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Classical probabilistic realisation of quantum double-slit interference

T0 review · 2 major / 3 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Classical field probabilities can host one-particle subsystems whose wave functions obey the Schrödinger equation and show double-slit interference.

desk verdict The algebraic map from classical Liouville evolution of a complex scalar field to one-particle Schrödinger dynamics is explicit and checks out; the detector-click and Bell claims remain interpretive assumptions. read the letter →

arxiv 2607.11580 v1 pith:2YAISBAA submitted 2026-07-13 quant-ph cond-mat.stat-mechhep-th

Christof Wetterich This is my paper · ORCID
classification quant-phcond-mat.stat-mechhep-th PACS 03.65.Ta03.65.Ud05.20.-y
keywords double-slitinterferenceclassicalprobabilitiesstatisticalobservablesconservedchargeone-particlesubsystemSchrödingerequationLiouvillecomplexscalarfield
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

This paper claims that the interference pattern of a quantum particle in the double-slit experiment can be realized entirely by a classical probability distribution over field configurations. The setting is a complex classical scalar field whose deterministic field equation is linear, with a mass term that may depend on position; probabilistic initial conditions turn the theory into a classical statistical system whose probability density evolves by the Liouville equation. Conserved charges, treated as statistical observables rather than functions of individual field values, label particle-number sectors. Inside the one-particle sector the probabilistic information is packaged into an ordinary complex wave function; the Liouville dynamics then forces that wave function to satisfy the Schrödinger equation for a particle in an arbitrary potential. Because the potential can be chosen freely, the construction reproduces interference, tunneling and discrete spectra. A sympathetic reader cares because the argument supplies an explicit classical-probability model for the paradigmatic quantum experiment and isolates the precise mathematical objects—statistical charges and closed subsystems—that make the embedding possible.

What carries the argument

The one-particle classical wave function q^(1)[ψ_S] (square root of the probability density), built by acting with creation operators on a Gaussian vacuum and expressed as a linear functional of the quantum density matrix ρ_S = ψ_S ψ_S^*; its Liouville evolution is identical to the von Neumann evolution of ρ_S.

What would settle it

Construct the explicit classical probability density w^(1) for a concrete double-slit potential, evolve it by the Liouville equation, and check whether the marginal probabilities for the detector observables reproduce the standard quantum interference fringes (or fail to).

Watch

Extended reading notes

Core claim

The Liouville equation for a specially constructed family of one-particle classical probability distributions w^(1) implies that the complex function ψ_S that parametrizes those distributions obeys the Schrödinger equation i∂_t ψ_S = H_S ψ_S, where H_S = √B and B is the spatial operator appearing in the classical field equation; in the non-relativistic limit this is the ordinary Hamiltonian for a particle in an arbitrary potential V(x) generated by a space-dependent mass.

Load-bearing premise

That the integer eigenvalues of the conserved charge operators (which are statistical, not classical, observables) correctly define particle-number subsystems whose detector clicks match ordinary quantum measurements, and that the lack of joint probabilities for those observables is enough to evade Bell constraints.

Editorial extensions

If this is right

  • Any potential that produces interference, tunneling or discrete spectra in ordinary quantum mechanics can be realized by a classical probability distribution over scalar fields.
  • The one-particle Schrödinger dynamics is a closed subsystem of a larger classical statistical field theory.
  • Particle discreteness arises as the integer spectrum of a statistical charge rather than as a fundamental particle ontology.
  • Bell inequalities need not constrain correlations that involve statistical observables, because those observables lack joint probabilities with classical field values.

