Time-reparameterisation invariant quantum evolution law: the lack of absolute time does not imply a stationary global state

Denis Bouvy; Ognyan Oreshkov

REVIEW 3 major objections 5 minor 20 references

Time-reparameterisation invariant quantum evolution law: the lack of absolute time does not imply a stationary global state

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Absence of absolute time need not freeze a quantum state: the paper constructs a time-reparameterisation invariant evolution law whose solutions are exactly the reparameterised Schrödinger solutions.

desk verdict A mathematically clean underdetermined evolution law shows that lacking absolute time need not force a stationary global state, provided you grant an ordered manifold of instants; the operational argument for full time-reparameterisation is the weakest link. read the letter →

arxiv 2608.08065 v1 pith:E4SMITPL submitted 2026-08-08 quant-ph gr-qc

Ognyan Oreshkov , Denis Bouvy This is my paper · ORCID

classification quant-phgr-qc

keywords time-reparameterisationinvarianceunderdeterminedquantumevolutionstationary-stateproblemPage-WoottersformalismHamiltonianconstraintrelationaltimeclocksgravity

topics Quantum Gravity

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 argues that a quantum system described without any external time parameter need not be frozen in a stationary state. The authors construct a time-reparameterisation invariant evolution law that, for a given initial condition, predicts the same trajectory through Hilbert space as the Schrödinger equation but leaves the speed of traversal underdetermined. Every solution of the law is a time-reparameterised Schrödinger solution, and relative to an internal clock the standard Schrödinger equation is recovered, while the global state remains nonstationary. The intended consequence is to reopen the question of how time appears in quantum gravity: the Page-Wootters formalism, an evolution-without-evolution mechanism, and the Hamiltonian-constraint picture owe their stationary global states to assumptions that, the paper argues, go beyond the mere absence of absolute time.

What carries the argument

The carrying object is the normalised-tangent evolution equation, obtained by dividing both sides of the Schrödinger equation by the norms of the two sides. This removes the magnitude of the velocity vector, leaving only the direction of motion, which is the same as dropping the speed along the trajectory. The companion identity $dt/d\tau = \|\partial_\tau|\psi\rangle\|/\|\hat H|\psi\rangle\|$ converts any solution back into Schrödinger gauge, and this is how the paper shows that the solution set consists exactly of reparameterised Schrödinger solutions. The clock-system model supplies the relational content: with the Page-Wootters form of the Hamiltonian, the system's state conditioned on clock reading $|t\rangle_C$ obeys the usual Schrödinger equation in $t$.

What would settle it

One concrete way to test the central claim is to prove a no-go theorem: if any time-reparameterisation invariant evolution law respecting standard sequential-measurement rules must force every global state to be stationary, the construction fails. Within the paper's own clock model, the falsifying observation would be that sequential measurements of the clock in its $|t\rangle$ basis, at unknown but ordered instants, produce the correlations of a stationary state—uniformly random, independent readings with no persistent drift—rather than the predicted correlations in which clock differences follow the Schrödinger time elapsed between instants.

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Extended reading notes

Core claim

The central claim is that time-reparameterisation invariance in quantum theory can be realised by an underdetermined differential equation rather than by a constraint that annihilates the state. For pure states the law is $\partial_\tau|\psi\rangle/\|\partial_\tau|\psi\rangle\| = -i\hat H|\psi\rangle/\|\hat H|\psi\rangle\|$ when $\hat H|\psi\rangle\neq 0$, and $\partial_\tau|\psi\rangle=0$ when $\hat H|\psi\rangle=0$; a density-matrix analogue with the trace norm replaces the commutator $[\hat H,\hat\rho]$. The equation fixes only the direction of change, so it is invariant under orientation-preserving reparameterisations of $\tau$. The paper shows that any solution is a reparameterised solution of the Schrödinger equation and vice versa. In a clock-system model with the Page-Wootters form of Hamiltonian, the joint state is $|t(\tau)\rangle_C\,e^{-i\hat H_S t(\tau)}|\psi_0\rangle_S$ for an arbitrary monotone $t(\tau)$, so the system follows Schrödinger evolution relative to the clock reading while the global state moves.

Load-bearing premise

The argument assumes that there is an ordered continuum of instants whose order is operationally meaningful, and that instantaneous states can be identified as belonging to specific instants, even though no time parameter is attached to those instants; if the ordering itself requires time structure, the predicted trajectory has no physical content.

