Entropic Dynamics approach to Relational Quantum Mechanics

Ariel Caticha; Hassaan Saleem

REVIEW 2 major objections 4 minor 23 references

Entropic Dynamics approach to Relational Quantum Mechanics

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Entropic Dynamics settles the quantized form of relational constraints: they act on expectation values, and parametrized relational quantum mechanics evades the problem of time.

desk verdict A well-built ED toy model whose central conclusion — expectation-value relational constraints — rests on a minimization branch that is not generally the true minimum, compounded by a phase-non-invariant mismatch measure. read the letter →

arxiv 2506.07921 v4 pith:VTDOC3CO submitted 2025-06-09 quant-ph

Ariel Caticha , Hassaan Saleem This is my paper

classification quant-ph

keywords Entropicdynamicsrelationalquantummechanicsbestmatchingconstraintsexpectationvaluesproblemoftimeparametrizedinformationgeometry

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

Entropic Dynamics derives quantum mechanics from inference principles, and this paper extends it to relational quantum mechanics: models in which only relative particle configurations matter, not absolute positions or orientations. The paper's central claim is that when relational symmetry is imposed by best matching successive quantum states, the resulting constraints must be imposed on expectation values, such as $\langle \psi|(\hat P - M\dot\lambda)|\psi\rangle = 0$ and $\langle \psi|(\hat L - \hat I\dot\zeta)|\psi\rangle = 0$. It also claims that Entropic Dynamics is already temporally relational, because the evolving system itself supplies the clock, and that writing the model in parametrized, generally covariant form avoids the problem of time that blocks canonical quantum gravity. The result matters because it offers a principled answer to a long-standing question — what constraints survive quantization — and a toy model for relational quantum gravity.

What carries the argument

The load-bearing object is the quantum mismatch measure $\delta(\dot\xi) = |\langle\psi_t|\psi_{t+dt}\rangle - 1|^2$, evaluated through the inner product $\langle\psi|\chi\rangle$ constructed from the information metric $G$ and symplectic form $\Omega$ of the epistemic phase space. Minimizing $\delta$ with respect to the shift velocity $\dot\xi$ determines equilocality between successive instants and produces constraints on expectation values. The dynamics that generates $\psi_{t+dt}$ is chosen to be a Hamilton-Killing flow, meaning it preserves both the symplectic and metric structures; this yields a bilinear Hamiltonian and a linear Schrödinger equation, with the best-matching shift appearing as a gauge-like term.

What would settle it

Take a simple two-particle state and compute the best-matching shift using an alternative mismatch measure, for example $\delta = 1 - |\langle\psi_t|\psi_{t+dt}\rangle|$ or a measure built from the Fisher information metric alone. If the minimizing shift is not $\dot\lambda^a = \langle\psi|\hat P^a|\psi\rangle/M$, then the expectation-value form of the constraints is an artifact of the chosen measure. This calculation can be done within the paper's own Entropic Dynamics framework without any new physics.

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Extended reading notes

Core claim

The paper claims that quantum best matching, the adaptation of classical best matching to quantum states, is governed by a mismatch measure $\delta(\dot\xi) = |\langle\psi_t|\psi_{t+dt}\rangle - 1|^2$ defined from the inner product that combines the information metric and the symplectic structure of the epistemic phase space. Minimizing this measure for rigid translations gives the shift velocity $\dot\lambda^a = \langle\psi|\hat P^a|\psi\rangle/M$, and for rigid rotations gives $\dot\zeta^a = (\tilde I^{-1})^a{}_b \tilde L^b$, where $\tilde I$ is the expected moment of inertia. The physical states are therefore restricted to subspaces satisfying $\langle\psi|(\hat P - M\dot\lambda)|\psi\rangle = 0$ and $\langle\psi|(\hat L - \hat I\dot\zeta)|\psi\rangle = 0$. The paper concludes that relational constraints in a quantum theory act on expectation values, not on operators or ontic states, and shows that the same Entropic Dynamics model, once parametrized, is invariant under relabeling of the time coordinate while still evolving in entropic time, thereby evading the Hamiltonian-constraint freeze that produces the problem of time.

