Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems

Alessio Belenchia; Esteban Castro-Ruiz; Flaminia Giacomini; \v{C}aslav Brukner

REVIEW 3 major objections 4 minor 8 cited by

Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read When quantum clocks interact gravitationally, whether an event is sharply localised in time is relative to the time reference frame, while the law of time evolution remains covariant.

desk verdict A serious and mostly sound contribution to quantum reference frames; the redshift-division step is the one genuine technical gap. read the letter →

arxiv 1908.10165 v2 pith:XDS7AM33 submitted 2019-08-27 quant-ph

Esteban Castro-Ruiz , Flaminia Giacomini , Alessio Belenchia , Časlav Brukner This is my paper · ORCID

classification quant-ph

keywords quantumclockstimereferenceframestemporallocalisationindefinitespacetimemetricgravitationaldilationswitchcovariantSchrödingerequationtimelessmechanics

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

The paper is trying to establish that the time at which an event happens can be a genuinely frame-dependent property once quantum clocks are allowed to gravitate. It develops an operational framework in which events are quantum operations triggered by clock readings, and time evolution is read off from a history state constrained by a 'timeless' equation. The central result is that even when the gravitational interaction entangles the clocks and makes the spacetime metric indefinite, each clock can still serve as a legitimate time reference frame in which evolution is unitary and events it defines are sharply localised in time. From another clock's frame, the same event can be smeared out in time, and the gravitational quantum switch's description as four spacetime points appears only in one such frame. A reader should care because the paper gives a concrete, calculable way to talk about causal order and time localisation without assuming a fixed background metric, and it makes a specific claim: when clocks gravitate, event localisability is relative.

What carries the argument

The central object is the history state $|\Psi\rangle$ solving a constraint $\hat{C}|\Psi\rangle = 0$, with each clock $I$ described by a time operator $\hat{T}_I$ and a conjugate Hamiltonian $\hat{H}_I$. A time reference frame is a choice of which clock to condition on; in that frame evolution is extracted from the reduced state $\langle \tau_I | \Psi \rangle$. The load-bearing piece is a quantum coordinate transformation, such as $\tau_A = t'_A + \tau_C(1 + \lambda \omega_B)$, that eliminates the redshift operator from the derivative and converts the evolution equation into Schrödinger form while making the event's trigger time a c-number in the defining clock's frame. The same transformation is what generates the frame dependence of event localisation.

What would settle it

Construct a physical state with non-negligible amplitude on clock energy eigenstates for which $1 + \lambda_{AC} \hat{H}_A + \lambda_{BC} \hat{H}_B$ has an eigenvalue crossing zero, for example by making the couplings large relative to the inverse energy scale; then Eq. (10) becomes singular, no unitary evolution in $C$'s frame exists, and the paper's universality claim for covariant unitary evolution would be refuted in that regime.

Watch

Extended reading notes

Core claim

The paper's central claim is that the localisability of events in time is relative to a quantum time reference frame whenever clocks interact gravitationally. In the frame of the clock that defines an event, the operation appears sharply localised and has the standard unitary dilation form; in the frame of another clock, the same event is spread over a time interval whose width is set by the energy uncertainty of the gravitating source. The paper further claims that the Schrödinger equation retains a universal covariant form under changes of time reference frames, and that this covariance holds even when the metric is indefinite. As a consequence, the familiar description of the gravitational quantum switch in terms of four spacetime points is not absolute but is a statement about one particular frame.

Load-bearing premise

The argument treats clock separations as fixed classical numbers and assumes the energies of the non-clock systems are small enough that the redshift denominator $1 + \lambda_{AC} \hat{H}_A + \lambda_{BC} \hat{H}_B$ never has a vanishing eigenvalue; if either assumption fails, the unitary Schrödinger form and the covariance result can break down.

Editorial extensions

If this is right

An event defined by a clock is always sharply localised in that clock's own time reference frame, even when the metric is indefinite.
The same event can be delocalised in another clock's frame, with a spread set by the energy uncertainty of the gravitating source; this relativity is absent when clocks do not interact.
Changing from one clock frame to another preserves the Schrödinger form of the evolution law, so physics is covariant under quantum time reference frame transformations.
The gravitational quantum switch's four-point causal description is frame-dependent; in the frame of one of the parties the events are sharply ordered relative to that party's clock.
A situation with an indefinite metric can be characterised operationally as one in which no time reference frame makes every event sharply localised.