Editorial extensions

If this is right

A closed system with no external clock can have a global state that traverses a Schrödinger trajectory at a rate the theory leaves unspecified; stationarity is not forced by time-reparameterisation invariance alone.
When a clock with a Page-Wootters form of Hamiltonian is available, the subsystem obeys the ordinary Schrödinger equation relative to the clock's reading, so standard quantum predictions are reproduced.
Hamiltonians that differ by a positive, possibly time-dependent scalar factor are indistinguishable under the new law, making the overall energy scale a gauge freedom.
Sequential measurements performed at ordered but unparametrised instants can reveal nontrivial global evolution, so the inability to observe a single time parameter does not by itself hide all evolution.
The underdetermined equations offer an alternative to imposing the Hamiltonian constraint in canonical quantum gravity, changing what is meant by a physical state of the universe.

Reading between the lines

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

The paper leaves a Heisenberg-picture formulation to future work; a natural extension would define reparameterisation equivalence classes of observable families rather than only of state trajectories.
The single-measurement versus sequential-measurement gap could be probed in the laboratory: with a high-quality clock of the Page-Wootters form, sequential clock readouts should show stable time-difference correlations even though each individual reading is random.
Applied to cosmology, the construction suggests replacing the Hamiltonian-constraint equation with an underdetermined law, which would yield a global history rather than a stationary wavefunction of the universe; this changes what initial-condition predictions look like without adding an external time.
The same reasoning may generalise to other symmetries: whenever a group acts on ordered families of events rather than on individual states, the invariant object can be a nontrivial trajectory rather than a fixed point.

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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

Colleague,

Read the Oreshkov-Bouvy paper. The headline: it is a clean existence proof that a time-reparameterisation-invariant quantum law need not force a stationary global state, and the core math is solid. The genuinely new thing is Eq. (10) (and its density-matrix analogue (13)): normalize the Schrödinger equation so the speed along the trajectory drops out. The proof that every solution is a re-time-parameterised Schrödinger solution is complete and easy to verify. The paper does what it claims and is transparent about what it assumes.

The main soft spot is also the main premise: the “ordered continuum of instants” in Sec. II. The law is invariant only under orientation-preserving diffeomorphisms, not arbitrary relabellings; the order of instants is carried in as physical content. The authors say this explicitly, so it is not a hidden assumption, but it means the non-stationary global state is a consequence of absent time plus an ordered event structure, not of absent time alone. If you do not grant that structure, the contrast with Page–Wootters weakens. The stress-test note makes this point accurately, and the paper would be improved by engaging with it more directly.

A second soft spot: the Appendix’s operational argument is fully worked for time translations, where the interval Δt between measurements is invariant. For full reparameterisation, the duration between instants is gauge and the joint probabilities are not fixed by the law. The final paragraph says “the situation is similar,” but that is a sketch, not an argument. The claimed reconstruction of the trajectory from sequential measurements under full reparameterisation needs more work. This does not kill the central construction, but it limits the operational story.

The critique of Dirac quantisation is suggestive but not a theorem: the authors argue that the Hamiltonian constraint should not be treated as a gauge generator when the time parameter is itself gauge. That is a legitimate challenge, but it is a position, not a derivation. The paper is best read as an existence proof and a challenge to a common inference, not as a full replacement formalism.

Who this is for: anyone working on the problem of time in quantum gravity, Page–Wootters, or relational quantum mechanics. It is short, readable, and honest. It deserves serious refereeing; the soft spots are caveats, not load-bearing flaws. I would engage with it and would cite it in a foundations context.

Referee Report

3 major / 5 minor

Summary. The paper constructs a time-reparameterisation-invariant quantum evolution law, Eqs. (10a)-(10b), by equating the normalized tangent vector of the state trajectory with the normalized action of -iH. It proves that every solution of the new equation is a time-reparameterised solution of the Schrödinger equation, via the reparameterisation defined in Eq. (11), and that the solution set is exactly the set of such reparameterised solutions. It also gives a trace-norm analogue for density matrices, Eq. (13), sketches a relational-clock recovery, Eqs. (14)-(15), and uses these results to argue that the absence of an external time parameter does not force a stationary global state, in contrast to the Page-Wootters formalism and Dirac constraint quantisation.

