{"id":"1b54e403-1324-4e9e-8c42-8e53479ee36d","arxiv_id":"2608.08065","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A time-reparameterisation-invariant quantum law is constructed whose solutions are exactly the re-timed Schrödinger trajectories, making a non-stationary global state consistent with the absence of absolute time.","lead":"The paper introduces a quantum evolution law that is invariant under re-timing, so the global state can change even when there is no absolute time parameter. It argues that this undercuts the common assumption that a clockless universe must sit in a stationary Hamiltonian eigenstate, with implications for the problem of time in quantum gravity.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-stationary global state is a gauge artifact unless ordered instants carry physical content; the Appendix's sequential-measurement argument works for time translations but not for full time reparameterisation.","rationale":"The mathematical core of the paper—that Eq. (10) is time-reparameterisation invariant and its solutions are exactly the reparameterised Schrodinger trajectories—is sound and I do not dispute it. The reader correctly identified the ordered-continuum assumption as the weakest point. My stress-test goes further: even granting the ordered continuum, the Appendix's operational distinction via sequential measurements is only demonstrated for the time-translation subgroup, where durations are invariant. Under the full reparameterisation group, durations between instants are pure gauge, so the joint probabilities that would distinguish a non-stationary global state from a stationary one are not predicted by the theory. The remaining invariant content is the orbit in state space, which is also encoded in the Page-Wootters stationary state. This makes the strongest claim physically weaker than the paper suggests, but it does not invalidate the paper's constructive contribution as a counterexample to the inference from 'no absolute time parameter' to 'necessarily stationary global state' under the explicitly stated premise of an ordered continuum. The authors are transparent about this premise, and the reader already regarded it as a significant caveat. Therefore the ACCEPT verdict with moderate confidence remains appropriate; no change is needed.","tokens_in":11442,"tokens_out":10772,"duration_ms":126134,"concrete_test":"Consider H = omega sigma_z on a qubit, initial state |+x>. Under Eqs. (10), pick two instants tau1 < tau2. For any Delta > 0, choose a smooth monotone t(tau) with t(tau1)=0 and t(tau2)=Delta (e.g., a smoothed interpolation). Compute the joint probability of outcomes for projective measurements sigma_x at tau1 and sigma_z at tau2 for two different values Delta and Delta'. If the joint probability depends on the arbitrary Delta, the proposed law makes no definite prediction for sequential correlations. Then compute the single-time averaged POVM from the Appendix and verify it commutes with H. This would show that, without an additional principle fixing Delta, the claimed non-stationary global state is empirically indistinguishable from the stationary Page-Wootters picture.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that a timeless theory can have a non-stationary global state is not established because it silently presupposes an ordered, differentiable, oriented one-dimensional continuum of instants (Sec. II, first two paragraphs and footnote 9). Eq. (10) is invariant only under orientation-preserving diffeomorphisms of tau, not under arbitrary relabellings; this remaining structure is a temporal scaffolding. The paper takes this order as operationally meaningful without an operational procedure that does not itself rely on a time parameter. The Appendix attempts to supply physical meaning via sequential measurements, but that argument is developed for time translations, where the duration Delta t between events is invariant. For the full time-reparameterisation group considered in Eqs. (10) and (13), the duration t(tau2)-t(tau1) between any two instants is pure gauge: for any desired positive value there exists a smooth monotone reparameterisation with the same initial condition. Consequently, the joint probabilities of sequential measurements are not fixed by the law; only single-time statistics are invariantly defined, and the Appendix shows these reduce to the stationary Page-Wootters form. The orbit itself is exactly the content already encoded in a Page-Wootters stationary state via clock-system correlations. Thus, unless the ordered instants themselves are granted physical reality and accessibility, the non-stationarity of the global state is a representational choice rather than a physical consequence of the absence of absolute time.