Load-bearing premise

The load-bearing premise is the specific choice of mismatch measure $\delta(\dot\xi) = |\langle\psi_t|\psi_{t+dt}\rangle - 1|^2$; the paper itself notes that other mismatch measures are possible, and a different measure would generally produce different relational constraints.

Editorial extensions

If this is right

Any relational symmetry in Entropic Dynamics, not just translations and rotations, should be imposed as expectation-value constraints on physical states.
A single particle can have a relational quantum dynamics, because what is best matched are infinite-dimensional wave functions rather than point configurations.
Parametrized Entropic Dynamics provides a toy model in which time remains a dynamical variable yet the Hamiltonian constraint does not stop evolution, offering an explicit strategy against the problem of time.
The same best-matching logic extends to any theory with redundancy of description, including electromagnetism, Yang-Mills theories, and possibly gravity.

Reading between the lines

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

A different measure of mismatch, such as one based purely on the Fisher information metric rather than the combined metric-symplectic inner product, would generally produce different best-matching constraints, so the expectation-value form is measure-dependent rather than forced by Entropic Dynamics alone.
Because the rotational constraint involves a time-dependent expected moment of inertia, the model predicts that relationality under rotations is dynamically nontrivial: the angular velocity $\dot\zeta$ changes even when angular momentum is conserved, which is the quantum analogue of the bucket argument and may require coupling rotational relationality to a gravitational field.
If the same best-matching logic is applied to gauge theories, the natural prediction is that gauge constraints should be imposed weakly on expectation values, giving a new reading of how redundant descriptions survive quantization.

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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

Short version: this is a real piece of work, but the central derivation has a hole that is bigger than the paper's own caveats suggest. The genuinely new item is the mismatch measure delta(xidot)=|langle psi_t|psi_{t+dt}rangle - 1|^2, adapted to the symplectic and metric structures of ED, and the explicit construction of translation- and rotation-relational models, including a parametrized action that exhibits general covariance and a discussion of why the Wheeler-DeWitt problem of time does not arise. All of this is set out clearly, with healthy self-awareness about the toy-model status and about the rotation case ending in an appeal to the equivalence principle.

What I cannot get past is the minimization. From Eqs. (55)-(57), delta=(dt^2/hbar^2) Htilde_xidot^2. The paper solves d Htilde/d xidot = 0. That is a stationary condition for delta only on the branch Htilde != 0. The actual minimization of Htilde^2 over the shift is different. Htilde is a quadratic, often non-positive, function of the shift. Where Htilde can vanish, the global minima are Htilde = 0 - a quadratic condition, not the linear expectation-value constraints in (62) or (72). Where Htilde stays positive and convex, d Htilde = 0 is indeed the minimum. But for bound states with negative energy expectation, the stationary point used in the paper is a local maximum of delta. So the headline answer to the quantization-of-constraints question is not established by the calculation as written.

There is also a gauge problem. Eq. (54) is not invariant under multiplying psi_{t+dt} by a global phase, even though the physical ray is unchanged. The natural gauge-invariant infinitesimal distance is the Fubini-Study one, and minimizing it gives covariance-type conditions such as Cov(H, Phat - M lambda) = 0 rather than = 0. Since the paper itself says other mismatch measures are possible, the language of settling the question is too strong. That is not fatal to the ED program, but it means the result is conditional on a post hoc postulate.

The parametrized-ED section is coherent but less novel than it sounds: ED constructs the quantum theory directly, so the problem of time never arises in the first place; the demonstration is useful as an illustration. The rotation treatment is partial and the authors admit it; the Newton-bucket/equivalence-principle paragraph is a promissory note.

Recommendation: send it to a serious referee. The framework and question matter, and the algebra is explicit enough to be checked and repaired. But a referee should focus on the minimization branch and on the gauge-invariance of the mismatch. I would not currently cite the expectation-value constraints as a settled consequence.