Reading between the lines

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

  • If the construction extends to interacting nonlinear field equations, ordinary multi-particle quantum mechanics with Coulomb forces could likewise sit inside a classical probability theory.
  • The same charge-subsystem logic may supply a classical-probability account of spin and entanglement once the field content is enlarged.
  • Explicit numerical solution of the Liouville equation for a double-slit mass profile would give a concrete classical-probability animation of fringe formation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper constructs a classical probabilistic field theory for a complex scalar field σ(t,x) obeying a linear second-order field equation with space-dependent mass term m+V(x). Probabilistic initial conditions are encoded in a real classical wave function q=√w whose Liouville evolution is generated by the operator L̂. Conserved statistical charge operators (built from functional derivatives) define integer particle-number subsystems. The vacuum is the Gaussian ground state of the associated quantum-field Hamiltonian H^(q); one-particle states are obtained by applying creation operators, yielding a family of classical probability distributions w^(1)[ρ_S] linear in the quantum density matrix ρ_S=ψ_S ψ_S*. Direct computation shows that Liouville evolution of w^(1) is equivalent to the von Neumann (or Schrödinger) equation generated by H_S=√B ≈ -Δ/(2m)+V(x). Arbitrary potentials, including those realizing double-slit interference, tunneling and discrete spectra, are thereby recovered as closed classical-statistical subsystems.

Significance. If the construction is accepted, it supplies an explicit, fully worked classical-statistical embedding of one-particle quantum mechanics (including interference) that evades standard no-go arguments by treating particle number and detector clicks as statistical observables lacking joint probabilities. The algebraic map from the classical Liouville operator to the one-particle Hamiltonian √B is parameter-free, written out in complete detail (eqs. 1–113 plus appendix), and immediately yields all standard quantum effects once a suitable V(x) is chosen. This is a concrete advance over earlier Koopman–von Neumann or operator formalisms for classical statistics, and the explicit formulae for w^(1) and the vacuum Gaussian constitute a falsifiable, reproducible contribution.

major comments (2)
  1. [Discrete particles, eqs. (134)–(138)] Section “Discrete particles”, eqs. (134)–(138) and Conclusions: the detector observable is defined as D̂(x)=Θ(∑_{y∈I(x)} n̂(y)) and the firing probability is simply set equal to ⟨D̂(x)⟩=∫_I |ψ_S|^2. The paper itself labels this an “assumption for an ideal measurement” and never derives it from the underlying classical measure w(σ,π). Because the central claim is a classical-probabilistic realisation of the double-slit experiment (including discrete detector clicks), this identification is load-bearing; either a derivation from the field-configuration measure or a sharper delimitation of the claim to the dynamical subsystem alone is required.
  2. [Introduction / after eq. (34)] Introduction and paragraph after eq. (34): the assertion that Bell inequalities need not apply because joint probabilities for statistical observables do not exist is left at the level of a general remark. No concrete calculation is given showing that the classical probability measure on (σ,π) forces the absence of the particular joint distributions that would enter a Bell test involving the charge or detector observables of the one-particle subsystem. A short explicit argument or counter-example would make the evasion claim rigorous rather than programmatic.
minor comments (3)
  1. [eqs. (6), (79)–(80)] Notation for the non-relativistic limit of H_S oscillates between √B and m+V-Δ/(2m); a single consistent expansion (including the constant shift of E_0) would improve readability.
  2. [Appendix] The appendix derivation of q^(1) is valuable but dense; a short intermediate identity relating the action of a† on the vacuum Gaussian to the bilinear form C(x,y) would help the reader verify eqs. (145)–(149).
  3. [References] Several arXiv preprints by the same author are cited as “to appear” or with future dates; updating the bibliography with final references (or arXiv identifiers) would aid readers.

Circularity Check

2 steps flagged · score 4.0 of 10

One-particle distributions are engineered as functionals of the quantum density matrix so Liouville evolution reproduces Schrödinger by intertwining; statistical-observable and Bell-evasion claims rest on the author's prior self-citations.

  1. self definitional [eqs. (2)–(10) and the paragraph containing “Our central result states…”]
    "We construct particular ”one-particle probability distributions” w^(1)(t)=(q^(1)(t))^2, where q^(1)(t)=∫ ψ_S(t,x)ψ*_S(t,y)C(x,y)q^(0)-q^(0)… Our central result states that the Liouville equation for w^(1) implies that ψ_S obeys the Schrödinger equation… Eq. (8) can be verified by direct computation for general quantum density matrices"

    w^(1) is introduced by definition as a linear functional of the quantum density matrix ho_S=ψ_Sψ*_S. The subsequent claim that Liouville evolution of this functional yields the von-Neumann evolution of ho_S is precisely the statement that the functional intertwines the two generators. The intertwining holds by the way the functional (built from the same creation operators that define the one-particle sector of the quantum theory) was constructed; the “direct computation” merely confirms the engineered map.

