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arxiv: 2605.15969 · v1 · pith:M3VCCD5Qnew · submitted 2026-05-15 · 🪐 quant-ph · cond-mat.stat-mech· nlin.CG

Quantum mechanics for classical transport equations

Christof Wetterich This is my paper

Pith reviewed 2026-05-20 18:15 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechnlin.CG
keywords classical transport equationsquantum mechanicsSchrödinger equationprobabilistic automataunitary evolutionnon-commuting operatorsfunctional integralclassical wave function
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The pith

Classical transport equations with probabilistic initial conditions realize quantum mechanics through a wave function obeying the Schrödinger equation.

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

The paper establishes that classical transport equations carrying probabilistic initial conditions can be recast as quantum systems. Time-local probabilities are captured by a classical wave function that evolves unitarily according to a Schrödinger equation. Statistical observables correspond to operators that fail to commute with classical ones, such as functions of energy or angular momentum which remain conserved. This framework constructs a complex functional integral and demonstrates that quantum features including superposition, interference, and phase dependence emerge directly from classical probabilistic flows.

Core claim

Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function whose unitary evolution obeys a Schrödinger equation. Statistical observables are represented by operators which do not commute with the ones associated to classical observables. Examples are functions of the quantum energy or the quantum angular momentum. We construct a complex functional integral for the quantum system which describes the probabilistic classical transport equation. The characteristic features of quantum mechanics, as the superposition of 1

What carries the argument

The classical wave function encoding time-local probabilistic information whose unitary evolution obeys a Schrödinger equation while permitting non-commuting operators for statistical observables.

If this is right

  • Functions of the quantum energy and quantum angular momentum act as conserved quantities represented by non-commuting operators.
  • Superposition of wave functions and interference effects appear directly in the probability flows of the classical system.
  • A complex functional integral supplies a path-integral formulation of the probabilistic transport dynamics.
  • Unitary time evolution and phase dependence govern the statistical evolution without extra postulates.

Where Pith is reading between the lines

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

  • Quantum mechanics may emerge as an effective description from suitably chosen classical probabilistic transport rules.
  • Discrete probabilistic automata could be used to simulate quantum interference and non-commuting observables in classical hardware.
  • The same mapping might extend to continuous or field-theoretic transport equations, linking quantum field theory to classical statistics.
  • Numerical tests on concrete models such as diffusion processes or lattice hopping would reveal whether the non-commuting operators produce observable statistical signatures.

Load-bearing premise

The probabilistic content of the classical transport equation can be faithfully encoded in a complex wave function whose unitary Schrödinger evolution exactly reproduces the original probability flow while allowing non-commuting operators for statistical observables without introducing inconsistencies in the classical interpretation.

What would settle it

A concrete computation on a discrete probabilistic automaton in which the wave-function evolution produces probability distributions that deviate from those required by the original transport rule or in which expectation values of non-commuting operators contradict the classical conservation laws.

read the original abstract

Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schr\"odinger equation. Statistical observables are represented by operators which do not commute with the ones associated to classical observables. Examples are functions of the quantum energy or the quantum angular momentum. They are important conserved quantities. We construct a complex functional integral for the quantum system which describes the probabilistic classical transport equation. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In the discrete case these are probabilistic automata. The time-local probabilistic information is encoded in a complex classical wave function whose unitary evolution obeys a Schrödinger equation. Statistical observables are represented by operators that do not commute with those associated to classical observables, with examples including functions of the quantum energy or quantum angular momentum as conserved quantities. A complex functional integral is constructed for the quantum system, and the authors argue that characteristic quantum features such as superposition, interference, phases, non-commuting operators and unitary time evolution are realized by probabilistic classical transport equations.

Significance. If the proposed construction were valid, the work would establish a direct correspondence allowing quantum-mechanical concepts and tools, including non-commuting operators and a functional-integral formulation, to be applied to classical probabilistic transport. This could provide new perspectives on conserved quantities in transport problems.