Significance. The mathematical core is sound and concise: the solution-set theorem is proved rather than assumed, norm preservation follows immediately from Hermiticity, and the comparison with the Page-Wootters stationary-state picture is clearly drawn. If the ordered-instant assumption is granted, the paper provides a genuine counterexample to the claim that reparameterisation invariance plus lack of external time implies a stationary global state. The paper is transparent about its main premise, which is a strength, and the proposed underdetermined equations are a useful alternative formulation for further work on time in quantum theory.

major comments (3)

[Sec. II, paragraph after Eq. (10); Appendix] The claim that the unparameterised trajectory can be reconstructed by sequential measurements whose order is known is not established for the full time-reparameterisation group. The Appendix treats only time translations, and its argument uses the invariance of the duration Delta t between two events; under the monotone diffeomorphisms considered for Eq. (10), the corresponding Schrödinger-time difference t(tau2)-t(tau1) is pure gauge, so the joint probabilities of two arbitrary instants are not fixed by the law. Please either prove an invariant extension, for example by conditioning on physical clock readings as in Eq. (15), or restrict the operational claim to a smaller symmetry group and state that restriction explicitly.
[Sec. II, first two paragraphs and footnote 9] The central conclusion that a timeless theory can have a non-stationary global state is conditional on the assumption that an ordered continuum of instantaneous states is operationally accessible. The paper states this premise clearly, but it does not provide a criterion that distinguishes this premise from having a time structure, and the conclusion collapses if only relational data among subsystems are available. I ask that the scope be tightened: the result shows consistency of non-stationarity with reparameterisation invariance under that premise, not that the premise follows from the absence of an external clock.
[Eqs. (14)-(15)] The relational-clock demonstration uses ideal continuous orthonormal clock states |t>_C that are not normalizable in a Hilbert space, as footnote 2 acknowledges. Since the solution-set theorem is formulated for Hilbert-space states, Eq. (15) should be presented either as a formal limiting argument or replaced by an explicit approximate-clock model with normalized states whose evolution is shown to satisfy Eq. (10). This is important because the claim that the global state is non-stationary in this model should be demonstrated for physically realizable states.

minor comments (5)

[Abstract and Sec. I] The abstract and Sec. I contain the typo 'global sate' instead of 'global state'; the conclusion likewise has 'invarinace' instead of 'invariance'.
[Sec. II, after Eq. (10)] The statement that Hamiltonians differing by an overall, possibly time-dependent, factor are indistinguishable should specify a positive real factor, since a negative time-dependent factor changes the sign of the normalized right-hand side of Eq. (10a).
[Eq. (13)] The density-matrix equation is asserted to have 'analogous properties' without an explicit proof of the solution-set statement; a short proof following the argument of Eqs. (11)-(12) would be helpful for the reader.
[References] Reference [12] has a typographical artifact ('187 (2007)/') and should be corrected to the standard citation of Bartlett, Rudolph, and Spekkens.
[Eq. (14)] Eq. (14) uses |t=0> (x) |psi_0> without subscripts; writing |t=0>_C (x) |psi_0>_S would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the evolution law in Eq. (10) is presented as an explicit construction, its solution-set theorem is proved in the text, and the main conclusion is conditional on a stated ordered-instant premise; the only self-citations are non-essential.

full rationale

The paper does not derive Eq. (10) from the absence of absolute time and then recover non-stationarity; it explicitly constructs the law so that its solutions are all time-reparameterised Schrödinger solutions. Section II states the construction criterion ('It is easy to construct an equation with these properties...') and then proves, via Eq. (11), that every solution is a reparameterised Schrödinger trajectory. This is a transparent definition of a proposed law, not a fitted input masquerading as a prediction, so it does not constitute circularity under the stated rules. The central philosophical conclusion ('the lack of absolute time does not imply a stationary global state') is conditional on the explicitly acknowledged premise of an ordered, orientable continuum of instants (Section II, first two paragraphs and footnote 9). That premise is an assumption rather than a circular reduction, and the paper flags it as such. The critique of Page-Wootters and Dirac quantisation relies on independent arguments about sequential measurements and the gauge role of the Hamiltonian, not on the constructed law alone. The Appendix's sequential-measurement argument is developed for time translations, where Δt is invariant; its extension to full time reparameterisation is asserted rather than proved, but that is a support gap, not circularity. Self-citations ([13], [18]) are minor and not load-bearing: [13] is immediately qualified as not directly applicable to time translation, and [18] is a pointer to future work. No fitted parameter is renamed as a prediction, and no external benchmark is involved.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. Its load-bearing assumptions are the standard quantum formalism, the identification of time reparameterisation as a gauge symmetry, the existence of an orderable continuum of instants, the Page-Wootters form of the Hamiltonian for the clock model, and the operational description of measurements without a time reference.

