Classical and Quantum Dynamics in an Information Theoretic Space

Christopher Griffin; Sean Golder

arxiv: 2604.09735 · v1 · submitted 2026-04-09 · 🪐 quant-ph

Classical and Quantum Dynamics in an Information Theoretic Space

Sean Golder , Christopher Griffin This is my paper

Pith reviewed 2026-05-10 16:49 UTC · model grok-4.3

classification 🪐 quant-ph

keywords information geometryBernoulli distributionKullback-Leibler divergenceLaplace-Beltrami operatorquantum oscillatorquantum penduluminformation space

0 comments

The pith

Quadratic approximation of the Kullback-Leibler potential makes the quantum oscillator in Bernoulli information space equivalent to the quantum pendulum in Euclidean space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops classical and quantum mechanics inside the information geometry of a Bernoulli random variable. It first finds the eigenvalues of the Laplace-Beltrami operator on this one-dimensional manifold and constructs Green's functions that solve the wave, heat, and Poisson equations. Momentum is then quantized, producing explicit energies and wavefunctions for a free particle and for several harmonic oscillators. The central result is that replacing the Kullback-Leibler divergence by its quadratic approximation converts the information-space oscillator into a system whose spectrum and eigenfunctions are identical to those of the ordinary quantum pendulum.

Core claim

In the information manifold of Bernoulli distributions the Laplace-Beltrami operator possesses a discrete spectrum that allows consistent quantization of momentum. When the Kullback-Leibler potential is replaced by its quadratic truncation, the resulting Schrödinger equation is unitarily equivalent to the equation for a quantum pendulum in Euclidean space, so the two systems share the same energies and wavefunctions.

What carries the argument

The Laplace-Beltrami operator induced by the Fisher information metric on the Bernoulli manifold, whose quadratic truncation of the Kullback-Leibler divergence supplies the potential term in the Hamiltonian.

If this is right

Green's functions for the Helmholtz equation in Bernoulli space solve the wave, heat, and Poisson equations.
Explicit energies and wavefunctions exist for the free particle and for multiple quantum oscillators in this space.
The quadratic-KL oscillator is spectrally identical to the Euclidean quantum pendulum.
Classical trajectories follow from the information-geometric spring-mass system already studied in prior work.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Known analytic results for the quantum pendulum, such as coherent states or tunneling amplitudes, transfer directly to information geometry.
Quadratic approximations of divergence potentials on other statistical manifolds may reveal analogous hidden equivalences.
Quantum dynamics in information space could supply a microscopic quantum layer beneath macroscopic free-energy principles.

Load-bearing premise

The Bernoulli probability manifold admits a consistent quantization of the momentum operator, and the quadratic truncation of the Kullback-Leibler potential preserves the essential spectral features without higher-order corrections altering the claimed equivalence.

What would settle it

Solve the Schrödinger equation on the Bernoulli manifold with the exact, non-quadratic Kullback-Leibler potential and check whether the low-lying eigenvalues still match the pendulum spectrum; any systematic deviation falsifies the equivalence.

Figures

Figures reproduced from arXiv: 2604.09735 by Christopher Griffin, Sean Golder.

**Figure 1.** Figure 1: FIG. 1. The first four eigenfunctions are illustrated. Their structure is a result of the Dirichlet boundary [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. The Green’s function for values of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (Left) The Mathieu function and its approximation. (Right) Exact and approximate energy levels for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

read the original abstract

We study elementary classical and quantum dynamics in an information geometric space corresponding to a Bernoulli random variable, extending work by Goehle and Griffin [Chaos, Solitons & Fractals, 188, 115535, (2024)], who study the information theoretic analog of the spring-mass system. Information geometric constructions are useful in both statistical physics and in physical interpretations of Friston's free energy principle, a form of the Bayesian brain hypothesis. In this letter, we derive the spectrum for the Laplace-Beltrami operator in Bernoulli space and find Green's functions for the Helmholtz equation, which provides solutions to the wave, heat, and Poisson equations. We then show how to quantize momentum in Bernoulli space and obtain energies and wavefunctions for both a free particle and a variety of quantum (harmonic) oscillators in this space. In particular, we show that quadratic approximation of the Kullback-Leibler potential used by Goehle and Griffin results in a quantum oscillator in information space that is equivalent to a quantum pendulum in Euclidean space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit spectra and quantization on the Bernoulli information manifold with a claimed pendulum equivalence, but boundary conditions on the interval likely break the match.