  2. self citation load bearing [paragraphs on statistical observables and Bell (after eq. 34) and “Discrete particles”]
    "A central new point in our approach is the use of statistical observables [32–34]. … Simultaneous probabilities are a central assumption of Bell’s inequalities. Since this assumption is not realized, one concludes that Bell’s inequalities do not need to hold for possible correlations involving statistical observables. … We will use this for a prediction of the probability p̄(x) that … the detector at x fires … p̄(x)=⟨D(x)⟩. … Our assumption is that standard particle detectors realize the relation (135)."

    The conceptual move that lets a classical probability distribution host genuine quantum interference—namely that conserved charges are statistical observables lacking joint probabilities, so Bell inequalities are inapplicable, and that detector clicks are simply the expectation values of the projector heta(∑n̂)—is justified solely by citations to the author’s own prior works [32–34, 35–37]. No independent derivation of these measurement postulates from the classical measure w(σ,π) is supplied in the present paper; the double-slit “realisation” therefore inherits its interpretive force from the self-citation chain.

full rationale

The paper is an explicit constructive embedding, not a pure tautology: the classical Liouville operator (9) is independent of the quantum Hamiltonian, the vacuum Gaussian (103)–(106) and creation operators (58)/(89) are written out, and the intertwining identity (8) is claimed to follow by direct (if tedious) differentiation. That calculation is self-contained and does not import fitted parameters or rename an external empirical pattern. Circularity appears only at two controlled points. First, the family w^(1) is defined from the outset as a linear functional of the quantum density matrix ρ_S (eqs. 2–7); once that map is chosen, the statement “Liouville on w^(1) implies Schrödinger/von Neumann on ρ_S” is the intertwining property of the map itself. Second, the interpretive claims that make the construction a “classical realisation of quantum particles” (statistical charges have no joint probabilities, therefore Bell inequalities need not hold; detector clicks are expectation values of the projector Θ(n̂)) rest load-bearingly on the author’s own earlier papers [32–34, 35–37] rather than on a derivation internal to the present text. These two features raise the score to 4 but do not collapse the central algebraic construction into a pure definitional identity. No uniqueness theorem is imported, no ansatz is smuggled via citation, and no data-fitting occurs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The central claim rests on standard classical statistical mechanics plus a small set of domain choices that select the one-particle sector and interpret statistical charges as particle numbers. No free parameters are fitted to data; the potential V(x) is an arbitrary input. The principal invented entities are the statistical charge observables and the classical real wave function q = √w.

assumptions (4)
  • domain assumption Classical probability distributions over field configurations (σ,π) evolve by the Liouville equation generated by the deterministic field equation ∂_t^{2}σ = −Bσ.
    Standard classical statistical mechanics of fields; invoked from the outset (eqs. 11–18).
  • standard math The eigenvalues of the global charge operator Q̂ are integers because a 2π phase rotation returns every field configuration to itself.
    Topological argument after eq. 47; used to obtain discrete particle numbers.
  • ad hoc to paper Statistical observables (operators involving functional derivatives) may be measured even though they lack definite values on individual microstates, and joint probabilities for pairs of non-commuting statistical observables need not exist.
    Core interpretive postulate that allows evasion of Bell inequalities; stated in the introduction and after eq. 34.
  • domain assumption The vacuum is the Gaussian annihilated by all a_n and b_n, and one-particle states are obtained by acting with a† on that vacuum.
    Standard Fock-space construction adapted to the classical setting (eqs. 73, 101–103).
invented entities (3)
  • Classical real wave function q = √w
    purpose: Encodes the probability distribution so that the Liouville equation becomes a Schrödinger-like equation with Hermitian generator H_L.
    Introduced after eq. 17; allows operator methods and complex structure to be imposed on classical statistics.
  • Statistical charge observables Q̂± built from functional derivatives
    purpose: Define integer particle numbers and closed subsystems without assigning a charge to each field configuration.
    Central to the particle concept (eqs. 40, 114–127); no independent experimental handle outside the framework.
  • One-particle classical probability distribution w^(1)[ψ_S]
    purpose: Closed subsystem whose Liouville evolution reproduces Schrödinger dynamics for arbitrary V(x).
    Constructed explicitly in eqs. 2–5 and 145–149; the double-slit claim rests on it.

