Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Quantum Systems as Indivisible Stochastic Processes

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper argues that every quantum system can be understood as an indivisible stochastic process in configuration space, so that Hilbert spaces and wave functions become optional mathematical bookkeeping.

desk verdict A clear, candid review of Barandes's own stochastic-quantum program, with a genuinely new but modest gauge result and a load-bearing unitarity claim that does not survive contact with the Stinespring dilation. read the letter →

arxiv 2507.21192 v1 pith:DKRLR2FA submitted 2025-07-27 quant-ph physics.hist-ph

Jacob A. Barandes This is my paper · ORCID
classification quant-phphysics.hist-ph
keywords indivisiblestochasticprocessesstochastic-quantumcorrespondencenon-MarkovianityquantumfoundationsgaugeinvarianceHilbert-spacedilationsmeasurementproblemunitarytimeevolution
topics The Measurement Problem
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 argues that quantum systems can be understood as indivisible stochastic processes: probabilistic dynamics on an ordinary configuration space whose laws are not required to decompose into independent steps over time. It claims that such processes reproduce all the empirical predictions of quantum theory, including interference, entanglement, noncommuting observables, and Born-rule statistics, while avoiding the measurement problem and the category problem. If the claim is right, Hilbert spaces, wave functions, and density matrices are demoted to convenient, gauge-dependent mathematical tools, and quantum theory becomes a realist stochastic theory in configuration space. The paper also presents new gauge invariances and Hilbert-space dilations that follow from this correspondence.

What carries the argument

The load-bearing object is the indivisible stochastic process, defined by first-order transition probabilities $\Gamma_{ij}(t\leftarrow t_0)$ connecting a target time to allowed division events, with no guarantee that $\Gamma$ factors through intermediate times. The dictionary $\Gamma_{ij}(t\leftarrow 0)=\mathrm{tr}(\Theta^\dagger(t\leftarrow 0)P_i\Theta(t\leftarrow 0)P_j)$ converts this process into Hilbert-space language; the modulus-squared factorization $\Gamma_{ij}=|\Theta_{ij}|^2$ and the Stinespring dilation theorem then turn $\Theta$ into a unitary operator, yielding linear, unitary quantum time evolution. Two gauge structures\u2014Schur-Hadamard entrywise phase redefinitions and Foldy-Wouthuysen time-dependent unitary transformations\u2014show that the Hilbert-space quantities are gauge-dependent encodings, while the stochastic process remains the invariant core.

What would settle it

A concrete counterexample would be a finite indivisible stochastic process\u2014defined by a stochastic matrix $\Gamma(t\leftarrow 0)$ with nonnegative entries and unit column sums, not factoring through intermediate times\u2014that is neither unistochastic nor embeddable as a subsystem of any finite-dimensional unistochastic matrix; finding one would falsify the claimed stochastic-quantum theorem. Conversely, a numerical search over small $N$ confirming that every such $\Gamma$ admits a unitary dilation would strengthen the correspondence.

Watch

Extended reading notes

Core claim

The central discovery is the stochastic-quantum correspondence: a bidirectional mapping between indivisible stochastic processes and quantum systems. In one direction, given transition probabilities $\Gamma_{ij}(t\leftarrow 0)=p(i,t|j,0)$ for an indivisible stochastic process, writing $\Gamma_{ij}=|\Theta_{ij}|^2$ and $p_i(t)=\mathrm{tr}(P_i\rho(t))$ yields the Born rule, linearity, and\u2014after a Stinespring dilation if needed\u2014a unitary time-evolution operator $U(t\leftarrow 0)$, from which the Schr\"odinger and von Neumann equations follow. In the other direction, any unitarily evolving quantum system is an indivisible stochastic process in disguise, with configurations as beables and measurements as ordinary stochastic interactions. The paper shows that the Hilbert-space description carries two independent gauge freedoms\u2014Schur-Hadamard phase transformations and Foldy-Wouthuysen transformations\u2014so its ingredients are not ontologically privileged, and that dilations of the Hilbert space can expose emergent observables such as spin without introducing preferred directions.

Load-bearing premise

The load-bearing premise is that any indivisible stochastic process can be dilated into one whose transition probabilities are the squared magnitudes of a unitary matrix; if some valid process cannot be unitarized in this way, the derivation of unitary quantum evolution collapses.

Editorial extensions

If this is right

  • If the correspondence holds, the unitary Schr\"odinger equation and the Born rule are derived consequences of a stochastic law of total probability, not independent postulates.
  • The measurement problem dissolves: measurements are just stochastic interactions between the system and a measuring device, with outcomes distributed by the same transition probabilities that govern all other dynamics.
  • The category problem dissolves as well: the same stochastic laws account for non-measurement phenomena, so there is no need to divide the world into measurements and ordinary evolutions.
  • Hilbert-space dilations become a free design resource: one can enlarge the internal Hilbert space to generate emergent observables such as spin, so spin need not be added as a primitive degree of freedom.
  • The framework predicts that closed quantum systems exhibit intrinsic non-Markovianity of the kind identified by Glick and Adami, and that this non-Markovianity is observable in suitable experiments.

