Large effects from quantum reference frames

Luis Martinez; Martin Bojowald

REVIEW 2 major objections 3 minor 27 references

Large effects from quantum reference frames

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper derives a large, Planck-constant-free shift in the measured position induced by a quantum reference frame reaching a turning point, and argues this gives a testable signature of relational quantum mechanics.

desk verdict The paper's headline effect is an artifact of comparing at equal unwrapped τ; in the physical frame variable φ, the quantum expectation exactly matches the classical value. read the letter →

arxiv 2506.14721 v1 pith:KEF35M6W submitted 2025-06-17 quant-ph

Martin Bojowald , Luis Martinez This is my paper · ORCID

classification quant-ph

keywords quantumreferenceframesrelationalmechanicsturningpointsnon-monotonicclockvariablesconstraintquantizationpositionshiftfundamentalclocks

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 argues that when the reference frame used to label a measurement has one turning point—a place where the frame variable slows to a stop and reverses—the measured position of the system acquires a shift that is a genuine quantum effect. For a system with $\hat H=\hat p$ and a frame subject to a linear potential, the overall shift is $\delta q = -4\langle\hat p\rangle^2/\lambda - 2(\Delta p)^2/\lambda$, and the leading term contains no $\hbar$ at all. The authors present this as an unexpectedly large quantum effect because it is not suppressed by the smallness of Planck's constant. If correct, it would turn quantum reference frames and relational quantum mechanics into something testable through a characteristic shift in a single observable, rather than through delicate measurements of correlations.

What carries the argument

The central object is the effective monotonic scale $\tau$, constructed from the non-monotonic frame coordinate $\phi$ by unwinding it at the turning point: $\phi(\tau)=\tau$ before the turning point and $\phi(\tau)=-\tau+2p^2/\lambda$ after it. This replacement makes $\tau$ run over all real values and keeps the square-root Hamiltonian $\sqrt{p^2-\lambda\phi(\tau)}$ real, which restores unitarity for evolution with respect to $\tau$. The sign in front of the square root is fixed by requiring positive energy for forward $\tau$-changes, and the phase of the wave function is made continuous between the two branches, introducing a cubic phase term proportional to $p^3/\lambda$. This $p$-dependence of the phase is what converts the turning-point event into a shift of $\langle\hat q\rangle$.

What would settle it

Measure the position before and after a single pass of the frame through its turning point and compare the late-time extrapolation: if the extrapolated displacement is not $\delta q = -4\langle\hat p\rangle^2/\lambda - 2(\Delta p)^2/\lambda$, or if the leading term scales with $\hbar$, the claim fails. A second check is to quantize the same model with a different junction condition at the turning point and see whether the $\hbar$-independent leading term survives.

Watch

Extended reading notes

Core claim

The central claim is a concrete formula for what happens to a measured position when a quantum reference frame with a single energy-dependent turning point is used. The frame variable $\phi$ feels the linear potential $V(\phi)=\lambda\phi\theta(\phi)$ with $\theta$ the Heaviside step function, and with system Hamiltonian $\hat H=\hat p$ the classical constraint $-p_\phi^2-\lambda\phi\theta(\phi)+p^2=0$ has a turning point at $\phi_t=p^2/\lambda$. The paper unwraps $\phi$ into an effective monotonic scale $\tau$, solves the resulting unitary evolution with a definite sign choice for the square-root Hamiltonian, and imposes continuity of the wave function across the turning point. The position expectation value extrapolated back from large $\tau$ is $q_0-2\langle\hat p^2\rangle/\lambda$, opposite in sign to the classical displacement $2\langle\hat p\rangle^2/\lambda$, so the overall shift is $\delta q = -4\langle\hat p\rangle^2/\lambda - 2(\Delta p)^2/\lambda$. The term $-4\langle\hat p\rangle^2/\lambda$ is independent of $\hbar$ and is identified as the large quantum effect.

Load-bearing premise

The numerical result depends on the paper's convention for unwinding the frame variable at the turning point and its choice of boundary condition there; a different unwrapping or junction condition could change or erase the shift.

Editorial extensions

If this is right

If correct, a single pass through the turning point produces a displacement $\delta q = -4\langle\hat p\rangle^2/\lambda - 2(\Delta p)^2/\lambda$ in the measured position extrapolated from late times.
Because the leading term has no $\hbar$, the effect is not a small semiclassical correction; it should be visible in a directly measured observable rather than only in phase interferometry.
A monotonic reference frame produces no such displacement, so observing the shift would specifically signal a non-monotonic quantum frame with an energy-dependent turning point.
In the gravitational realization with $\lambda=m^2g$, the paper's scaling estimates give a shift of about 10 m for a 100-amu atom at $T\sim1$ K, about $10^{-5}$ m for microkelvin atom traps, and coherence times near 1 ms in the latter case.
Replacing the gravitational force by a large electric force on an ion shortens the required coherence time at the expense of the expected shift.