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Cite this review

Pith. "Pith review of Classical probabilistic realisation of quantum double-slit interference." pith.science (2026). https://pith.science/paper/2YAISBAA

@misc{pith2026260711580,
  author       = {Pith},
  title        = {Pith review of: Classical probabilistic realisation of quantum double-slit interference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YAISBAA}},
  note         = {Machine review of arXiv:2607.11580}
}
read the original abstract

We demonstrate how the interference effects for a quantum particle in the double-slit experiment can be described by classical probabilities. We investigate a classical field theory for a complex scalar field with probabilistic initial conditions. A central element are conserved charges leading to the concept of particles. These are statistical observables which describe properties of the probability distribution for field configurations. The conserved charges define subsystems for particle excitations of a vacuum state. We encode the probabilistic information for the one-particle subsystem in a complex wave function. The Liouville equation for the classical probability distribution implies that the time evolution of this wave function obeys the Schr\"odinger equation for a quantum particle in a potential. The potential arises from a space-dependence of the mass term in the otherwise relativistic classical field theory. It can be chosen arbitrarily, realizing the typical quantum effects of interference, tunneling or discrete energy spectra.

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Forward citations

Cited by 1 Pith paper

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

  1. Quantum observables for probabilistic classical particles

    quant-ph 2026-07 conditional novelty 5.0 of 10

    Solutions of the Liouville equation can be rewritten as a Schrödinger equation whose observables are non-commuting 'quantum' operators, reproducing the harmonic oscillator and hydrogen atom spectra as special subsystems.

Reference graph

Works this paper leans on

50 extracted references · 29 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Wetterich, Quantum Correlations in Classical Statis- tics, inLecture Notes in Physics, Berlin Springer Ver- lag, Vol

    C. Wetterich, Quantum Correlations in Classical Statis- tics, inLecture Notes in Physics, Berlin Springer Ver- lag, Vol. 633, edited by H.-T. Elze (2004) p. 180, arXiv: quant-ph/0212031

    arXiv 2004

  2. [2]

    Wetterich, Probabilistic observables, conditional cor- relations, and quantum physics, Annalen der Physik622, 467 (2010), arXiv:0810.0985 [quant-ph]

    C. Wetterich, Probabilistic observables, conditional cor- relations, and quantum physics, Annalen der Physik622, 467 (2010), arXiv:0810.0985 [quant-ph]

    arXiv 2010

  3. [3]

    Wetterich, Emergence of quantum mechanics from classical statistics, inJournal of Physics Conference Se- ries, Journal of Physics Conference Series, Vol

    C. Wetterich, Emergence of quantum mechanics from classical statistics, inJournal of Physics Conference Se- ries, Journal of Physics Conference Series, Vol. 174 (IOP,

  4. [4]

    012008, arXiv:0811.0927 [quant-ph]

    p. 012008, arXiv:0811.0927 [quant-ph]

    arXiv

  5. [5]

    B. O. Koopman, Hamiltonian systems and trans- formation in hilbert space, Proceedings of the National Academy of Sciences17, 315 (1931), https://www.pnas.org/doi/pdf/10.1073/pnas.17.5.315

    doi:10.1073/pnas.17.5.315 1931

  6. [6]

    J. v. Neumann, Zur operatorenmethode in der klassischen mechanik, Annals of Mathematics33, 587 (1932)

    1932

  7. [7]

    H. E. Kandrup, Phase mixing in time-independent hamil- tonian systems, Monthly Notices of the Royal Astronom- ical Society301, 960 (1998)