Reading between the lines

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

  • If indivisible stochastic processes are the primitive ontology, quantum theory may be formulated as a classical stochastic process with memory, opening a direct route to stochastic simulation algorithms that avoid the sign problem of path-integral Monte Carlo.
  • The same dictionary suggests a concrete research program: search for finite transition matrices that are valid indivisible processes but provably not unistochastic even after dilation; a proof that none exist would complete the claimed correspondence, while an example would delimit it.
  • The gauge treatment of the Hamiltonian suggests that energy is a gauge-dependent quantity in this framework, which may have consequences for how energy conservation is understood in cosmological or gravitational settings.
  • Because the paper's dynamics need not be Markovian, Bell-type locality arguments may need re-examination; the author's promised future work on this point could either rescue locality in space at the cost of temporal non-Markovianity or produce new empirical signatures.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes that quantum systems can be understood as 'indivisible stochastic processes' on a configuration space, governed only by first-order transition probabilities and the law of total probability. It introduces a complex 'time-evolution operator' Θ via Γ_ij = |Θ_ij|^2, rewrites the transition matrix in Hilbert-space form as tr(Θ† P_i Θ P_j), and claims that after a Stinespring dilation one may take Θ to be unitary. From this it derives the Schrödinger and von Neumann equations, identifies Schur-Hadamard and Foldy-Wouthuysen gauge invariances, discusses dynamical symmetries, and proposes Hilbert-space dilations as a way to represent emergent observables such as spin. The paper argues that wave functions and Hilbert spaces are gauge-dependent mathematical tools and that the stochastic formulation evades the measurement and category problems.

Significance. If the central claims were correct, the paper would offer a realist, configuration-space reinterpretation of quantum theory with no fundamental wave function, and it would supply a first-principles derivation of unitary time evolution. The paper also contains some correct and possibly useful formal material: the trace identity in Eq. (39) is an algebraic rewriting of Eq. (25), the Schur-Hadamard gauge freedom in Eq. (29) is a genuine invariance of the dictionary, and the dilation construction in Sec. 4.2 generalizes the representation in a coherent way. However, the advertised derivation of unitarity is unsupported and the correspondence is largely built by definition rather than derived. As a result, the foundational significance claimed in the abstract and introduction is not achieved; what remains is a formal dictionary whose interpretive force is much weaker than presented.

major comments (3)
  1. [§3.5, Eqs. (58)–(62)] The inference from the Kraus decomposition and Stinespring dilation to unitarity of the original time-evolution operator is invalid. The Kraus operators K_β = ΘP_β define a completely positive trace-preserving map, not a unitary operator. Stinespring dilation constructs a unitary on a larger Hilbert space of dimension at most N^3; it does not make the original N×N matrix Θ unitary. Indeed, every unistochastic matrix is doubly stochastic, so a column-stochastic matrix with unequal row sums, such as Γ = [[1, 1/2], [0, 1/2]], cannot be written as |U_ij|^2 for any 2×2 unitary U. Therefore Eq. (64) and the statement that 'one can always assume that the system's time-evolution operator Θ(t←0) is unitary' do not follow. The correct statement is the partial-trace form in Eq. (110), which describes the original process as a subsystem of a unistochastic process and does not imply that the original system evolves unitarily. This error is load-bearing because the claimed 'first-principles motivation for unitary time evolution' and the stochastic-quantum theorem as stated in §3.5 rest on it.
  2. [§3.1–§3.3, Eqs. (25), (39)] The central relation Γ_ij = |Θ_ij|^2 and the derived dictionary tr(Θ† P_i Θ P_j) are algebraic identities, not dynamical postulates with independent content. For any column-stochastic matrix Γ, one can always choose Θ_ij = √Γ_ij times arbitrary phases, so the correspondence is satisfied by construction. This means the paper does not derive that quantum systems must be indivisible stochastic processes; it shows only that the two-time transition probabilities of any quantum evolution can be repackaged in the stochastic language. The interpretation claim that quantum theory 'becomes' a classical-looking stochastic theory therefore rests on a definitional maneuver unless additional physical constraints are supplied. The paper should explicitly separate the mathematical representation theorem from the stronger physical claim that the stochastic process is the fundamental ontology.
  3. [§4.2, Eq. (110) and §5, 'Dynamical axiom'] The paper's own later caveat in §4.2 contradicts the unqualified unitarity claim in §3.5. Eq. (110) expresses Γ_ij through a unitary on a dilated space after a partial trace, which is the standard Stinespring form and applies to arbitrary CPTP maps. This does not make the original system's evolution unitary. The 'Dynamical axiom' in §5 and the claim that the framework provides 'a first-principles way to understand ... why the time evolution of closed quantum systems is ... unitary' are therefore unsupported. The manuscript should either retract the unitarity derivation or reformulate the main theorem as a subsystem statement throughout, and should assess whether the advertised consequences for the measurement problem and the category problem survive that weakened statement.
minor comments (5)
  1. [§3.5, 'as shown in other work'] The key stochastic-quantum theorem and the Stinespring step are deferred to previous papers (Barandes 2023, 2025). Since the present paper claims to 'initiate a deeper investigation' and presents the theorem as central, the proof should be either reproduced or stated with enough detail to be checked in this manuscript.
  2. [§4.1, Eq. (89)] The anti-unitary symmetry condition is written as VΘV† = Θ with an overline, but the overline is not defined in the surrounding text. The subsequent redefinition V→V and Eq. (90) clarify the intent, but the notation should be introduced explicitly.
  3. [§3.4, 'by an re-appropriation of notation'] This phrase contains a grammatical error; it should be 'by a re-appropriation of notation.'
  4. [§4.2, 'Dilation-emergeables'] The notion of dilation-emergeables is introduced but not defined with the same precision as beables and emergeables in §3.4. A formal definition and at least one concrete example would help the reader assess whether this concept carries new content.
  5. [§1, 'non-Markovianity'] The relationship between the paper's indivisibility and the standard notions of non-Markovianity from open quantum systems is discussed only informally. A precise statement of how indivisibility differs from divisibility of quantum channels would improve accessibility.