read the letter

The main thing to know is that this work quantizes classical and quantum dynamics on the information geometry of a Bernoulli random variable, extending their prior classical spring-mass analog, and claims that a quadratic approximation to the Kullback-Leibler potential produces an oscillator equivalent to the quantum pendulum. They derive the Laplace-Beltrami spectrum, Green's functions for the Helmholtz equation, and then quantize momentum to obtain energies and wavefunctions for free particles and oscillators in this space. The explicit pendulum link is the new concrete result not in the earlier paper. They do this for a single low-dimensional family, which keeps everything calculable and connects directly to statistical physics and free-energy-principle ideas. That is useful when you want a worked example rather than abstract generalities. The derivations follow standard information-geometric and quantization steps, and the citation to their own prior work is appropriate since it supplies the classical starting point. The soft spot is the equivalence itself. The Fisher metric turns the Bernoulli simplex into an interval [0, π] with singular endpoints, so the self-adjoint Laplace-Beltrami operator needs boundary conditions at the ends, typically vanishing or reflecting. The quantum pendulum, however, is defined on the circle with periodic conditions and Mathieu eigenfunctions. The coordinate change that flattens the metric pulls the quadratic potential back to a cosine term, but the domains and allowed functions do not coincide, so the spectra and wavefunctions differ. The abstract states the equivalence without showing the boundary handling, and if the full text does not resolve this the claim does not hold. This is narrow but technically grounded work aimed at people already using information geometry for dynamics or Bayesian brain models. It deserves a serious referee to verify the quantization details and clarify whether the equivalence survives with proper boundary conditions or needs to be restated as an approximation on the interior.

Referee Report

2 major / 2 minor

Summary. The manuscript studies classical and quantum dynamics on the information-geometric manifold of Bernoulli distributions, extending Goehle and Griffin. It derives the spectrum of the Laplace-Beltrami operator, constructs Green's functions for the Helmholtz equation (yielding solutions to wave, heat, and Poisson equations), quantizes the momentum operator, and obtains energies/wavefunctions for a free particle and quantum oscillators. The central claim is that the quadratic approximation of the Kullback-Leibler potential produces a quantum oscillator in information space equivalent to the quantum pendulum in Euclidean space.

Significance. If the derivations and equivalence hold under consistent boundary conditions, the work supplies concrete spectra and Green's functions on a curved information manifold, strengthening links between information geometry, statistical physics, and Friston's free energy principle. Explicit quantization and the claimed mapping to a standard quantum system (pendulum) would be a useful technical contribution.

major comments (2)

[Quantization section and KL-approximation derivation] The equivalence claim (abstract and the section deriving the quantum oscillator via quadratic KL approximation) does not address domain and boundary conditions. The Fisher metric on the Bernoulli simplex is isometric to the interval [0, π] via φ = 2 arcsin(√p), making the Laplace-Beltrami operator d²/dφ²; self-adjointness at the singular endpoints φ=0,π typically requires Dirichlet or Neumann conditions, whereas the quantum pendulum is defined on S¹ with periodic boundary conditions whose eigenfunctions are Mathieu functions. The quadratic KL term pulls back to a cos φ potential, but the differing BC mean the eigenvalue problems are not identical even after the coordinate change.
[KL potential approximation] § on the quadratic approximation of the KL potential: the manuscript asserts that this truncation yields spectral equivalence to the pendulum without showing that higher-order terms in the KL expansion do not alter the low-lying eigenvalues or the claimed mapping. An explicit comparison of the first few eigenvalues (or an error bound) between the truncated and full potential is needed to support the equivalence.

minor comments (2)

[Abstract] The abstract describes derivations but supplies no equations, coordinate transformations, or error estimates; adding the explicit form of the quadratic KL term and the φ coordinate change would improve readability.
[Quantization of momentum] Notation for the quantized momentum operator and the precise definition of the information-geometric inner product should be stated once and used consistently when transitioning from the classical to the quantum case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify important technical points in our work on dynamics in Bernoulli information space. We address each major comment below and will incorporate revisions accordingly.

read point-by-point responses

Referee: [Quantization section and KL-approximation derivation] The equivalence claim (abstract and the section deriving the quantum oscillator via quadratic KL approximation) does not address domain and boundary conditions. The Fisher metric on the Bernoulli simplex is isometric to the interval [0, π] via φ = 2 arcsin(√p), making the Laplace-Beltrami operator d²/dφ²; self-adjointness at the singular endpoints φ=0,π typically requires Dirichlet or Neumann conditions, whereas the quantum pendulum is defined on S¹ with periodic boundary conditions whose eigenfunctions are Mathieu functions. The quadratic KL term pulls back to a cos φ potential, but the differing BC mean the eigenvalue problems are not identical even after the coordinate change.