Referee Report

2 major / 4 minor

Summary. The paper develops relational quantum mechanics within the entropic dynamics (ED) framework. It constructs non-relativistic quantum models that are relational under rigid translations and rotations by introducing a best-matching criterion based on a mismatch measure between successive quantum states. The authors derive constraints that are imposed on expectation values, e.g., ⟨ψ|(P̂ − Mλ̇)|ψ⟩ = 0 and ⟨ψ|(L̂ − Iζ̇)|ψ⟩ = 0 (Eq. 72), and they claim that this settles the question of how classical constraints should be quantized. The paper also parametrizes the model to make temporal relationality explicit and argues that this evades the analogue of the problem of time in quantum gravity.

Significance. The paper is clearly written and builds on a well-developed ED framework, providing explicit derivations of the best-matching conditions and connecting them to the Barbour–Bertotti tradition. The idea of deriving quantum relational constraints from an information-geometric matching principle is interesting and could be a useful testing ground for relational quantum mechanics. However, the central claim that ED settles the quantization-of-constraints question is heavily dependent on a specific choice of mismatch measure, and the minimization argument has a technical gap. These issues need to be addressed before the conclusions can be fully accepted.

major comments (2)

[Section 4, Eq. (54)] The mismatch measure δ(ξ̇) = |⟨ψ_t|ψ_{t+dt}⟩ − 1|² is not invariant under the U(1) gauge freedom of quantum states: multiplying ψ_{t+dt} by e^{iα} changes δ even though the physical ray is unchanged. The paper itself notes that other mismatch measures are possible (Section 4). A gauge-invariant measure such as the Fubini-Study metric gives, to O(dt²), 1 − |⟨ψ_t|ψ_{t+dt}⟩|² = dt² Var(H_{ξ̇})/ℏ², and minimizing this alternative measure yields covariance-type constraints (e.g., Cov(H, P̂_a − Mλ̇_a) = 0) rather than the expectation-value constraints in Eq. (72). Thus the conclusion that relational constraints must be imposed on expectation values is not a consequence of ED itself but of this particular phase-non-invariant choice. The abstract's claim that the ED approach 'settles' the quantization-of-constraints question is therefore too strong; at most it settles the question relative to a specific postulate.
[Section 4, Eqs. (56)–(57)] The paper asserts that δ(ξ̇) 'does have a minimum' and derives the BM conditions from ∂δ/∂ξ̇ = 0. However, δ = dt² ⟨H⟩²/ℏ² is the square of the energy expectation. The stationarity condition ∂δ/∂ξ̇ = 0 is satisfied either when ⟨H⟩ = 0 or when ∂⟨H⟩/∂ξ̇ = 0; the paper only considers the second branch. At a point where ∂⟨H⟩/∂ξ̇ = 0, the second derivative of δ is proportional to ⟨H⟩ ∂²⟨H⟩/∂ξ̇² plus a nonnegative term. If ⟨H⟩ < 0 and ∂²⟨H⟩/∂ξ̇² > 0, the stationary point is a local maximum, not a minimum. The paper does not check the second-order condition nor discuss the ⟨H⟩ = 0 branch. This undermines the 'minimization' justification for the BM constraints; the derivation at best establishes stationary points, not minima.

minor comments (4)

[Section 5.2] The claim that the parametrized ED model 'evades the analogue of the problem of time' is overstated. The super-Hamiltonian constraint in Eq. (78) is a constraint on the Hamiltonian functional, not on quantum states, and the model never quantizes this constraint. The problem of time in canonical quantum gravity arises precisely when the constraint is imposed on the quantum state (ĤΨ = 0). The toy model avoids this because it is a first-quantized system with an external label x0; it does not demonstrate how a full ED quantum gravity would handle the constraint. The paper should temper this claim to 'illustrates a strategy' rather than 'evades all these problems'.
[Section 4.1] There is a typo: 'we can can go a bit further' should read 'we can go a bit further'.
[Section 4, property (c)] The statement that δ(ξ̇) 'does have a minimum' is not sufficient; the specific stationary point obtained from ∂δ/∂ξ̇ = 0 must be checked to actually be a minimum (see major comment 2). The paper should either provide the second-order analysis or rephrase the argument in terms of stationary points.
[Section 3.2] The sentence 'the system is its own clock' is slightly misleading: in ED, the clock is defined by the transition probabilities and the chosen scaling of duration, not by the physical system's internal state. A more precise wording would avoid confusion with Page–Wootters-type clocks.