Circularity Check

3 steps flagged · score 8.0 of 10

The stochastic-quantum correspondence is built by definition (Gamma = |Theta|^2), while the claimed first-principles derivation of unitarity imports Stinespring dilation and self-cites the stochastic-quantum theorem instead of deriving it.

  1. self definitional [Section 3.1, Eq. (25); Section 3.3, Eq. (39)]
    "Then solve the inequality Γij(t←0)≥0 appearing in (9) by introducing a “potential” consisting of complex-valued matrix elements Θij(t←0) related to Γ ij(t←0) according to the modulus-squaring operation: Γij(t←0) =|Θ ij(t←0)| 2. Note that this formula is not a postulate, but an identity... Γij(t←0) = tr(Θ †(t←0)P iΘ(t←0)P j). This formula provides the dictionary that translates between indivisible stochastic processes... and the formalism of quantum-theoretic Hilbert spaces."

    The dictionary (39) is just the identity (25) rewritten with configuration projectors: any nonnegative Γ admits a Θ by taking entrywise square roots, and any unitary U yields a stochastic Γ via Γ_ij=|U_ij|^2. Therefore the 'correspondence' is an identity by construction; it cannot fail to accommodate quantum mechanics and it derives no constraint from the stochastic axioms. What is presented as a discovery, the stochastic-quantum correspondence, is actually the definition of Θ, so the central representation claim is self-definitional.

  2. ansatz smuggled in via citation [Section 3.5, Eqs. (62)-(64); cf. Section 4.2, Eq. (110)]
    "As shown in other work (Barandes 2025, 2023), and as will be explained in detail later in the present paper, the existence of these Kraus decompositions has an important implication. Specifically, after an application, if necessary, of the Stinespring dilation theorem (Stinespring 1955, Keyl 2002), which involves expanding the original N-dimensional Hilbert space to a ‘dilated’ Hilbert space of dimension no greater than N3, one can always assume that the system’s time-evolution operator Θ(t←0) is unitary... It follows that the basic relationship (25) now takes the form Γij(t←0) =|U ij(t←0)| 2"

    Stinespring dilation unitarizes an enlarged, dilated system; it does not make the original N×N matrix Θ unitary. Most column-stochastic matrices are not unistochastic, so Eq. (64), which asserts Γ_ij=|U_ij|^2 for an N×N unitary U, does not follow. The paper itself later retreats to the partial-trace form (110) and to calling the process only a 'subsystem of a unistochastic process.' Thus the advertised 'first-principles motivation for unitary time evolution' is not derived from indivisibility; unitarity is imported as an ansatz via a dilation theorem and via citation to the author's prior work.

1 more flagged steps
  1. self citation load bearing [Section 3.5, after Eq. (65)]
    "The preceding analysis implies that an indivisible stochastic process can be viewed either as a unistochastic process itself, or (if a nontrivial dilation was required) as a subsystem of a unistochastic process. This statement is called the stochastic-quantum theorem (Barandes 2023)."