    1998

  8. [8]

    V. I. Man’ko and G. Marmo, Alternative commutation relations, star products and tomography, Physica Scripta 60, 111 (1999), quant-ph/9903021

    arXiv 1999

Show all 50 references
  1. [9]

    Mauro, On koopman–von neumann waves, Interna- tional Journal of Modern Physics A17, 1301 (2002), arXiv:quant-ph/0105112 [quant-ph]

    D. Mauro, On koopman–von neumann waves, Interna- tional Journal of Modern Physics A17, 1301 (2002), arXiv:quant-ph/0105112 [quant-ph]

    2002 arXiv

  2. [10]

    Gozzi and D

    E. Gozzi and D. Mauro, On koopman–von neumann waves ii, International Journal of Modern Physics A19, 1475 (2004), arXiv:quant-ph/0306029 [quant-ph]

    2004 arXiv

  3. [11]

    Nikoli´ c, Classical mechanics without determin- ism, Foundations of Physics Letters19, 553 (2006), arXiv:quant-ph/0505143 [quant-ph]

    H. Nikoli´ c, Classical mechanics without determin- ism, Foundations of Physics Letters19, 553 (2006), arXiv:quant-ph/0505143 [quant-ph]

    2006 arXiv

  4. [12]

    Nikoli´ c, Classical mechanics as nonlinear quantum me- chanics, arXiv:quant-ph/0707.2319 (2007)

    H. Nikoli´ c, Classical mechanics as nonlinear quantum me- chanics, arXiv:quant-ph/0707.2319 (2007)

    2007 arXiv

  5. [13]

    I. V. Volovich, Randomness in classical mechanics and quantum mechanics, Foundations of Physics41, 516 (2011), arXiv:0910.5391 [quant-ph]

    2011 arXiv

  6. [14]

    D. I. Bondar, R. Cabrera, R. R. Lompay, M. Y. Ivanov, and H. A. Rabitz, Operational dynamic modeling tran- scending quantum and classical mechanics, Physical Re- view Letters109, 190403 (2012), arXiv:1107.5139 [quant- ph]

    2012 arXiv

  7. [15]

    Arbabi and I

    H. Arbabi and I. Mezi´ c, Ergodic theory, dynamic mode decomposition and computation of spectral properties of the koopman operator, SIAM Journal on Applied Dynamical Systems16, 2096 (2017), arXiv:1611.06664 [math.DS]

    arXiv

  8. [16]

    Korda, M

    M. Korda, M. Putinar, and I. Mezi´ c, Data-driven spec- tral analysis of the koopman operator, arXiv:1710.06532 (2017)

    2017 arXiv

  9. [17]

    X. D. Arsiwalla, D. Chester, and L. H. Kauffman, On the operator origins of classical and quantum wave func- tions, Quantum Studies: Mathematics and Foundations 11, 193 (2024), arXiv:2211.01838 [math-ph]

    2024 arXiv

  10. [18]

    Chru´ sci´ nski, Koopman’s approach to dissipation, Re- ports on Mathematical Physics57, 319 (2006)

    D. Chru´ sci´ nski, Koopman’s approach to dissipation, Re- ports on Mathematical Physics57, 319 (2006)

    2006

  11. [19]

    Klein, From koopman–von neumann theory to quan- tum theory, Quantum Studies: Mathematics and Foun- dations5, 219 (2018), arXiv:1705.07427 [math.DS]

    U. Klein, From koopman–von neumann theory to quan- tum theory, Quantum Studies: Mathematics and Foun- dations5, 219 (2018), arXiv:1705.07427 [math.DS]

    2018 arXiv

  12. [20]

    Mezi´ c, On numerical approximations of the koopman operator, Mathematics10, 1180 (2022), arXiv:2009.05883 [math.DS]

    I. Mezi´ c, On numerical approximations of the koopman operator, Mathematics10, 1180 (2022), arXiv:2009.05883 [math.DS]

    2022 arXiv

  13. [21]