Reading between the lines

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

If clock separations are promoted from classical numbers to quantum operators, the redshift operator becomes operator-valued in a stronger sense, and the frame changes may generate additional entanglement or fail to be unitary; the covariant chain is likely to be modified outside the semiclassical-trajectory regime.
The same covariant-form argument may extend to higher-order gravitational corrections if the constraint is expanded beyond first order in $1/c^2$, though the paper only treats the first-order case.
The framework suggests that a tabletop experiment with a mass in spatial superposition could in principle reveal the predicted frame-dependence through the statistics of clock-ancilla correlations, but the paper stops short of proposing a concrete setup.

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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

Here's my read on the Castro-Ruiz et al. clock paper. It is worth a serious look. The paper does something real: it shows that once quantum clocks interact gravitationally, whether an event is sharply localised in time depends on which clock's frame you use, and that the gravitational quantum switch looks like a superposition of causal orders only from a distant frame. The derivations are explicit and follow the Page-Wootters tradition honestly; I didn't find fitted parameters or hidden circularity. The switch analysis (Eqs. 30-33) is a nice consistency check, and the authors are careful to flag their assumptions.

The main soft spot is exactly what the stress-test flags: the step from Eq. (7) to Eq. (10) divides by the redshift operator s = 1 + λ_AC H_A + λ_BC H_B, and for perfect clocks that operator has zero modes on the physical state space. The stress-test says the inverse is not densely defined; I would phrase it more carefully—the operator is densely defined on states that vanish fast enough near the singular set, but the paper's own Gaussian clock states are not in its domain. So the Schrödinger equation (10) is not a well-defined generator on the full state space the paper works with. The caveat about small energies is not sufficient as written, because a perfect clock with finite time width has energy support on the whole real line. This is a genuine gap, but it's repairable: one can restrict to energy-filtered states, or add a regulator, or work with realistic clocks. It affects the universality claim for Eq. (10), but notice that the event-localisation result (Eqs. 13-14) assumes C is far away, so the denominator is trivial there. The divisibility problem is thus not load-bearing for the paper's central event-relativity result, but it is load-bearing for the stronger covariance/unitarity claim.

Two smaller caveats. First, the headline 'there is always a frame where the event is sharp' is partly definitional, since an event is defined by the clock that triggers it; the substantive claim is the delocalisation in other frames, which is real. Second, the characterisation of the situation as an indefinite metric is interpretive; the paper shows an operational asymmetry, not a metric operator. These are minor in proportion.

Bottom line: this deserves a serious referee. I'd send it out. The main request to the authors should be to close the domain gap in Eq. (10) or state clearly the restricted state space on which it holds. If that can be done, the paper is a strong contribution to the quantum-clocks and quantum-reference-frame literature. I would cite it if working in that area.

Referee Report

3 major / 4 minor

Summary. The paper develops a quantum time-reference-frame formalism based on the Page-Wootters timeless approach. It defines an event as a quantum operation triggered by a clock reading, constructs history states for both non-interacting and gravitationally interacting clocks, and derives the reduced evolution in each clock frame. The central claims are that (i) unitary time evolution exists relative to any gravitationally interacting clock, (ii) the evolution law is covariant under changes of time reference frames, and (iii) the temporal localisability of events is reference-frame dependent when the metric is indefinite, as illustrated by the gravitational quantum switch. The derivations are formal manipulations within the constraint formalism, and the authors are explicit about their idealisations: perfect clocks, classical and time-independent clock separations, and an unproven assumption that the redshift operator can be divided without producing divergences.