    The paper's central existence claim, the stochastic-quantum theorem, is asserted by citing the author's own earlier work rather than proved in this paper. It is load-bearing because it is what licenses the claim that every indivisible stochastic process corresponds to a quantum system. Moreover, the cited theorem's statement is strictly weaker than Eq. (62): it ends at 'subsystem of a unistochastic process,' which is the partial-trace form (110), not the unqualified unitary evolution of the original system. The self-citation therefore carries the weight of the paper's strongest conclusion without independent verification here.

full rationale

The central derivation chain is circular in two related ways. First, the stochastic-quantum dictionary is not a substantive bridge law but an identity: Γ_ij = |Θ_ij|^2 is always solvable by choosing Θ_ij = sqrt(Γ_ij), and the projector trace form (39) is exactly that same identity in Hilbert-space clothing. Consequently, every stochastic transition matrix automatically has a Hilbert-space 'time-evolution operator,' and every unitary matrix automatically produces a stochastic process via Γ_ij=|U_ij|^2. The claimed correspondence is therefore guaranteed by construction and adds no independent empirical or dynamical constraint; any quantum unitary qualifies, so the framework cannot be falsified by accommodation. Second, the paper's signature first-principles result, that unitarity follows from indivisible stochasticity, is not obtained from the stochastic axioms. The step at Eq. (62) applies Stinespring dilation and then asserts that the system's own Θ is unitary, but Stinespring only yields a unitary on a dilated space; Eq. (64) would require the original transition matrix to be unistochastic, which is false for generic column-stochastic matrices. The paper's own later Eq. (110) gives only a partial-trace form, undercutting the unqualified Eq. (64). That step is further supported by the self-cited 'stochastic-quantum theorem' (Barandes 2023), which is load-bearing and not proved in this paper. The paper does contain independent mathematical material, such as Schur-Hadamard and Foldy-Wouthuysen gauge structures and dilation constructions, but these are elaborations on the already definitional dictionary rather than evidence that the correspondence or unitarity is derived from first principles. Overall, the strongest claims reduce to definitions plus self-citation, warranting a score of 8.

Assumptions & free parameters 1 free parameters · 4 assumptions · 3 invented entities

The central correspondence rests on a definitional identity (Gamma_ij = |Theta_ij|^2) plus standard theorems (Kraus, Stinespring). The interpretive axioms (configuration-space realism, transition probabilities only at division events, epistemic standalone probabilities) are domain assumptions. No numerical parameters are fitted, but the full dynamics is carried by the chosen transition matrix, which is an input rather than a derivation.

free parameters (1)
  • Transition matrix Gamma(t<-0) and potential Theta(t<-0) = unspecified; chosen so that Gamma_ij(t<-0)=|U_ij(t<-0)|^2 for the target unitary U
    Eq. (64) sets the stochastic dynamics equal to the modulus-squared unitary of the quantum system being modeled. The empirical content lives in this choice, which is an input from ordinary quantum theory, not a prediction of the stochastic framework.
assumptions (4)
  • standard math Stinespring dilation theorem and Kraus representation theorem.
    Invoked in Section 3.5 to write any CPTP map as a unitary on a dilated Hilbert space and to justify the stochastic-quantum theorem.
  • domain assumption Configuration space and standalone probabilities are the fundamental ontology; transition probabilities have objective chance status.
    The kinematical and epistemic axioms in Section 5 are stated, not derived, and carry the realist interpretation.
  • domain assumption Transition probabilities need be defined only for conditioning times called division events, not for all intermediate times.
    Section 2.2 defines indivisible processes this way; this assumption is what prevents divisibility and makes the process non-Markovian.
  • ad hoc to paper A given quantum system is fully captured by its two-time transition matrix in the configuration basis.
    The dictionary and all subsequent claims assume this reduction; the paper does not prove it for arbitrary measurements of emergeables except by reference to prior work.
invented entities (3)
  • Indivisible stochastic process
    purpose: A stochastic law with only first-order transition probabilities at division events, lacking the divisibility condition.
    This is a new category of process introduced to house quantum systems; no independent measurement distinguishes it from an ordinary non-Markovian process.
  • Division events
    purpose: Special conditioning times at which the transition matrix is defined; their existence is assumed and may be generated by interactions.
    Invented to allow a sparse set of transition laws without a full Kolmogorov tower; no independent empirical signature is specified.
  • Dilation-emergeables
    purpose: Observables (e.g., spin) that appear only after dilating the Hilbert space but are claimed to pre-exist as patterns in the stochastic dynamics.
    The paper gives no criterion for identifying which dilation observables correspond to real patterns; the spin example is illustrative.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Systems as Indivisible Stochastic Processes." pith.science (2026). https://pith.science/paper/DKRLR2FA

@misc{pith2026250721192,
  author       = {Pith},
  title        = {Pith review of: Quantum Systems as Indivisible Stochastic Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DKRLR2FA}},
  note         = {Machine review of arXiv:2507.21192}
}
read the original abstract

According to the stochastic-quantum correspondence, a quantum system can be understood as a stochastic process unfolding in an old-fashioned configuration space based on ordinary notions of probability and `indivisible' stochastic laws, which are a non-Markovian generalization of the laws that describe a textbook stochastic process. The Hilbert spaces of quantum theory and their ingredients, including wave functions, can then be relegated to secondary roles as convenient mathematical appurtenances. In addition to providing an arguably more transparent way to understand and modify quantum theory, this indivisible-stochastic formulation may lead to new possible applications of the theory. This paper initiates a deeper investigation into the conceptual foundations and structure of the stochastic-quantum correspondence, with a particular focus on novel forms of gauge invariance, dynamical symmetries, and Hilbert-space dilations.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

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. Interpreting Quantum Learning Models via Stochastic Processes

    quant-ph 2026-07 conditional novelty 5.0 of 10

    Quantum dynamics admits two complementary stochastic representations — Markovian with negative transition weights under complete measurements, or positive but memoryful under projective measurements — a trade-off illu...