Reading between the lines

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

A periodic clock built from repeated half-cycles of this model would accumulate the turning-point phase multiple times; whether the $\hbar$-independent shift adds per half-cycle or saturates is a natural extension the paper does not address.
The formula separates the shift into a mean-momentum part and a fluctuation part; an experiment with asymmetric or squeezed momentum distributions could isolate $-(\Delta p)^2/\lambda$ and test the quantum-correction term independently.
If the shift proves insensitive to alternative junction conditions at the turning point, the observable could serve as a calibration standard for engineered quantum reference frames in trapped-ion or cold-atom laboratories.

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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

First thing to know: the central claim does not hold up. The large shift in Eq. (21) is an artifact of the unwrapped variable τ. When the state is expressed in the physical frame variable φ, the conditional expectation ⟨q⟩_φ equals the classical ensemble-averaged q(φ) at every φ. I checked this from the paper's own wavefunction (18). For φ≤0 after the turning point, ψ(φ,p)=f(p)e^{i(pφ−4p³/(3λ))/ℏ}, giving ⟨q⟩_φ=q0−φ+4⟨p²⟩/λ, exactly the classical value. For the return branch 0≤φ≤p²/λ, the same cancellation occurs. So the −4⟨p⟩²/λ term in (21) appears only because quantum and classical expectations are compared at equal τ rather than equal φ, and τ is not an observable: it is defined using the system momentum p, which varies across the superposition.

The paper does have real merits. It is a clean, analytically solvable model of a quantum reference frame with a single energy-dependent turning point. The quantization method from [13,14] is applied consistently, the wavefunction and the expectation values are derived without hidden assumptions, and the numerics in Fig. 2 match Eq. (21). If the goal were to demonstrate a technical solution of this toy model, it succeeds.

The soft spots beyond the main issue are minor. The experimental estimates in Section 3 mix massless and massive descriptions—the m in λ=m²g is not the same as the m in the momenta—though it is only an order-of-magnitude argument. There is a likely typo in Eq. (11), where the third branch repeats 'p²/λ≥φ≥0'. And the claim that the qualitative effect is independent of the model choices is unsupported; it rests on the same unwrapping convention.

Who should read this? People working on relational quantum dynamics will find the explicit calculation useful, but the advertised testable signature is not there. This paper deserves a serious referee because the flaw is subtle and the mathematics is solid; a referee could push the authors to specify an operational way to observe τ, which I suspect cannot be done without already knowing the momentum distribution. I would not cite it in my own work, but I would not let it vanish without a proper review.

Referee Report

2 major / 3 minor

Summary. The paper constructs a solvable model of a quantum reference frame with a single, energy-dependent turning point. The frame variable φ is subject to a piecewise linear potential V(φ)=λφθ(φ), and the system is described by H=p, so that the constraint C=−p_φ²−λφθ(φ)+H(q,p)²=0 yields explicit two-branch classical solutions q(φ). The authors introduce an unwrapped monotonic parameter τ and solve the constraint quantum mechanically in the p-representation, obtaining the piecewise wave function (19). From the asymptotic τ-dependence of ⟨q⟩(τ) they derive a quantum shift δq_quantum=−2⟨p²⟩/λ, compare it with a classical shift 2⟨p⟩²/λ, and conclude that the total shift δq=−4⟨p⟩²/λ−2(Δp)²/λ is a surprisingly large, ℏ-independent quantum effect with possible experimental signatures in coherent atom ensembles.

Significance. If the central claim held, the paper would be significant: it would provide a rare analytical model of a non-monotonic quantum reference frame, identify a ℏ-independent observable effect of quantum reference frames, and offer a concrete experimental signature. The algebraic derivation is internally consistent, the wave function is explicit, and Fig. 2 numerically reproduces Eq. (21) within the τ-parameterization. However, the advertised effect does not survive evaluation in the physical relational variable φ: using the paper's own wave function, the conditional position expectation ⟨q⟩_φ equals the classical q(φ) on every branch, so the large shift in Eq. (21) is an artifact of the p-dependent unwrapping convention rather than a genuine quantum reference frame effect. The central testable prediction is therefore not supported, and the remaining content is a technical exercise in clock unwrapping rather than a demonstration of large observable quantum corrections.

major comments (2)