Significance. If the claims survive scrutiny, the framework offers an operational way to talk about events, unitary evolution, and localisation without a fixed spacetime metric, and it gives a concrete bridge between the Page-Wootters picture and the indefinite-causal-order literature. The explicit history states (Eqs. 4, 13, 14, 30–33) and the frame-change formulas (Eqs. 24–27) are valuable technical tools, and the gravitational-switch analysis makes a sharp, checkable statement: the four-point description of the switch is frame-dependent. The paper is commendably open about its assumptions and limitations. The principal risk is mathematical: the formal division by the redshift operator that underlies Eq. (10) and the covariance claim is not justified on the physical Hilbert space for the perfect-clock states used in the paper; closing this gap is essential before the central claims can be accepted.

major comments (3)

[Evolution with respect to gravitationally interacting clocks, Eq. (10)] The derivation of C's-frame Schrödinger equation divides Eq. (7) by the redshift operator s = 1 + λ_AC H_A + λ_BC H_B, with the caveat that "the energies of the state of the system C-bar are small enough such that no divergences occur." This caveat is load-bearing and is not proven. For the perfect clocks used throughout, each H_I has the full real line as its spectrum. On the physical Hilbert space defined by constraint (6), the conditions s = 0 and H_A + H_B + H_C + λ_AB H_A H_B + λ_AC H_A H_C + λ_BC H_B H_C = 0 admit joint solutions for generic negative couplings, and a clock state with finite width in T has non-vanishing overlap with this singular set because its energy wavefunction has full support. Consequently s^{-1} is not a densely defined operator on the physical state space, Eq. (10) is not a well-defined generator of unitary evolution, and the frame-change formulas (24)–(25) inherit this problem. The later gravitational-switch analysis uses clock states "sharply localised" around t_A = t_B = 0 (text near Eq. (30)), whose broad energy support makes the singular set relevant, so the caveat is not satisfied even in the paper's own examples. The authors must either prove that the physical states used have s bounded away from zero uniformly, or reformulate the division as a regulated operator and show that the relevant limits exist.
[Eq. (6) and the Discussion] The assumption that the relative distances x_IJ (and hence the couplings λ_IJ) are c-numbers and time-independent is acknowledged by the authors as an idealisation, but it is essential to the algebraic structure used in all derivations. The paper's motivation is the regime where the metric is indefinite because matter is quantum; treating clock separations classically removes the quantum spatial superposition that is the source of the indefinite metric in the gravitational switch. The central claim that, for gravitationally interacting clocks, the localisability of events in time is relative is therefore established only for a restricted class of semiclassical clock trajectories. The paper should state clearly how the conclusion is expected to extend, or to fail, when the x_IJ are promoted to quantum operators.
[Events with respect to gravitationally interacting clocks, Eqs. (12)–(14)] The statement that "the localisability of events in time is relative, and depends on the time reference frame which defines the events" is partly true by construction: an event triggered by clock A is, by definition, sharp when expressed in A's frame. The non-trivial gravitational effect is that the same event can become fuzzy in C's frame even for an initially sharp clock state, due to the operator-valued redshift factor in Eq. (13). The paper also shows a similar relative fuzziness for non-interacting clocks with unsharp initial states. The authors should more explicitly separate the definitional sharpness (which is built into the event definition) from the physically substantive, interaction-induced delocalisation, otherwise the novelty of the gravitational claim can be obscured.

minor comments (4)

[Equations (3), (6), (28)] Several constraint equations are garbled in the submission (e.g., Eq. (6) renders as "( (|Ψ⟩=0" and Eq. (28) has an unclosed parenthesis). These display equations need to be corrected before publication.
[Gravitational quantum switch, Eqs. (30)–(33)] The comparison between C's-frame and A's-frame descriptions of the switch would be much easier to follow if the event times in each frame and in each mass configuration were summarised in a table (e.g., t*, t*/Δ^Q(A,C), t*/Δ^Q(B,C), and the corresponding entries in A's frame). Currently the reader must reconstruct this information from the prose.
[General framework] The term "covariant" is used in the nonstandard sense of form-invariance under quantum time-reference-frame transformations. Although the text says "form invariant," it would help to define this explicitly at first use and to contrast it with the usual relativistic notion of covariance.
[Methods, Changing time reference frames] In Eq. (24), the derivation of the evolution operator in A's frame from that in C's frame assumes the integration measure in the time representation is simply dt_C; the paper notes this is an assumption but does not discuss the conditions under which it holds. A brief comment on when this measure is non-trivial would make the method more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained; the redshift-operator division is an explicit assumption representing correctness risk, not a circular step.