Reference graph

Works this paper leans on

75 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [1]

    Quantum Computing and Hidden Variables

    S. Aaronson. “Quantum Computing and Hidden Variables”.Physical Review A, 71(3):032325, March 2005.doi:10.1103/PhysRevA.71.032325

    work page doi:10.1103/physreva.71.032325 2005

  2. [2]

    The Stochastic-Quantum Theorem

    J. A. Barandes. “The Stochastic-Quantum Theorem”, 2023. URL:https://arxiv. org/abs/2309.03085,arXiv:2309.03085

    arXiv 2023

  3. [3]

    The Stochastic-Quantum Correspondence

    J. A. Barandes. “The Stochastic-Quantum Correspondence”.Philosophy of Physics, 3(1):8, June 2025.arXiv:2302.10778,doi:10.31389/pop.186

    arXiv 2025

  4. [4]

    J. D. Bjorken and S. D. Drell.Relativistic Quantum Mechanics. International Series in Pure and Applied Physics. McGraw-Hill Book Company, New York, 1964

    work page 1964

  5. [5]

    On the Einstein-Podolsky-Rosen Paradox

    J. S. Bell. “On the Einstein-Podolsky-Rosen Paradox”.Physics, 1(3):195–200, 1964

    work page 1964

  6. [6]

    On the Problem of Hidden Variables in Quantum Mechanics

    J. S. Bell. “On the Problem of Hidden Variables in Quantum Mechanics”.Reviews of Modern Physics, 38(3):447–452, July 1966.doi:10.1103/RevModPhys.38.447

    doi:10.1103/revmodphys.38.447 1966

  7. [7]

    Introduction to the Hidden-Variable Question

    J. S. Bell. “Introduction to the Hidden-Variable Question”. In B. d’Espagnat, editor, Foundations of Quantum Mechanics, Volume Course IL of49th International School of Physics “Enrico Fermi”, pages 171–181. Academic Press, Varenna, Italy, 1971. Report No. CERN-TH-1220

    work page 1971

  8. [8]

    Subject and Object

    J. S. Bell. “Subject and Object”. In J. Mehra, editor,The Physicist’s Conception of Nature, pages 687–690. D. Reidel, Dordrecht, 1973. Report No. CERN-TH-1582. 30

    work page 1973

Show all 75 references
  1. [9]

    The Theory of Local Beables

    J. S. Bell. “The Theory of Local Beables”.Epistemological Letters, 9:11–24, 1976

    1976

  2. [10]

    On the Impossible Pilot Wave

    J. S. Bell. “On the Impossible Pilot Wave”.Foundations of Physics, 12:989–999, October 1982.doi:10.1007/BF01889272

    1982 doi

  3. [11]

    La Nouvelle Cuisine

    J. S. Bell. “La Nouvelle Cuisine”. In A. Sarlemijn and P. Kroes, editors,Between Science and Technology, pages 97–115. Elsevier, 1990

    1990

  4. [12]

    Dynamical Reduction Models

    A. Bassi and G. Ghirardi. “Dynamical Reduction Models”.Physical Reports, 379(5):257–426, 2003. URL:https://www.sciencedirect.com/science/article/ pii/S0370157303001030,arXiv:quant-ph/0302164,doi:10.1016/S0370-1573(03) 00103-0

    2003 arXiv

  5. [13]

    Can Quantum-Mechanical Description of Physical Reality be Considered Complete?

    N. Bohr. “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”.Physical Review, 48(8):696, 1935.doi:10.1103/PhysRev.48.696

    1935 doi

  6. [14]

    Quantenmechanische Statistik und Korrelationsrechnung

    F. A. Bopp. “Quantenmechanische Statistik und Korrelationsrechnung”.Zeitschrift für Naturforschung A, 2(4):202–216, 1947.doi:10.1515/zna-1947-0402

    1947 doi

  7. [15]

    Ein für die Quantenmechanik bemerkenswerter Satz der Korrela- tionsrechnung

    F. A. Bopp. “Ein für die Quantenmechanik bemerkenswerter Satz der Korrela- tionsrechnung”.Zeitschrift für Naturforschung A, 7(1):82–87, 1952.doi:10.1515/ zna-1952-0117

    1952

  8. [16]

    Statistische Untersuchung des Grundprozesses der Quantentheorie der Elementarteilchen

