably a tilde), but the displayed equation omits the tilde. Please harmonize the notation.
[4.3 (Eq. 4.85)] If eq. (4.84) is corrected to include the cP term, the unitary transformation in eq. (4.85) should be re-derived; the present version gives only the scaling part of the relation between |ℓ,0> and |ℓ,k>.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the York Hamiltonian is derived by differentiating an independently computed wavefunction rather than imposed by construction.

full rationale

The derivation chain is self-contained. The generalized Hartle-Hawking wavefunction in eq. (4.71) is obtained from the JT path integral/boundary-particle formalism in Section 4.2, with the Hamiltonian treatment in Section 4.1 providing an independent cross-check; no property of the York Hamiltonian is assumed in obtaining this wavefunction. The Schrodinger equation (4.77) is then derived by explicitly differentiating the known k-dependence of the wavefunction, eqs. (4.72)-(4.76), rather than by fitting an operator to force the result. The only uncontrolled step is the stated large-ell approximation (4.76), which is a correctness/validation caveat rather than a circular reduction: the exact derivative contains an additional -2k/(1+k^2)^{3/2} P term, so eq. (4.84) is approximate, but this does not make the claim equivalent to its inputs. The manifestly Hermitian form (4.84) follows from the explicitly stated normalization choice (4.81) and is checked against the independently derived WDW constraint in Appendix C; that check is only leading-order/large-ell, again a validation gap rather than circularity. The self-citations in the paper ([5], [23], [32], [51]) are contextual, speculative, or non-load-bearing, and the core derivation relies on standard external machinery of JT gravity. No circular step can be exhibited.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

All dynamical parameters in the computations are boundary data (phi_b, beta, k, L) or the fixed JT coupling; nothing is fitted to an external dataset. The hand-chosen items are the corner counterterm coefficient -pi/2 in eq. (2.22), added so that overlaps of wavefunctions close correctly in Appendix B.3, and the state normalization factor (1+k^2)^{1/4} in eq. (4.81), chosen to give the length basis a k-independent norm. Both are explicit conventions that the paper states; they enter the Hermiticity interpretation but not the physical content of the wavefunction. The background axioms are the standard JT action, the boundary particle reformulation of Yang, the uniqueness of constant-K slices at fixed boundary time, the k -> ik continuation, and the large-q (large dilaton) limit.

free parameters (1)

Corner counterterm coefficient (2.22) = -pi/2 per corner
Introduced by hand so that the net corner contribution vanishes when computing overlaps of wavefunctions (Appendix B.3); it does not enter the central wavefunction in Sections 3 and 4.

assumptions (7)

domain assumption JT gravity action (2.1) with GHY term on the asymptotic boundary, York terms on the K-boundary, and Hayward corner terms (2.21) gives a consistent variational principle.
Boundary term analysis in Section 2.2 and Appendix A; standard within the JT literature but assumed without a fully general proof.
domain assumption The JT gravity path integral is equivalent to a non-relativistic charged particle on H2 (boundary particle formalism).
Invoked in Section 4.2 to evaluate the wavefunction via the propagator (4.29); transferred from Yang [17] and Kitaev-Suh [39].
domain assumption For any boundary time slice there exists a unique spacelike constant-K slice anchored at the boundary.
Asserted in Section 4.1 before eq. (4.8); demonstrated classically by the explicit solution (3.9), used to define the length variable and the |ell,k> states.
domain assumption Euclidean path integral results are connected to Lorentzian canonical results by the analytic continuation k -> ik.
Used in Section 4.2 (comparison after eq. 4.53) and Appendix B.2; the continuation of the phase e^{-2q arcsin(k)} to e^{2iq sinh^{-1}(k)} is assumed rather than derived.
domain assumption The large-q limit (q = phi_B ~ phi_b/epsilon) identifies the particle boundary with the asymptotic AdS boundary.
Standard holographic renormalization limit used in Section 4.2 to obtain eq. (4.51); the subleading corrections are not controlled.
standard math The physical phase space of JT gravity is two-dimensional, with symplectic form d ell_ren ^ dP.
Reviewed in Section 2.1; used in Section 4.1 to write the ADM Hamiltonian (4.15) in length variables.
standard math Bessel function orthogonality identities (4.55) and (B.36) hold as distributional identities.
Used to fix the completeness relations (4.54) and (B.37) and to derive the delta-function structure of the two-York-boundary amplitude.

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Pith. "Pith review of York time in JT gravity." pith.science (2026). https://pith.science/paper/DQ4RXACB

@misc{pith2026250519231,
  author       = {Pith},
  title        = {Pith review of: York time in JT gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQ4RXACB}},
  note         = {Machine review of arXiv:2505.19231}
}

abstract

The notion of time in general relativity must arise from an internal clock, i.e., a degree of freedom in the gravitational theory internal to the system that can serve the role of a physical clock. One such internal notion of time is the York time, corresponding to constant extrinsic curvature slicing of spacetime. We study the Hartle-Hawking wavefunction of asymptotically $AdS_2$ JT gravity as a function of York time. Using both canonical quantization and the JT gravity path integral, we explicitly calculate this wavefunction and show that it satisfies a Schrodinger equation with respect to York time. We find the corresponding York Hamiltonian, which turns out to be manifestly Hermitian. Our analysis cleanly avoids operator ordering ambiguities. The dependence of the wavefunction on York time should be thought of as emerging from a unitary transformation of the gravitational length basis states, and not from a physical time evolution of the state in the dual boundary theory.

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