full rationale

The paper contains no fitted parameters, no calibrated quantities renamed as predictions, and no load-bearing self-citation chain. Equations (7), (10), (13) and (14) follow from the stated constraints (6) and (12) by acting with a clock bra and effecting the coordinate change tau_A := t'_A + tau_C(1 + lambda omega_B). The one mathematically delicate step is the formal division by the redshift operator 1 + lambda_AC H_A + lambda_BC H_B in going from Eq. (7) to Eq. (10). The authors explicitly flag this as an assumption ('we formally divide by the redshift factor operator... with the assumption that the energies of the state of the system C-bar are small enough such that no divergences occur'), so the possible non-invertibility is an unproven premise for unitary evolution, not a circularity: the conclusion is not assumed and then read back as a derivation. The frame-dependence of event localisation also has independent content. An 'event' is defined operationally as an operation conditioned on a clock reading; that an A-triggered event is sharp in A's own frame is built into the choice of A's time coordinate, but the nontrivial claim is that the same event is delocalised in C's frame through the operator-valued argument in Eq. (13), which is a derived consequence of the gravitational coupling and the energy spread of clock B. This asymmetry is not present in the non-interacting case, so it does not reduce to the definition of 'event' by construction. Citations to prior work such as [13], [16] and [25]-[27] are contextual, interpretational, or concern the physical motivation of the coupling; none is invoked as an unverified uniqueness theorem or as the sole justification of a central result. Accordingly, no circular step is identifiable under the required standard.

Assumptions & free parameters 3 free parameters · 6 assumptions · 2 invented entities

No constants are fitted to data: the quantitative content is carried by scenario parameters (clock distances, trigger times, clock widths) and by imported physical assumptions (perfect clocks, first-order post-Newtonian coupling, non-vanishing redshift denominator). The central claim, that event localisation is frame-relative under gravitational interaction, is derived from these inputs rather than tuned to match a target result.

free parameters (3)

relative clock distances x_IJ = not fitted (scenario inputs)
The couplings λ_IJ = -G/(c^4 x_IJ) are fixed by the assumed geometry. Results scale with them; they are chosen by hand for each experiment.
event trigger times t*_A, t*_B = not fitted (scenario inputs)
The clock readings at which operations fire; free experimental parameters in the model.
clock wave-packet width σ = not fitted (scenario input)
Sets the initial clock fuzziness; via Fourier transform it sets the energy spread 1/σ that drives the delocalisation effects in Eqs. (5) and (13).

assumptions (6)

domain assumption Perfect clocks: [T_I, H_I] = i for I = A, B, C, with Hamiltonians unbounded from below.
Used throughout to define time eigenstates and the constraint; acknowledged by the authors as an idealisation (Discussion, Methods 'Review of the timeless approach').
domain assumption First-order post-Newtonian gravitational interaction between clocks, H_int = (1/2) sum_{I≠J} λ_IJ H_I H_J, with λ_IJ = -G/(c^4 x_IJ) and c-number, time-independent distances x_IJ.
Imported from [13] and [16]; this is the mechanism that produces frame-relative event localisation. The c-number treatment of x_IJ is explicitly assumed in the same section.
domain assumption The operation term in the constraint is gravitationally coupled: f(T_A) is multiplied by (1 + λ H_B) in Eq. (12).
Justified by the universality of gravity (any operation energy couples to the other clock). Without this term the gravitational relativity of event localisation disappears; it is stated, not derived.
ad hoc to paper The redshift operator 1 + λ_AC H_A + λ_BC H_B is invertible on the physical states: energies are small enough that no divergences occur.
Required to pass from Eq. (7) to Eq. (10); only a qualitative no-divergence condition is given, with details deferred to Methods.
standard math Page-Wootters formalism: history states via group averaging P = ∫ dα e^{-iαC}, probabilities from the physical inner product.
Background theory from [20, 21, 38], assumed throughout.
domain assumption All gravitational effects are kept to first order in 1/c^2; higher order terms are neglected.
The clock-clock coupling and the redshift factors are truncated at this order; the paper lists higher-order generalisation as future work.

invented entities (2)

Time reference frame (a quantum clock as temporal reference)
purpose: Operationally define events, time evolution, and event localisation without a fixed metric
A conceptual re-framing of Page-Wootters plus quantum reference frame ideas; no new physical entity, no independent falsifiable handle.
Operator-valued coordinate time d/dt̂ = (1 + λ_AC H_A + λ_BC H_B) d/dτ (Eq. 9)
purpose: Interpret the left-hand side of Eq. (7) as a derivative with respect to quantum coordinate time
A formal device inside the framework, referenced to a mathematical theory of operator derivatives [32]; no external empirical signature.