    F. A. Bopp. “Statistische Untersuchung des Grundprozesses der Quantentheorie der Elementarteilchen”.Zeitschrift für Naturforschung A, 8(1):6–13, 1953.doi:10.1515/ zna-1953-0103

    1953

  9. [17]

    Aspects of Objectivity in Quantum Mechanics

    H. Brown. “Aspects of Objectivity in Quantum Mechanics”. In J. Butterfield and C. Pagonis, editors,From Physics to Philosophy, pages 45–70. Cambridge University Press, 1999. URL:https://philpapers.org/rec/BROAOO-2

    1999

  10. [18]

    AreRindlerQuantaReal? InequivalentParticleConcepts in Quantum Field Theory

    R.CliftonandH.Halvorson. “AreRindlerQuantaReal? InequivalentParticleConcepts in Quantum Field Theory”.British Journal for the Philosophy of Science, 52(3):417– 470, September 2001. URL:https://www.jstor.org/stable/3541925

    2001

  11. [19]

    Quantum Error Correction

    I. L. Chuang. “Quantum Error Correction”. InQuantum Machines: Measurement and Control of Engineered Quantum Systems: Lecture Notes of the Les Houches Summer School: Volume 96, July 2011. Oxford University Press, June 2014.doi:10.1093/ acprof:oso/9780199681181.003.0007

    2011

  12. [20]

    On Pseudo-Stochastic Matrices and Pseudo-Positive Maps

    D. Chruściński, V. Man’ko, G. Marmo, and F. Ventriglia. “On Pseudo-Stochastic Matrices and Pseudo-Positive Maps”.Physica Scripta, 90(11):115202, October 2015. doi:10.1088/0031-8949/90/11/115202. 31

    2015 doi

  13. [21]

    Naive Realism about Operators

    M. Daumer, D. Dürr, S. Goldstein, and N. Zanghì. “Naive Realism about Operators”. Erkenntnis, 45(2):379–397, November 1996.arXiv:quant-ph/9601013,doi:10.1007/ BF00276801

    1996 arXiv

  14. [22]

    Non-Markovian Beables vs. Massive Parallelism

    E. Dennis. “Non-Markovian Beables vs. Massive Parallelism”.arXiv preprint arXiv:1003.3192, March 2010.arXiv:1003.3192,doi:10.48550/arXiv.1003.3192

    2010 doi arXiv

  15. [23]

    P. A. M. Dirac.The Principles of Quantum Mechanics. Oxford University Press, 1st edition, 1930

    1930

  16. [24]

    Classical Stochastic Representation of Quantum Mechanics

    M. J. de Oliveira. “Classical Stochastic Representation of Quantum Mechanics”. Brazilian Journal of Physics, 55(1):13, 2025.arXiv:2308.00151,doi:10.1007/ s13538-024-01649-x

    2025 arXiv

  17. [25]

    J. L. Doob.Stochastic Processes. Wiley-Interscience, 1990

    1990

  18. [26]

    The Theory of the Universal Wave Function

    H. Everett III. “The Theory of the Universal Wave Function”. Unpublished draft Ph.D. thesis (137 pp.), Princeton University, 1956

    1956

  19. [27]

    B. H. Feintzeig.On Algebraic Methods in Quantum Theories. PhD thesis, 2016. URL: https://escholarship.org/uc/item/42t27123

    2016

  20. [28]

    Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik

    I. Fényes. “Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik”.Zeitschrift für Physik, 132(1):81–106, February 1952.doi:10. 1007/BF01338578

    1952

  21. [29]

    Quantum Mechanics is the Square Root of a Stochastic Process

    M. Frasca. “Quantum Mechanics is the Square Root of a Stochastic Process”.arXiv,

  22. [30]

    On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit

    L. L. Foldy and S. A. Wouthuysen. “On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit”.Physical Review, 78(1):29–36, April 1950.doi:10.1103/ PhysRev.78.29

    1950

  23. [31]

    Markovian and Non-Markovian Quantum Measure- ments

    J. R. Glick and C. Adami. “Markovian and Non-Markovian Quantum Measure- ments”.Foundations of Physics, July 2020.arXiv:1701.05636,doi:10.1007/ s10701-020-00362-4

    2020

  24. [32]

    Comment on “Describing the State of a Stochastic Process,

    D. T. Gillespie, W. O. Alltop, and J. M. Martin. “Comment on “Describing the State of a Stochastic Process,” by D. T. Gillespie [Am. J. Phys. 66 (6), 533–536 (1998)]”. American Journal of Physics, 67(3):258–259, March 1999.doi:10.1119/1.19235

    1998 doi

  25. [33]

    Modeling Quantum Measurement Probability as a Classical Stochastic Process

    D. T. Gillespie, W. O. Alltop, and J. M. Martin. “Modeling Quantum Measurement Probability as a Classical Stochastic Process”.Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(3):548–562, August 2001.doi:10.1063/1.1378791. 32

    2001 doi

  26. [34]