Circularity Check

1 steps flagged · score 6.0 of 10

Expectation-value constraint result is an artifact of the chosen mismatch measure (Eq. 54), not a derivation.

self definitional [Sec. 4 (Eqs. 54–57) and Sec. 4.1 (Eqs. 62, 72)]
"We propose the following candidate for a measure of mismatch, δ(ξ̇)=|⟨ψ_t|ψ_{t+dt}⟩−1|² ... Other mismatch measures are in principle possible ... Then we find δ(ξ̇)=|⟨ψ_t|δψ_t⟩|²=dt²/ℏ² ⟨ψ_t| Ĥ_ξ̇|ψ_t⟩² ... The ED approach to BM provides a crisp answer: the quantum constraints that express relationality are to be imposed on expectation values."

The mismatch measure of Eq. (54) is defined through the overlap ⟨ψ_t|ψ_{t+dt}⟩. Using Eq. (55), Eq. (56) reduces it to dt²/ℏ² ⟨Ĥ_ξ̇⟩², so minimizing δ(ξ̇) with respect to ξ̇ sets derivatives of the expectation value of Ĥ to zero. This is exactly the expectation-value constraint of Eqs. (62) and (72). Thus the central conclusion is not derived from ED principles but is built into the specific choice of δ. The paper itself states that 'Other mismatch measures are in principle possible'; for example, a Fubini–Study-type measure would lead to variance or covariance constraints instead. The 'crisp answer' to the quantization-of-constraints question is therefore an artifact of the selected measure, not a unique outcome of the framework.

full rationale

The paper's headline claim—that relational quantum constraints must be imposed on expectation values—follows directly from the proposed mismatch measure δ(ξ̇)=|⟨ψ_t|ψ_{t+dt}⟩−1|². Because δ reduces to the squared expectation value of the Hamiltonian, minimizing it with respect to shifts yields expectation-value constraints by construction. The paper explicitly acknowledges that other mismatch measures are possible, which means this result is not forced by the ED framework itself. This is a partial circularity: the input measure already selects the expectation-value structure that is later presented as the answer to a longstanding question. The remaining developments, including the Hamilton–Killing derivation of the Schrödinger equation and the parametrized ED treatment of time, build on the authors' prior ED work [12–14,16] through normal self-citation; those citations are cumulative framework support rather than a circular reduction of the present conclusion. No machine-checked or external falsification of the central measure choice is provided, but the paper's own admission of alternative measures is the decisive limitation. Overall score 6: the central 'prediction' reduces, to a significant degree, to the choice of mismatch measure.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The new content rests on the ED framework from the same author's prior publications, plus a newly chosen mismatch measure. The main free choice is the mismatch measure, which directly determines the form of the relational constraints.

free parameters (3)

Particle masses m_n = identified with empirical masses
Introduced in Eqs. (16)-(17) to define the duration of entropic time; later identified as masses. Their values are inputs, not derived.
Diffusion constant η = not stated in this paper
Introduced in Section 3.2 (α' = 1/η). Determines the fluctuation scale in Eq. (23). The relation to ℏ is not given in this paper; standard ED sets η = 2ℏ.
Mismatch measure δ(ξ̇) = |⟨ψ_t|ψ_{t+dt}⟩ − 1|²
Chosen in Eq. (54) as a candidate. The paper admits other measures are possible. The derived BM constraints depend on this choice.