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11648,"tokens_out":9521,"duration_ms":111283,"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":[{"comment":"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.","section":"Sec. II, paragraph after Eq. (10); Appendix"},{"comment":"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.","section":"Sec. II, first two paragraphs and footnote 9"},{"comment":"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.","section":"Eqs. (14)-(15)"}],"minor_comments":[{"comment":"The abstract and Sec. I contain the typo 'global sate' instead of 'global state'; the conclusion likewise has 'invarinace' instead of 'invariance'.","section":"Abstract and Sec. I"},{"comment":"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).","section":"Sec. II, after Eq. (10)"},{"comment":"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.","section":"Eq. (13)"},{"comment":"Reference [12] has a typographical artifact ('187 (2007)/') and should be corrected to the standard citation of Bartlett, Rudolph, and Spekkens.","section":"References"},{"comment":"Eq. (14) uses |t=0> (x) |psi_0> without subscripts; writing |t=0>_C (x) |psi_0>_S would improve readability.","section":"Eq. (14)"}],"recommendation":"major_revision","confidential_remarks":"The main risk of this paper is not technical correctness but the status of the ordered-instant assumption; the authors are upfront about it. I see no novelty or disclosure concern: the construction is original and related work is cited. The paper would benefit from one revision round addressing the full-reparameterisation operational claim and the ideal-clock gap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead 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.\n\nThe 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.\n\nA 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":12215,"tokens_out":3221,"would_cite":true,"duration_ms":34639,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["time-reparameterisation invariance","underdetermined quantum evolution","stationary-state problem","Page-Wootters formalism","Hamiltonian constraint","relational time","quantum clocks","quantum gravity"],"falsifier":"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.","tokens_in":11193,"feed_emoji":"🕐","tokens_out":12344,"duration_ms":117218,"temperature":0.7,"pith_summary":"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.","feed_headline":"Timeless quantum states can evolve, just not on a schedule","feed_subtitle":"A new evolution law reproduces Schrödinger trajectories without fixing speed, challenging stationary-universe pictures.","key_machinery":"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 $t$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Page-Wootters stationary-observable mechanism whose stationary global state the paper challenges.","marker":"[1]"},{"why":"Defines the Hamiltonian-constraint quantisation procedure that the paper argues is unjustified when the time parameter itself is the gauge variable.","marker":"[4]"},{"why":"Relates the constraint quantisation to the Wheeler-DeWitt equation, the canonical quantum-gravity stationary-state picture the paper contrasts with its own law.","marker":"[5]"},{"why":"Gives the Hamiltonian constraint $p_C+H_S=0$ for an unparameterised system, the concrete constraint-quantisation example whose stationary global state the paper reinterprets.","marker":"[6]"},{"why":"Provides the general solution $|\\psi\\rangle=\\int dt\\,|t\\rangle_C|\\psi(t)\\rangle_S$ used to display the stationary encoding of Schrödinger evolution.","marker":"[7]"},{"why":"Supports the reference-frame claim that losing access to a symmetry makes only invariant observables accessible, an assumption the paper then revisits for time translations and reparameterisations.","marker":"[13]"}],"fun_headline_variants":["Quantum law without absolute time still allows evolution","No absolute time? Quantum state can still evolve, just not on schedule","Direction-only quantum evolution: timelessness doesn't freeze the state","Reparameterisation-invariant law: global state need not be stationary","New law: quantum trajectories survive without an absolute time"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quantum law without absolute time still allows evolution","No absolute time? Quantum state can still evolve, just not on schedule","Direction-only quantum evolution: timelessness doesn't freeze the state","Reparameterisation-invariant law: global state need not be stationary","New law: quantum trajectories survive without an absolute time"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000191,"raw_usage":{"total_tokens":1350,"prompt_tokens":956,"completion_tokens":394,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":310}},"tokens_in":572,"tokens_out":394,"duration_ms":4315,"temperature":1.0,"reasoning_tokens":310,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:28:55.118339+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Page-Wootters stationary-observable mechanism whose stationary global state the paper challenges."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Hamiltonian-constraint quantisation procedure that the paper argues is unjustified when the time parameter itself is the gauge variable."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Relates the constraint quantisation to the Wheeler-DeWitt equation, the canonical quantum-gravity stationary-state picture the paper contrasts with its own law."},{"cited_title":"Henneaux and C","cited_arxiv_id":null,"evidence_quote":"Gives the Hamiltonian constraint $p_C+H_S=0$ for an unparameterised system, the concrete constraint-quantisation example whose stationary global state the paper reinterprets."},{"cited_title":"Giovanetti, S","cited_arxiv_id":null,"evidence_quote":"Provides the general solution $|\\psi\\rangle=\\int dt\\,|t\\rangle_C|\\psi(t)\\rangle_S$ used to display the stationary encoding of Schrödinger evolution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supports the reference-frame claim that losing access to a symmetry makes only invariant observables accessible, an assumption the paper then revisits for time translations and reparameterisations."}],"review_version":1}