[§3, Eq. (21); §2.3, Eqs. (12), (18)–(19)] The claimed large shift is an artifact of the p-dependent unwrapping variable τ. Using the manuscript's own wave function (19) and the relation (12), the position expectation at fixed physical frame reading φ equals the classical q(φ) on every branch. For φ≤0 after the turning point, ψ=f(p)exp(i(pφ−4p³/(3λ))/ℏ), so ⟨q⟩_φ=q0−φ+4⟨p²⟩/λ, identical to the classical last line of (11). For 0≤φ≤p²/λ on the return branch, the phase S=−(2/(3λ))(p³+(p²−λφ)^{3/2}) gives ⟨q⟩_φ=q0+2λ^{−1}⟨p(p+√(p²−λφ))⟩, again identical to the classical expression in (11). Thus the −4⟨p⟩²/λ term in Eq. (21) appears only because different momentum components are compared at equal unwrapped τ, not because of a quantum effect in the relational observable q(φ). The central testable prediction is therefore not supported by the model as presented.
[§2.3, Eq. (12); §3] The parameter τ is not a physical observable of the relational system. Its definition depends on the system momentum p through the turning point p²/λ and through the branch choice (before or after the turning point). In a superposition of different p, a given value of the frame reading φ corresponds to different τ for different p-components, and conversely, at a fixed τ different components sit at different φ. Consequently, an experiment that records (φ,q) cannot reconstruct the asymptotic shift (20) without supplying an external, p-dependent bookkeeping of which branch each component is on. If τ is instead interpreted as an external time parameter, the model reduces to unitary evolution with respect to a background parameter, and the result is no longer a test of quantum reference frames. The operational meaning of τ must be specified before the effect can be considered measurable.

minor comments (3)

[§3, Eq. (22)] The displayed estimate has a dimensional inconsistency as printed: δq/m has units length/mass, whereas the subsequent estimate δq∼10 m at T∼1 K for m∼100 amu follows from δq∼k_BT/(mg). Please correct the displayed formula and the surrounding text.
[§3] The statement that ℏ-independence of the leading term makes it 'an unexpectedly large quantum effect' is not by itself persuasive, since the classical shift in Eq. (14) is also ℏ-independent; the argument would need to be based on the physical-variable analysis rather than on the absence of ℏ.
[§3, final paragraph] The generalization to a general system Hamiltonian, replacing p by an energy operator and using a POVM time, is not a derivation; the computation of the shift in Eq. (20) uses the canonical conjugate of p to define ⟨q⟩. This part should be presented as conjectural rather than as a straightforward substitution.

Circularity Check

1 steps flagged · score 6.0 of 10

Headline quantum shift in Eq. (21) is an artifact of the p-dependent unwrapping variable τ; in the physical frame variable φ, quantum and classical position agree.

self definitional [Section 2.2, Eq. (12); Section 3, Eqs. (20)-(21)]
"In order to keep track of when the turning point has been encountered, we introduce an effective monotonic scale τ constructed from the frame variable ϕ... After the turning point, τ should continue to increase while ϕ decreases, such that ϕ=−τ+c with a constant c. Continuity then uniquely determines ... ϕ(τ)= τ if τ≤p^2/λ; −τ+2p^2/λ if τ≥p^2/λ. ... Our prime observable of interest is the displacement of the position at τ=0 obtained by extrapolating the asymptotic ⟨ˆq⟩(τ) for large τ back to τ=0: δq_quantum = −2⟨ˆp^2⟩/λ."

The claimed large effect is computed with respect to the auxiliary parameter τ, not the physical frame variable φ. By the paper's own Eq. (12), after the turning point τ is defined with a p-dependent offset, τ = −φ + 2p^2/λ. Substituting this into the asymptotic wave function in Eq. (19) gives ψ(φ,p)=f(p)exp(iℏ^{-1}(pφ−4p^3/(3λ))), and the position expectation becomes ⟨q⟩_φ = q0 − φ + 4⟨p^2⟩/λ, which is exactly the classical q(φ) in Eq. (11) (last line). The extra −4⟨p⟩^2/λ in Eq. (21) therefore comes solely from equating τ across different momenta after the turning point; in the relational variable φ, quantum and classical predictions coincide. The headline shift is thus built into the definition of the observable, not independently derived from the physics.

full rationale

The paper is self-contained: no parameters are fitted to data, and the quantum evolution and expectation values follow from explicit wave functions. However, the central prediction of a 'surprisingly large' shift is self-definitional. The observable is defined by extrapolating ⟨q⟩(τ) to τ=0, and the monotonic parameter τ is constructed, in Eq. (12), with a momentum-dependent post-turn offset 2p^2/λ. Transforming the paper's own asymptotic solution from τ to the physical frame variable φ makes the quantum expectation equal to the classical q(φ), so the large term in Eq. (21) is an artifact of the chosen unwrapping coordinate rather than a physical effect in the relational variable. The self-citations [12,13,14,17] are used as methodological precedent and are not load-bearing once the explicit construction in Section 2.3 is given. Because the derivation itself is analytic and the criticism is about the construction of the observable, the appropriate circularity score is partial: the headline shift reduces by definition, while the framework and wave-function calculations retain independent content.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The model is explicit and parametrically minimal, with one slope lambda plus the initial state, but it rests on a specific quantization convention for non-monotonic frames and on a toy Hamiltonian. No new physical entities are postulated.