major comments (1)
  1. [Abstract] Abstract, paragraph 2 and the central construction: the claim that a complex wave function ψ with p = |ψ|^2 evolves under a unitary Schrödinger equation i∂tψ = Hψ such that the induced dynamics on p exactly reproduces the original classical transport operator L (∂tp = Lp) for arbitrary initial phases is not supported. For a unitary matrix M the expansion of |Mψ|^2 contains diagonal terms ∑|Mkj|^2 pj plus off-diagonal interference terms 2Re(Mki conj(Mkj) ψi conj(ψj)) for i≠j. These phase-dependent cross terms vanish for all initial phases only if M is a monomial (permutation) matrix, i.e., the dynamics is deterministic. This contradicts the probabilistic character of the transport equations under consideration and renders the mapping internally inconsistent.
minor comments (1)
  1. The relation between the classical wave function and the underlying transport equation would benefit from an explicit low-dimensional example (e.g., a two-state probabilistic automaton) showing the explicit form of H and verification that probability is conserved without phase dependence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their thorough review and for highlighting an important aspect of our central construction. We address the major comment in detail below and propose revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2 and the central construction: the claim that a complex wave function ψ with p = |ψ|^2 evolves under a unitary Schrödinger equation i∂tψ = Hψ such that the induced dynamics on p exactly reproduces the original classical transport operator L (∂tp = Lp) for arbitrary initial phases is not supported. For a unitary matrix M the expansion of |Mψ|^2 contains diagonal terms ∑|Mkj|^2 pj plus off-diagonal interference terms 2Re(Mki conj(Mkj) ψi conj(ψj)) for i≠j. These phase-dependent cross terms vanish for all initial phases only if M is a monomial (permutation) matrix, i.e., the dynamics is deterministic. This contradicts the probabilistic character of the transport equations under consideration and renders the mapping internally inconsistent.

    Authors: We thank the referee for this precise mathematical observation, which correctly identifies that a generic unitary evolution would introduce phase-dependent interference in the probability distribution. In our manuscript, the complex wave function is not an arbitrary complex vector; rather, it is specifically constructed to encode the probabilistic information together with phases that are compatible with the transport dynamics. The Hamiltonian is designed such that the unitary operator effectively acts as a stochastic map on the probabilities while preserving unitarity on the wave function level. However, we recognize that the original presentation in the abstract could be interpreted as implying independence from initial phases. To address this, we will revise the abstract and the main text to explicitly state that the phases are chosen in accordance with the classical transport process, ensuring that the induced probability evolution matches the classical operator without interference artifacts. This clarification maintains the quantum-like features for the chosen encoding while acknowledging the referee's point on the general case. revision: yes

Circularity Check

2 steps flagged

Unitary Schrödinger evolution and non-commuting operators introduced by wave-function encoding rather than derived from classical transport

specific steps
  1. self definitional [Abstract, paragraph 2]
    "The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables are represented by operators which do not commute with the ones associated to classical observables."

    The wave function is defined so that its modulus squared supplies the classical probability density; the unitary Schrödinger evolution is then asserted to hold for this wave function, making the Schrödinger equation and the non-commuting operator algebra hold by the choice of encoding rather than by independent derivation from the classical transport operator L.

  2. fitted input called prediction [Abstract, paragraph 2 and construction of functional integral]
    "We construct a complex functional integral for the quantum system which describes the probabilistic classical transport equation. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations."

    The functional integral and the listed quantum features are obtained by mapping the classical transport onto the chosen wave-function representation; the reproduction of the original probability flow is therefore enforced by construction of the map rather than emerging as a prediction from unmodified classical dynamics.

full rationale

The paper encodes classical probabilities p in a complex wave function ψ with p = |ψ|^2 and asserts that this ψ obeys a unitary Schrödinger equation whose induced dynamics on p reproduces the original linear transport operator L. This encoding choice directly supplies the unitary evolution, superposition, phases, and non-commuting operators; the claimed reproduction of arbitrary probabilistic transport cannot hold for generic stochastic T because interference cross-terms appear in |Mψ|^2 unless M is monomial (deterministic). The central features are therefore imposed by the representation rather than independently derived, producing partial circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that probabilistic information admits a complex encoding whose unitary dynamics matches the original transport equation exactly. No free parameters or new invented entities are introduced in the abstract; the mapping itself supplies the quantum structure.

axioms (1)
  • domain assumption Probabilistic initial conditions of a classical transport equation can be encoded in a complex wave function without loss of classical content.
    This encoding is the step that allows the Schrödinger equation and non-commuting operators to be defined.

pith-pipeline@v0.9.0 · 5645 in / 1292 out tokens · 52166 ms · 2026-05-20T18:15:24.233073+00:00 · methodology

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Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 23 internal anchors

  1. [1]

    Quantum correlations in classical statistics

    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

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  2. [2]

    Probabilistic observables, conditional correlations, and quantum physics

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

    work page internal anchor Pith review Pith/arXiv 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,

    work page

  4. [4]

    Emergence of quantum mechanics from classical statistics

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

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    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)

    work page arXiv 2024

  6. [6]

    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]

    work page arXiv 2022

  7. [7]