    WhyQuantumMechanicsCannotbeFormulatedasaMarkovProcess

    D.T.Gillespie. “WhyQuantumMechanicsCannotbeFormulatedasaMarkovProcess”. Physical Review A, 49(3):1607–1612, March 1994.doi:10.1103/PhysRevA.49.1607

    1994 doi

  27. [35]

    The Multivariate Langevin and Fokker–Planck Equations

    D. T. Gillespie. “The Multivariate Langevin and Fokker–Planck Equations”.American Journal of Physics, 64(10):1246–1257, October 1996.doi:10.1119/1.18387

    1996 doi

  28. [36]

    Reply to “Comment on ‘Why Quantum Mechanics Cannot be For- mulated as a Markov Process’

    D. T. Gillespie. “Reply to “Comment on ‘Why Quantum Mechanics Cannot be For- mulated as a Markov Process’ ””.Physical Review A, 56(4):3304–3306, October 1997. doi:10.1103/PhysRevA.56.3304

    1997 doi

  29. [37]

    Describing the State of a Stochastic Process

    D. T. Gillespie. “Describing the State of a Stochastic Process”.American Journal of Physics, 66(6):533–536, June 1998.doi:10.1119/1.18895

    1998 doi

  30. [38]

    Non-Markovian Stochastic Processes

    D. T. Gillespie. “Non-Markovian Stochastic Processes”. InAIP Conference Proceedings, Volume 511, pages 49–56. American Institute of Physics, July 2000.doi:10.1063/1. 60002

    doi

  31. [39]

    Gauge Invariance, Time-Dependent Foldy-Wouthuysen Transforma- tions, and the Pauli Hamiltonian

    T. J. Goldman. “Gauge Invariance, Time-Dependent Foldy-Wouthuysen Transforma- tions, and the Pauli Hamiltonian”.Physical Review D, 15(4):1063–1067, February 1977. doi:10.1103/PhysRevD.15.1063

    1977 doi

  32. [40]

    D. J. Griffiths.Introduction to Electrodynamics. Cambridge University Press, 5th edition, 2023

    2023

  33. [41]

    Unified Dynamics for Microscopic and Macro- scopic Systems

    G. Ghirardi, A. Rimini, and T. Weber. “Unified Dynamics for Microscopic and Macro- scopic Systems”.Physical Review D, 34(2):470–491, 1986.doi:10.1103/PhysRevD.34. 470

    1986 doi

  34. [42]

    Haag.Local Quantum Physics: Fields, Particles, Algebras

    R. Haag.Local Quantum Physics: Fields, Particles, Algebras. Springer-Verlag, 1993

    1993

  35. [43]

    P. R. Halmos.Finite-Dimensional Vector Spaces. Van Nostrand, 2nd edition, 1958

    1958

  36. [44]

    The Hadamard Product

    R. A. Horn. “The Hadamard Product”. InMatrix Theory and Applications, Volume 40, pages 87–169, 1990.doi:10.1090/psapm/040/1059485

    1990 doi

  37. [45]

    J. D. Jackson.Classical Electrodynamics. Wiley, 3rd edition, 1998

    1998

  38. [46]

    Fundamentals of Quantum Information Theory

    M. Keyl. “Fundamentals of Quantum Information Theory”.Physics Reports, 369(5):431–548, October 2002.arXiv:0202122,doi:10.1016/S0370-1573(02) 00266-1

    2002 doi

  39. [47]

    General State Changes in Quantum Theory

    K. Kraus. “General State Changes in Quantum Theory”.Annals of Physics, 64(2):311– 335, June 1971. URL:https://www.sciencedirect.com/science/article/pii/ 0003491671901084,doi:10.1016/0003-4916(71)90108-4

    1971 doi

  40. [48]

    The Problem of Hidden Variables in Quantum Mechanics

    S. B. Kochen and E. P. Specker. “The Problem of Hidden Variables in Quantum Mechanics”.Indiana University Mathematics Journal, 17(1):59–87, 1967. 33

    1967

  41. [49]

    Induced Representations of Locally Compact Groups I

    G. W. Mackey. “Induced Representations of Locally Compact Groups I”.Annals of Mathematics, 55(1):101–139, 1952. URL:https://www.jstor.org/stable/1969423, doi:10.2307/1969423

    1952

  42. [50]

    Quantum Mechanics and Hilbert Space

    G. W. Mackey. “Quantum Mechanics and Hilbert Space”.American Mathematical Monthly, 64(8):45–57, 1957. URL:https://www.jstor.org/stable/2308516,doi: 10.2307/2308516

    1957

  43. [51]

    Messiah.Quantum Mechanics, Volume 1

    A. Messiah.Quantum Mechanics, Volume 1. North-Holland Publishing Company and Interscience Publishers, Amsterdam and New York, 1961. Translated from the French Mécanique Quantiqueby G. M. Temmer

    1961

  44. [52]