    Darling and L

    K. Darling and L. M. Widrow, Linear operator theory of phase mixing, Monthly Notices of the Royal Astronomi- cal Society533, 79 (2024)

    2024

  14. [22]

    Stengl, P

    M. Stengl, P. Gelß, S. Klus, and S. Pokutta, Exis- tence and uniqueness of solutions of the Koopman- von Neumann equation on bounded domains, Journal of Physics A Mathematical General57, 395302 (2024), arXiv:2306.13504 [math.AP]

    2024 arXiv

  15. [23]

    Nakao and I

    H. Nakao and I. Mezi´ c, Spectral analysis of the Koop- man operator for partial differential equations, Chaos30, 113131 (2020), arXiv:2004.10074 [nlin.PS]

    2020 arXiv

  16. [24]

    Kaiser, J

    E. Kaiser, J. N. Kutz, and S. L. Brunton, Data-driven ap- 14 proximations of dynamical systems operators for control, arXiv e-prints , arXiv:1902.10239 (2019)

    1902 arXiv

  17. [25]

    Hamzi, H

    B. Hamzi, H. Owhadi, and U. Vaidya, Kernel Methods for Some Transport Equations with Application to Learn- ing Kernels for the Approximation of Koopman Eigen- functions: A Unified Approach via Variational Methods, Green’s Functions and the Method of Characteristics, arXiv e-pr...

    2026

  18. [26]

    Novikau and I

    I. Novikau and I. Joseph, Quantum algorithm for the advection-diffusion equation and the koopman-von neu- mann approach to nonlinear dynamical systems, Com- put. Phys. Commun.309, 109498 (2025)

    2025

  19. [27]

    I. Mezi´ c, Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State-Space Ge- ometry, Journal of NonLinear Science30, 2091 (2020), arXiv:1702.07597 [nlin.CD]

    arXiv

  20. [28]

    Mezi´ c, Analysis of fluid flows via spectral properties of the koopman operator, Annual Review of Fluid Mechan- ics45, 357 (2013)

    I. Mezi´ c, Analysis of fluid flows via spectral properties of the koopman operator, Annual Review of Fluid Mechan- ics45, 357 (2013)

    2013

  21. [29]

    Mauroy, Koopman operator framework for spectral analysis and identification of infinite-dimensional sys- tems, Mathematics9, 2495 (2021)

    A. Mauroy, Koopman operator framework for spectral analysis and identification of infinite-dimensional sys- tems, Mathematics9, 2495 (2021)

    2021

  22. [30]

    Giannakis and C

    D. Giannakis and C. Valva, Consistent spectral approxi- mation of Koopman operators using resolvent compactifi- cation, Nonlinearity37, 075021 (2024), arXiv:2309.00732 [math.DS]

    2024 arXiv

  23. [31]

    J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika1, 195 (1964)

    1964

  24. [32]

    J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theo- ries, Phys. Rev. Lett.23, 880 (1969)

    1969

  25. [33]

    Wetterich, Quantum field theory for classical fields, arXiv e-prints , arXiv:2603.05061 (2026)

    C. Wetterich, Quantum field theory for classical fields, arXiv e-prints , arXiv:2603.05061 (2026)

    2026

  26. [34]

    Wetterich, Quantum mechanics for classical trans- port equations (2026), arXiv preprint, arXiv:2605.15969 [quant-ph]

    C. Wetterich, Quantum mechanics for classical trans- port equations (2026), arXiv preprint, arXiv:2605.15969 [quant-ph]

    2026 arXiv

  27. [35]

    Wetterich (2026), to appear

    C. Wetterich (2026), to appear

    2026

  28. [36]

    Wetterich, Quantum particles from classical probabil- ities in phase space, International Journal of Theoretical Physics51, 3236 (2012), arXiv:1003.0772 [quant-ph]

    C. Wetterich, Quantum particles from classical probabil- ities in phase space, International Journal of Theoretical Physics51, 3236 (2012), arXiv:1003.0772 [quant-ph]

    2012 arXiv

  29. [37]