    The Cellular Automaton Interpretation of Quantum Mechanics

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

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  8. [8]

    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]

    work page arXiv 2026

  9. [9]

    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)

    work page arXiv 2026

  10. [10]

    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

    work page doi:10.1073/pnas.17.5.315 1931

  11. [11]

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

    work page 1932

  12. [12]

    Gozzi, Hidden brs invariance in classical mechanics, Phys

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

    work page 1988

  13. [13]

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

    work page 1998

  14. [14]

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

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  15. [15]

    On Koopman-von Neumann Waves

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

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  16. [16]

    On Koopman-von Neumann Waves II

    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]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  17. [17]

    Classical mechanics without determinism

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

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  18. [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)

    work page 2006

  19. [19]

    Classical mechanics as nonlinear quantum mechanics

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

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  20. [20]

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

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  21. [21]

    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]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  22. [22]

    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)

    work page 2013

  23. [23]

    Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator

    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]

    work page internal anchor Pith review Pith/arXiv arXiv 2096

  24. [24]

    Data-driven spectral analysis of the Koopman operator

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

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  25. [25]

    From Koopman-von Neumann Theory to Quantum Theory

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

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  26. [26]

    Data-driven approximations of dynamical systems operators for control

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

    work page internal anchor Pith review Pith/arXiv arXiv 1902

  27. [27]

    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]

    work page arXiv 2020

  28. [28]

    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]

    work page arXiv 2091

  29. [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)

    work page 2021

  30. [30]

    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]

    work page arXiv 2022

  31. [31]

    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 9 11, 193 (2024), arXiv:2211.01838 [math-ph]

    work page arXiv 2024

  32. [32]

    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)

    work page 2024

  33. [33]

    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]

    work page arXiv 2024

  34. [34]

    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]

    work page arXiv 2024

  35. [35]

    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)

    work page 2025

  36. [36]

    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-prints , arXiv:2603.06872 (2026)

    work page arXiv 2026

  37. [37]

    Gozzi, M

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

    work page 1989

  38. [38]

    Time Evolution of Non-Equilibrium Effective Action

    C. Wetterich, Time evolution of non-equilibrium effec- tive action, Physical Review Letters78, 3598 (1997), arXiv:hep-th/9612206 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  39. [39]

    Non-Equilibrium Time Evolution in Quantum Field Theory

    C. Wetterich, Non-equilibrium time evolution in quan- tum field theory, Physical Review E56, 2687 (1997), arXiv:hep-th/9703006 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  40. [40]

    Nonequilibrium Quantum Fields: From Cold Atoms to Cosmology

    J. Berges, Nonequilibrium Quantum Fields: From Cold Atoms to Cosmology, arXiv e-prints , arXiv:1503.02907 [hep-ph] (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  41. [41]

    Quantum correlations from incomplete classical statistics

    C. Wetterich, Quantum correlations from incomplete classical statistics, arXiv e-prints , hep-th/0104074 (2001)

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  42. [42]

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

    work page 1964

  43. [43]

    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)

    work page 1969

  44. [44]

    Quantum particles from classical probabilities in phase space

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

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  45. [45]

    Quantum particles from coarse grained classical probabilities in phase space

    C. Wetterich, Quantum particles from coarse grained classical probabilities in phase space, Annals Phys.325, 1359 (2010), arXiv:1003.3351 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  46. [46]

    Information transport in classical statistical systems

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

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  47. [47]

    Quantum formalism for classical statistics

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

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  48. [48]

    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]

    work page arXiv 2025

  49. [49]

    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

    work page 1950

  50. [50]

    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

    work page 1951

  51. [51]

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

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

    work page 1969

  52. [52]

    Wolfram, Statistical mechanics of cellular automata, Rev

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

    work page 1983

  53. [53]

    P. C. Martin, E. D. Siggia, and H. A. Rose, Statistical dy- namics of classical systems, Phys. Rev. A8, 423 (1973)

    work page 1973

  54. [54]

    H.-K. Janssen, On a lagrangean for classical field dynam- ics and renormalization group calculations of dynamical critical properties, Zeitschrift f¨ ur Physik B: Condensed Matter23, 377 (1976)

    work page 1976

  55. [55]

    De Dominicis, Techniques de renormalisation de la th´ eorie des champs et dynamique des ph´ enom` enes cri- tiques, Journal de Physique Colloques37, C1 (1976)

    C. De Dominicis, Techniques de renormalisation de la th´ eorie des champs et dynamique des ph´ enom` enes cri- tiques, Journal de Physique Colloques37, C1 (1976)

    work page 1976