    Quantum Stochastic Processes and Quantum Non-Markovian Phenomena

    S. Milz and K. Modi. “Quantum Stochastic Processes and Quantum Non-Markovian Phenomena”.PRX Quantum, 2:030201, May 2021.arXiv:2012.01894v2,doi:10. 1103/PRXQuantum.2.030201

    2021 arXiv

  45. [53]

    Quantum Mechanics on a Real Hilbert Space

    J. Myrheim. “Quantum Mechanics on a Real Hilbert Space”. 1999. URL:https: //arxiv.org/abs/quant-ph/9905037,arXiv:quant-ph/9905037

    1999 arXiv

  46. [54]

    Dynamical Theories of Brownian Motion

    E. Nelson. “Dynamical Theories of Brownian Motion”. 1967

    1967

  47. [55]

    Quantum Fluctuations

    E. Nelson. “Quantum Fluctuations”. 1985

    1985

  48. [56]

    Parzen.Stochastic Processes

    E. Parzen.Stochastic Processes. Holden-Day, 1962

    1962

  49. [57]

    General Probabilistic Theories: An Introduction

    M. Plávala. “General Probabilistic Theories: An Introduction”.Physics Reports, 1033:1–64, September 2023. General probabilistic theories: An introduction.arXiv: 2103.07469,doi:10.1016/j.physrep.2023.09.001

    2023 arXiv

  50. [58]

    M. E. Peskin and D. V. Schroeder.An Introduction to Quantum Field Theory. Westview Press, 1995

    1995

  51. [59]

    Rosenblatt.Random Processes

    M. Rosenblatt.Random Processes. Oxford University Press, 1962

    1962

  52. [60]

    Ross.Stochastic Processes

    S. Ross.Stochastic Processes. John Wiley and Sons, 2nd edition, 1995

    1995

  53. [61]

    Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen

    I. Schur. “Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen”.Journal für die reine und angewandte Mathematik, 140:1–28,

  54. [62]

    Shankar.Principles of Quantum Mechanics

    R. Shankar.Principles of Quantum Mechanics. Plenum Press, 2nd edition, 1994

    1994

  55. [63]

    Quantum Mechanics can be Formulated as a Non-Markovian Stochastic Process

    G. A. Skorobogatov and S. I. Svertilov. “Quantum Mechanics can be Formulated as a Non-Markovian Stochastic Process”.Physical Review A, 58(5):3426–3432, November 1998.doi:10.1103/PhysRevA.58.3426. 34

    1998 doi

  56. [64]

    Positive Functions on C*-algebras

    W. F. Stinespring. “Positive Functions on C*-algebras”.Proceedings of the American Mathematical Society, 6(2):211–216, April 1955.doi:10.2307/2032342

    1955 doi

  57. [65]

    Strocchi.An Introduction to the Mathematical Structure of Quantum Mechanics

    F. Strocchi.An Introduction to the Mathematical Structure of Quantum Mechanics. World Scientific, 2nd edition, 2008

    2008

  58. [66]

    Quantum Theory in Real Hilbert Space

    E. C. G. Stueckelberg. “Quantum Theory in Real Hilbert Space”.Helvetica Physica Acta, 33(4):727–752, 1960. URL:https://www.e-periodica.ch/digbib/view?pid= hpa-001:1960:33::715#735

    1960

  59. [67]

    von Neumann.Mathematische Grundlagen der Quantenmechanik

    J. von Neumann.Mathematische Grundlagen der Quantenmechanik. Berlin: Springer, 1932

    1932

  60. [68]

    Dividing Quantum Channels

    M. M. Wolf and J. I. Cirac. “Dividing Quantum Channels”.Communications in Mathematical Physics, 279:147–168, 2008.arXiv:math-ph/0611057,doi:10.1007/ s00220-008-0411-y

    2008 arXiv

  61. [69]

    Weinberg.The Quantum Theory of Fields, Volume 2

    S. Weinberg.The Quantum Theory of Fields, Volume 2. Cambridge University Press, 1996

    1996

  62. [70]

    Weinberg.The Quantum Theory of Fields, Volume 1

    S. Weinberg.The Quantum Theory of Fields, Volume 1. Cambridge University Press, 1996

    1996

  63. [71]

    Quantum Representation of Classical Statistical Ensembles

    C. Wetterich. “Quantum Representation of Classical Statistical Ensembles”. Un- published draft, 2009. URL:https://www.thphys.uni-heidelberg.de/~wetterich/ Material/quantum_representation.pdf

    2009

  64. [72]

    E. P. Wigner.Gruppentheorie und ihre Anwendung auf die Quanten mechanik der Atomspektren. Friedrich Vieweg und Sohn, 1931

    1931

  65. [73]

    Zangwill.Modern Electrodynamics

    A. Zangwill.Modern Electrodynamics. Cambridge University Press, 2012. 35

    2012

  66. [1911]

    URL:https://eudml.org/doc/149352

  67. [2012]

    URL:https://arxiv.org/abs/1201.5091,arXiv:1201.5091

    arXiv

Pith tools

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