    Wetterich, Information transport in classical sta- tistical systems, Nucl

    C. Wetterich, Information transport in classical sta- tistical systems, Nucl. Phys. B927, 35 (2018), arXiv:1611.04820 [cond-mat.stat-mech]

    2018 arXiv

  30. [38]

    Wetterich, Quantum formalism for classical statis- tics, Annals of Physics393, 1 (2018), arXiv:1706.01772 [quant-ph]

    C. Wetterich, Quantum formalism for classical statis- tics, Annals of Physics393, 1 (2018), arXiv:1706.01772 [quant-ph]

    2018 arXiv

  31. [39]

    Wetterich,The Probabilistic World(Springer Nature, Heidelberg, 2025) arXiv:2011.02867 [quant-ph]

    C. Wetterich,The Probabilistic World(Springer Nature, Heidelberg, 2025) arXiv:2011.02867 [quant-ph]

    2025 arXiv

  32. [40]

    Wetterich, The probabilistic world II : Quantum mechanics from classical statistics, arXiv e-prints , arXiv:2408.06379 (2024)

    C. Wetterich, The probabilistic world II : Quantum mechanics from classical statistics, arXiv e-prints , arXiv:2408.06379 (2024)

    2024 arXiv

  33. [41]

    Ulam, Random processes and transformations, inPro- ceedings of the International Congress of Mathemati- cians, Vol

    S. Ulam, Random processes and transformations, inPro- ceedings of the International Congress of Mathemati- cians, Vol. 2 (1950) pp. 264–275

    1950

  34. [42]

    von Neumann, The general and logical theory of au- tomata, inCerebral Mechanisms in Behavior; The Hixon Symposium(Wiley, Oxford, England, 1951) pp

    J. von Neumann, The general and logical theory of au- tomata, inCerebral Mechanisms in Behavior; The Hixon Symposium(Wiley, Oxford, England, 1951) pp. 1–41

    1951

  35. [43]

    Zuse,Rechnender Raum(Vieweg, Teubner Verlag, 1969)

    K. Zuse,Rechnender Raum(Vieweg, Teubner Verlag, 1969)

    1969

  36. [44]

    Wolfram, Statistical mechanics of cellular automata, Rev

    S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys.55, 601 (1983)

    1983

  37. [45]

    t’ Hooft, The Cellular Automaton Interpretation of Quantum Mechanics, arXiv e-prints , arXiv:1405.1548 [quant-ph] (2014)

    G. t’ Hooft, The Cellular Automaton Interpretation of Quantum Mechanics, arXiv e-prints , arXiv:1405.1548 [quant-ph] (2014)

    2014 arXiv

  38. [46]

    Gozzi, Hidden brs invariance in classical mechanics, Phys

    E. Gozzi, Hidden brs invariance in classical mechanics, Phys. Lett. B201, 525 (1988)

    1988

  39. [47]

    Gozzi, M

    E. Gozzi, M. Reuter, and W. D. Thacker, Hidden brs invariance in classical mechanics. ii, Phys. Rev. D40, 3363 (1989)

    1989

  40. [48]

    Wetterich, Fermions as generalized ising models, Nu- clear Physics B917, 241 (2017), arXiv:1612.06695 [cond- mat.stat-mech]

    C. Wetterich, Fermions as generalized ising models, Nu- clear Physics B917, 241 (2017), arXiv:1612.06695 [cond- mat.stat-mech]

    2017 arXiv

  41. [49]

    Wetterich, Fermionic quantum field theories as prob- abilistic cellular automata, Phys

    C. Wetterich, Fermionic quantum field theories as prob- abilistic cellular automata, Phys. Rev. D105, 074502 (2022), arXiv:2111.06728 [hep-lat]

    2022 arXiv

  42. [50]

    Wetterich, Complex wave functions, CPT and quantum field theory for classical generalized Ising models, Nuclear Physics B1022, 117204 (2026), arXiv:2505.24392 [quant-ph]

    C. Wetterich, Complex wave functions, CPT and quantum field theory for classical generalized Ising models, Nuclear Physics B1022, 117204 (2026), arXiv:2505.24392 [quant-ph]

    2026

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