assumptions (5)

domain assumption Standard quantum postulates: states are vectors in Hilbert space, observables are Hermitian, the Born rule holds, and ordinary Schrödinger evolution is valid in the Schrödinger gauge.
Used throughout; the paper assumes the standard probabilistic interpretation for the new law (Sec. II, 'The probabilistic interpretation...') and uses the Schrödinger equation as the reference dynamics.
domain assumption Time reparameterisation is a gauge symmetry and there is no external time parameter.
Central premise stated in the Introduction and Sec. II; without it the construction is irrelevant.
domain assumption There exists an ordered continuum of instantaneous states or events, identifiable and orderable without an external time label.
Sec. II, first two paragraphs and footnote 9; this enables the trajectory in state space to be physically meaningful.
domain assumption The Page-Wootters form of the Hamiltonian, H_CS = H_C tensor 1 + 1 tensor H_S, with an approximate ideal clock basis |t>_C, is used for the relational recovery.
Sec. II near Eq. (14); follows Page-Wootters. The ideal |t> basis is not normalizable, so physical clocks are only approximate (footnote 2).
domain assumption Measurements can be performed at unknown instants with the usual quantum measurement update, and the average over time translations is obtained as the T to infinity limit of uniform averages over finite intervals.
Appendix; supports the claim that single measurements look Page-Wootters-like while sequential measurements reveal nonstationarity. The limit procedure on the noncompact time-translation group is a modeling choice.

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Cite this review

Pith. "Pith review of Time-reparameterisation invariant quantum evolution law: the lack of absolute time does not imply a stationary global state." pith.science (2026). https://pith.science/paper/E4SMITPL

@misc{pith2026260808065,
  author       = {Pith},
  title        = {Pith review of: Time-reparameterisation invariant quantum evolution law: the lack of absolute time does not imply a stationary global state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E4SMITPL}},
  note         = {Machine review of arXiv:2608.08065}
}

read the original abstract

We challenge the common belief that if there is no absolute time parameter in physics, a quantum system described without reference to an external clock can be assumed to be in a stationary state of its Hamiltonian. We present a time-reparameterisation invariant quantum evolution law, which for a given initial condition predicts the same trajectory in state space as the Schr\"odinger equation, except that for nontrivial trajectories it does not predict the speed at which the trajectory is traversed. The solutions of this evolution law are all time-reparameterised solutions of the Schr\"odinger equation. We show how the predictions of the Schr\"odinger equation are recovered relative to an internal clock in this framework. In contrast to the Page-Wootters formalism or Dirac's quantisation of the Hamiltonian constraint, here the global sate is not stationary. We discuss the assumptions leading to the common conclusion that the state can be taken stationary and suggest that they need to be revisited.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

D. N. Page and W. K. Wootters, Evolution without evolution: Dynamics described by stationary observables,Phys. Rev. D. 27, 2885 (1983). 6

work page 1983

[2] [2]

There is no Hamiltonian bounded from below that can generate this ideal translation for a continuous orthonormal basis|t⟩[3]. In practice, a realistic clock can only keep track of time approxi- mately since its normalised, physical states|s(t)⟩,⟨s(t)|s(t)⟩=1, ∀t, cannot be strictly orthogonal to each other for differentt, ⟨s(t)|s(t′)⟩,0,∀t,t ′, and their ...

work page

[3] [3]

Pauli, General Principles of Quantum Mechanics (Springer- V erlag, 1980)

W. Pauli, General Principles of Quantum Mechanics (Springer- V erlag, 1980). Original article published in German in 1933

work page 1980

[4] [4]

P. A. M. Dirac, Lectures on Quantum Mechanics, published by Belfer Graduate School of Science, Yeshiva University, 1964; (Dover Publications, 2001)

work page 1964

[5] [5]

B. S. DeWitt, Quantum Theory of Gravity. I. The Canonical Theory,Physical Review160, 1113 (1967)

work page 1967

[6] [6]

Henneaux and C

M. Henneaux and C. Teitelboim, Quantization of Gauge Sys- tems, (Princeton University Press, 1992)

work page 1992

[7] [7]

Giovanetti, S

V . Giovanetti, S. Lloyd, and L. Maccone, Quantum Time,Phys. Rev. D92, 045033 (2015)

work page 2015

[8] [8]

Colosi and C

D. Colosi and C. Rovelli, A simple background-independent hamiltonian quantum model,Phys.Rev. D68, 104008 (2003)

work page 2003