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Cite this review

Pith. "Pith review of Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems." pith.science (2026). https://pith.science/paper/XDS7AM33

@misc{pith2026190810165,
  author       = {Pith},
  title        = {Pith review of: Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDS7AM33}},
  note         = {Machine review of arXiv:1908.10165}
}

read the original abstract

The standard formulation of quantum theory relies on a fixed space-time metric determining the localisation and causal order of events. In general relativity, the metric is influenced by matter, and is expected to become indefinite when matter behaves quantum mechanically. Here, we develop a framework to operationally define events and their localisation with respect to a quantum clock reference frame, also in the presence of gravitating quantum systems. We find that, when clocks interact gravitationally, the time localisability of events becomes relative, depending on the reference frame. This relativity is a signature of an indefinite metric, where events can occur in an indefinite causal order. Even if the metric is indefinite, for any event we can find a reference frame where local quantum operations take their standard unitary dilation form. This form is preserved when changing clock reference frames, yielding physics covariant with respect to quantum reference frame transformations.

Figures

Figures reproduced from arXiv: 1908.10165 by the authors.

**Figure 2.** FIG. 2. Relative localisability of events for non interacting clocks with unsharp initial states. This figure describes the history [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Gravitating quantum clocks from the point of view of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Experimental set up of the gravitational switch. The mass [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: FIG. 5. Space-time diagram for gravitational switch thought experiment as described in the time reference frame of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6. The gravitational switch thought experiment as described in the time reference frame of [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 8 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Relational path integral, effective actions and quantum frame covariance in gravity
hep-th 2026-07 conditional novelty 7.0 of 10

The gravitational path integral is reformulated with dynamical reference frames, producing gauge-invariant correlators, frame-dependent vacua, and effective actions, with sharpness of events becoming frame-relative.
How many degrees of freedom describe a quantum N-particle state?
quant-ph 2026-07 conditional novelty 7.0 of 10

For closed quantum N-particle systems all 3N canonical degrees of freedom are physical; the frame degrees of freedom that relational models discard reappear as non-Heisenberg terms in generalised uncertainty relations...
Relational entanglement entropies and quantum reference frames in gauge theories
hep-th 2025-06 accept novelty 7.0 of 10

Quantum reference frames built from Wilson lines give lattice gauge theories gauge-invariant subsystem factorizations and a hierarchy of relational entanglement entropies.
Subsystem decompositions of quantum evolutions and transformations between causal perspectives
quant-ph 2024-11 accept novelty 7.0 of 10

Alice's and Bob's causal perspectives in the quantum switch cannot be related by any fixed change of subsystem decomposition, so they are not equivalent descriptions of the same evolution.
On the relation between perspective-neutral, algebraic, and effective quantum reference frames
quant-ph 2025-07 conditional novelty 6.0 of 10

For ideal quantum reference frames with a single constraint, the perspective-neutral, algebraic, and effective semiclassical approaches describe the same physics and the same frame-switching rules.
Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy
hep-th 2024-12 accept novelty 6.0 of 10

Gravitational subregion entropy is observer-dependent: different quantum clocks produce different von Neumann algebras and different entropy functionals for the same global state.
Doubly Quantum Mechanics
quant-ph 2024-12 conditional novelty 6.0 of 10

Replacing SU(2) with SU_q(2) turns measurement probabilities into operators and makes reference-frame alignment between two observers fundamentally imprecise even in the limit of infinitely many exchanged spins.
Agency under indefinite causality: operational eternalism in higher-order quantum theory
quant-ph 2025-12 conditional novelty 5.0 of 10

In indefinite-causal-order quantum theory, an observer is a perspective-dependent grouping of input/output data, with a new 'friendliness' criterion for causally compatible agents.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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