assumptions (7)

domain assumption Maximum entropy principle applies to transition probabilities
Foundation of ED, used in Eqs. (8)-(11) to derive the Gaussian transition probability.
domain assumption Prior Q(x'|x) in Eq. (9) encodes only short steps and translational/rotational invariance
Assumed maximally uninformative prior; reflects ignorance and is itself derivable from MaxEnt only under stated symmetries.
ad hoc to paper The drift constraint Eq. (10) with potential φ(x) captures the relevant physical information
Standard ED device; the form of the constraint is postulated to obtain a drift term.
ad hoc to paper E-phase space is assigned flat metric (41)
Section 3.4: 'without loss of generality [14]... they are flat'. This choice determines the complex structure and the wave-function coordinates.
domain assumption Hamilton-Killing flows (46) preserve both symplectic and metric structures
Taken from prior ED work [13]; restricts Hamiltonians to bilinear form (47) and hence linear Schrödinger equation.
domain assumption Potential V depends only on interparticle distances (51)
Imposed in Section 3.5 to make ST1 transformations a symmetry; needed for momentum conservation.
ad hoc to paper The mismatch measure (54) is the correct measure of state mismatch
Proposed in Section 4 as a candidate. The central result (expectation-value constraints) is derived from this choice.

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Cite this review

Pith. "Pith review of Entropic Dynamics approach to Relational Quantum Mechanics." pith.science (2026). https://pith.science/paper/VTDOC3CO

@misc{pith2026250607921,
  author       = {Pith},
  title        = {Pith review of: Entropic Dynamics approach to Relational Quantum Mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTDOC3CO}},
  note         = {Machine review of arXiv:2506.07921}
}

read the original abstract

The general framework of Entropic Dynamics (ED) is used to construct non-relativistic models of relational quantum mechanics from well known inference principles -- probability, entropy and information geometry. Although only partially relational -- the absolute structures of simultaneity and Euclidean geometry are still retained -- these models provide a useful testing ground for ideas that will prove useful in the context of more realistic relativistic theories. The fact that in ED the positions of particles have definite values, just as in classical mechanics, has allowed us to adapt to the quantum case some intuitions from Barbour and Bertotti's classical framework. Here, however, we propose a new measure of the mismatch between successive states that is adapted to the information metric and the symplectic structures of the quantum phase space. We make explicit that ED is temporally relational and we construct non-relativistic quantum models that are spatially relational with respect to rigid translations and rotations. The ED approach settles the longstanding question of what form should the constraints of a classical theory take after quantization: the quantum constraints that express relationality are to be imposed on expectation values. To highlight the potential impact of these developments, the non-relativistic quantum model is parametrized into a generally covariant form and we show that the ED approach evades the analogue of what in quantum gravity has been called the problem of time.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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The theory of gravitation in Hamiltonian form,

P.A.M. Dirac, “The theory of gravitation in Hamiltonian form, ” Proc. R. Soc. Lond. A246, 333-343 (1958)

work page 1958

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R. Arnowitt, S. Deser, and C.W. Misner, “The Dynamics of General Rela- tivity.” Gravitation: an introduction to current research, ed. by L. Witten, (Wiley, New York, 1962); arXiv:gr-qc/0405109

arXiv 1962

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B. S. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory,” Phys. Rev. 160, 1113-1148 (1967)

work page 1967

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S. A. Hojman, K. Kuchaˇ r, and C. Teitelboim, “Geometrodynamics re- gained,” Ann. Phys. 96.1, 88-135 (1976)

work page 1976

[5] [5]

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J. B. Barbour and B. Bertotti, “Mach’s Principle and the Structure of Dynamical Theories”, Proc. Roy. Soc. A 382, 295 (1982)

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J. Barbour, “The Nature of Time,” written for the Foundational Questions Institute 2009 essay competition on The Nature of Time; arXiv:0903.3489. 20

arXiv 2009

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J. Barbour, “The definition of Mach’s Principle,” Found. Phys. 40, 1263- 1284 (2010)

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