free parameters (1)

lambda (linear potential slope) = Not fitted; model input, estimated as m^2 g in Section 3
The central shift scales as 1/lambda, and the value of lambda is an input of the model, not derived from first principles. In the experimental estimate it is set to m^2 g by hand.

assumptions (4)

domain assumption Constraint quantization with a quadratic constraint C = -p_phi^2 - lambda phi theta(phi) + H(q,p)^2 = 0 and the physical Hilbert space construction from references [14,13].
Section 2.1: 'The approach of [14], together with further extensions given in [13], makes it possible to deal with such models.' The central result inherits this construction.
ad hoc to paper The reference frame potential has the Heaviside-bounded linear form V(phi) = lambda phi theta(phi), producing exactly one energy-dependent turning point.
Section 2.1, Eq. (4). Chosen for mathematical tractability; the generality claim in Section 3 is asserted, not derived.
ad hoc to paper The system Hamiltonian is chosen as H = p, a massless-like operator, to make the model solvable.
Section 2.2: 'we use the simplest non-trivial case, H=p'. Later experimental estimates treat the system as a massive atom, creating an inconsistency.
ad hoc to paper An effective monotonic scale tau winds the non-monotonic phi and fixes the sign of the square-root Hamiltonian for forward tau-evolution.
Section 2.3, Eqs. (12), (16)-(19). The phase and the derived shift depend on this unwrapping and phase-continuity convention.

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Cite this review

Pith. "Pith review of Large effects from quantum reference frames." pith.science (2026). https://pith.science/paper/KEF35M6W

@misc{pith2026250614721,
  author       = {Pith},
  title        = {Pith review of: Large effects from quantum reference frames},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KEF35M6W}},
  note         = {Machine review of arXiv:2506.14721}
}

read the original abstract

Reference frames are used to parameterize measurements of physical effects, but since their practical realization uses material objects, they may affect observations performed in a combined quantum state of the measured system together with the frame. Here, a procedure is used that makes it possible to describe non-monotonic reference scales in a quantum treatment, revealing large quantum effects in the measured system whenever a reference frame encounters a turning point. Subtle quantum correlations in the combined state of system and frame, and more broadly the concept of relational quantum mechanics, can be tested via a characteristic and surprisingly large shift in the measured value.

Figures

Figures reproduced from arXiv: 2506.14721 by the authors.

**Figure 2.** Classical solution and expectation value for the position in a numerically solved [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

Review history (2 revisions) →

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Aharonov and T

Y. Aharonov and T. Kaufherr, Quantum frames of reference,Phys. Rev. D30 (1984) 368–385

work page 1984

[2] [2]

Rovelli, Relational Quantum Mechanics,Int

C. Rovelli, Relational Quantum Mechanics,Int. J. Theor. Phys.35 (1996) 1637, [quant-ph/9609002]

arXiv 1996

[3] [3]

Entanglement of quantum clocks through gravity

E. Castro-Ruiz, F. Giacomini, and C. Brukner, Entanglement of quantum clocks through gravity,PNAS114 (2017) E2303, [arXiv:1507.01955]

work page Pith review arXiv 2017

[4] [4]

Giacomini, E

F. Giacomini, E. Castro-Ruiz, and C. Brukner, Quantum mechanics and the covari- ance of physical laws in quantum reference frames,Nat. Commun.10 (2019) 494, [arXiv:1712.07207]

arXiv 2019

[5] [5]

Giacomini, E

F. Giacomini, E. Castro-Ruiz, and C. Brukner, Relativistic quantum reference frames: the operational meaning of spin,Phys. Rev. Lett.123 (2019) 090404, [arXiv:1811.08228]

arXiv 2019

[6] [6]

Vanrietvelde, P

A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and E. Castro-Ruiz, A change of per- spective: switching quantum reference frames via a perspective-neutral framework, Quantum4 (2020) 225, [arXiv:1809.00556]

arXiv 2020

[7] [7]

Vanrietvelde, P

A. Vanrietvelde, P. A. Hoehn, and F. Giacomini, Switching quantum reference frames in theN-body problem and the absence of global relational perspectives, [arXiv:1809.05093]

arXiv

[8] [8]

P. A. Hoehn, A. R. H. Smith, and M. P. E. Lock, The Trinity of Relational Quantum Dynamics,Phys. Rev. D104 (2021) 066001, [arXiv:1912.00033]

arXiv 2021