Authors: We appreciate the referee's precise identification of the boundary condition distinction. Our derivation proceeds on the information manifold using the coordinate φ = 2 arcsin(√p) ∈ [0, π], where the Laplace-Beltrami operator is indeed d²/dφ². Self-adjointness on this compact interval with singular endpoints at the probability simplex boundaries naturally leads us to Dirichlet conditions (wavefunctions vanishing at φ = 0, π), which are the physically appropriate choice for the Bernoulli distribution space. The quadratic KL approximation yields the cos φ potential, producing a Schrödinger equation of pendulum form. While we acknowledge that this differs from the periodic boundary conditions on S¹ (yielding Mathieu functions), the core equivalence we claim is in the potential and the resulting differential operator structure within the information-geometric setting. We will revise the manuscript to explicitly discuss the domain, the choice of Dirichlet conditions, and the resulting differences from the standard quantum pendulum, while preserving the mapping as an analogy in the information space context. revision: yes
Referee: [KL potential approximation] § on the quadratic approximation of the KL potential: the manuscript asserts that this truncation yields spectral equivalence to the pendulum without showing that higher-order terms in the KL expansion do not alter the low-lying eigenvalues or the claimed mapping. An explicit comparison of the first few eigenvalues (or an error bound) between the truncated and full potential is needed to support the equivalence.

Authors: We agree that an explicit verification strengthens the claim. In the revised manuscript we will add a comparison of the lowest eigenvalues (ground state and first excited states) obtained from the quadratic KL truncation versus the full KL potential, together with a first-order perturbation estimate bounding the shift induced by higher-order terms. This will confirm that the truncation preserves the low-lying spectrum to high accuracy for the regimes of interest, thereby supporting the equivalence under the quadratic approximation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper extends the KL-potential approximation from the cited Goehle-Griffin work but performs independent derivations of the Laplace-Beltrami spectrum, Green's functions for the Helmholtz equation, momentum quantization in Bernoulli space, and the claimed equivalence of the quadratic truncation to the quantum pendulum. No load-bearing step reduces by the paper's own equations to a self-definition, fitted input renamed as prediction, or unverified self-citation chain. The self-citation is limited to referencing the prior potential form; the new spectral and equivalence results are constructed from the information-geometric manifold and quantization procedure stated in the present manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework assumes the standard Riemannian structure of information geometry and introduces a specific quantization rule and quadratic truncation without new postulated entities.

axioms (2)

domain assumption The space of Bernoulli distributions equipped with the Fisher metric forms a Riemannian manifold on which the Laplace-Beltrami operator and momentum quantization are well-defined.
Invoked to extend classical dynamics to the quantum regime in information space.
ad hoc to paper The quadratic term of the Kullback-Leibler divergence supplies a valid potential whose spectrum matches that of the quantum pendulum.
Central step that produces the claimed equivalence; higher-order terms are discarded without justification in the abstract.

pith-pipeline@v0.9.0 · 5470 in / 1446 out tokens · 83103 ms · 2026-05-10T16:49:27.930971+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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[1] [1]

merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

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[2] [2]

merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

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[3] [3]

merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

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merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

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merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

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author author P. Mathews \ and\ author M. Lakshmanan ,\ title title On a unique nonlinear oscillator , \ @noop journal journal Quarterly of Applied Mathematics \ volume 32 ,\ pages 215--218 ( year 1974 ) NoStop

work page 1974

[7] [7]

author author J. F. \ Carinena , author M. F. \ Ranada , author M. Santander , \ and\ author M. Senthilvelan ,\ title title A non-linear oscillator with quasi-harmonic behaviour: two-and n-dimensional oscillators , \ @noop journal journal Nonlinearity \ volume 17 ,\ pages 1941 ( year 2004 ) NoStop

work page 1941

[8] [8]

author author J. F. \ Cari \ n ena , author M. F. \ Ranada , \ and\ author M. Santander ,\ title title The quantum harmonic oscillator on the sphere and the hyperbolic plane: -dependent formalism, polar coordinates, and hypergeometric functions , \ @noop journal journal Journal of mathematical physics \ volume 48 ( year 2007 ) NoStop

work page 2007

[9] [9]

author author J. F. \ Cari \ n ena , author M. F. \ Ranada , \ and\ author M. Santander ,\ title title One-dimensional model of a quantum nonlinear harmonic oscillator , \ @noop journal journal Reports on Mathematical Physics \ volume 54 ,\ pages 285--293 ( year 2004 ) NoStop

work page 2004

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Calogero \ and\ author S

author author F. Calogero \ and\ author S. Graffi ,\ title title On the quantization of a nonlinear hamiltonian oscillator , \ @noop journal journal Physics Letters A \ volume 313 ,\ pages 356--362 ( year 2003 ) NoStop

work page 2003

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author author F. Calogero ,\ title title On the quantization of yet another two nonlinear harmonic oscillators , \ @noop journal journal Journal of Nonlinear Mathematical Physics \ volume 11 ,\ pages 1--6 ( year 2004 ) NoStop

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