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REVIEW 4 major objections 3 minor 16 references

Phenomenological quantum mechanics II: deducing the formalism from experimental observations

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that the full Hilbert-space formalism of quantum mechanics can be deduced from sequential-measurement statistics, with a master bi-trajectory measure replacing the quantum state as the fundamental object.

desk verdict A clean, carefully assembled reconstruction of Hilbert-space QM from bi-probabilities, undermined only by the unproved master measure that carries the paper's most ambitious claim. read the letter →

arxiv 2507.04812 v2 pith:F72QAJIQ submitted 2025-07-07 quant-ph math-phmath.MPphysics.hist-ph

Piotr Szańkowski , Davide Lonigro , Fattah Sakuldee , {L}ukasz Cywiński , Dariusz Chruściński This is my paper · ORCID
classification quant-phmath-phmath.MPphysics.hist-ph
keywords quantumfoundationsbi-trajectoryformalismmulti-timeprobabilitydistributionssequentialmeasurementsHilbertspacerepresentationdensitymatrixHamiltonianHeisenbergcut
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

The paper attempts to derive the mathematical formalism of quantum mechanics from scratch, taking as the only input the multi-time probability distributions that sequential measurements of quantum observables produce. From these observed distributions it reconstructs Hilbert spaces, projection-valued measurements, density matrices, unitary time evolution, Hamiltonians, and the tensor-product rule for composite systems. The central object that emerges is not the quantum state but a master complex-valued “bi-trajectory measure” on pairs of possible histories of a system; the state is demoted to a metric representing how the experiment was initialized. If the deduction works, standard quantum mechanics is a representation of this more fundamental measure, and long-standing problems tied to the Heisenberg cut could be reposed as problems about when a bi-trajectory collapses to a classical uni-trajectory.

What carries the argument

The load-bearing object is the master bi-trajectory measure $Q(H,\hat H)[\eta_+,\eta_-][D\eta_+][D\eta_-]$, a complex-valued measure on pairs of histories of a reference-basis index $\eta$ over “space-time” coordinates $(t,\varphi)$, where $t$ is time and $\varphi$ parameterizes unitary rotations of a reference basis. Its discrete restrictions, Eq. (7.6), are traces of alternating products of projectors; all multi-observable bi-probabilities are pushforwards of it, and all standard-formalism correlation functions are its moments. The derivation chain that carries the argument is: positive-semidefinite bi-probabilities → Hilbert-space inner products (via a cited representation theorem) → projector-valued measurement link → initialization as density-matrix metric → unitary evolution and Hamiltonian from uncertainty relations and Zeno behavior → tensor-product Hilbert space for composite systems → the master measure as the unique source of all the above.

What would settle it

Compute the discrete system bi-probabilities (7.6) for a concrete finite-dimensional system, say a qubit with a fixed Hamiltonian and two inequivalent fine-grained observables, and check whether the family satisfies the consistency conditions required for a single measure on all bi-trajectories; a set of multi-observable probabilities that violates those conditions would disprove the master-measure claim. The corresponding experimental test is a sequential measurement of three fine-grained observables whose joint bi-probabilities cannot be reproduced by any single bi-trajectory measure.

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Extended reading notes

Core claim

The paper's central claim is that the bi-trajectory formalism is not an alternative interpretation but a phenomenological deduction: the full Hilbert-space machinery follows from the observed rules of sequential measurement, and the resulting master object is the bi-trajectory measure $Q(H,\hat H)[\eta_+,\eta_-][D\eta_+][D\eta_-]$ on pairs of trajectories $(t,\varphi) \mapsto (\eta_+(t,\varphi),\eta_-(t,\varphi))$. The deduction proceeds by using an inner-product representation theorem to read bi-probabilities as inner products, then showing that single-device phenomenology forces the measurement–projector link and the density-matrix form of initialization, that uncertainty relations and the Zeno effect force unitary evolution generated by a Hermitian Hamiltonian, and that factorization for independent systems forces tensor products. The paper argues that every empirically testable prediction of standard quantum mechanics, including multi-time correlation functions, is a moment of this master measure, so the two formulations agree on all observations while differing in what they regard as fundamental.

Load-bearing premise

The derivation stands on the assumed existence of one master complex-valued bi-trajectory measure over all observables and all times (Eq. 7.8); the paper explicitly states it cannot prove this with the method used for the single-observable case and can only offer a strong completely-positive-map argument, so if that measure fails to exist the master-object claim collapses.

Editorial extensions

If this is right

  • Every multi-time correlation function of the standard formalism becomes a moment of the master bi-trajectory measure, so the two formulations are empirically indistinguishable by construction.
  • The quantum state loses its foundational role and becomes only a metric representing an initialization event, which removes the formal need for a Heisenberg cut and for a collapse rule.
  • The classical limit is identified with the collapse of a bi-trajectory measure onto its diagonal, reducing it to a single classical trajectory measure; measurement outcomes then satisfy classical consistency conditions.
  • Composite systems acquire the tensor-product structure automatically, and interactions appear as coupling terms in the Hamiltonian, giving a direct derivation of Hermitian operators as observables.
  • Modeling a measurement device becomes, in principle, a technical problem of constructing quantum-classical hybrids, not a conceptual obstruction.

Reading between the lines

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

  • Editorial: Even if the master measure's existence is not yet proven, the paper's completely-positive dynamical-map construction suggests an operational substitute: define the theory by its moment hierarchy and use experimental multi-time statistics to test whether higher moments match the measure's predictions.
  • Editorial: The bi-trajectory picture gives a concrete criterion for the quantum-to-classical transition — an observable is classical exactly when its reduced bi-trajectory measure becomes diagonal — which could be tested in models of decoherence and in experiments on sequentially measured coarse-grained observables.
  • Editorial: A natural next step the paper leaves open is to search for a finite-dimensional counterexample to the master-measure extension; finding one would not invalidate the derived Hilbert-space machinery, but would force the master object to be replaced by a family of observable-specific measures.
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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

4 major / 3 minor

Summary. The manuscript attempts to derive the Hilbert-space formalism of quantum mechanics from phenomenological multi-time probability distributions obtained in sequential measurements. Building on Part I, the authors formulate bi-probability distributions Q and use Gudder's theorem to represent them as inner products in a Hilbert space. They then introduce projectors and a metric, derive unitary evolution and a Hamiltonian from the uncertainty relations and the Zeno effect, infer tensor-product structure for composite systems, and finally propose a master bi-trajectory measure Q[η+,η−] as the fundamental object, asserting that the quantum system is synonymous with this measure. The paper is framed as a reconstruction of standard quantum mechanics rather than a new empirical theory.

Significance. The paper has several genuine strengths: it offers a clear axiomatic basis (Q1)–(Q8), a rigorous use of Gudder's inner-product representation, a simple recurrence for sequential probabilities leading to Eq. (4.27), and an explicit demonstration that the measurement–projector link fails for sequences. The proposed bi-trajectory formalism is conceptually interesting and could, if completed, provide a fresh perspective on the measurement problem and the classical limit. However, the advertised deduction is incomplete: it relies on an extra-empirical 'economy of entities' postulate in Sec. 4.2, the off-diagonal representation in Sec. 4.4 is not uniquely derived, and the master-measure existence in Sec. 7 is an unproved conjecture. The paper's observational agreement with standard quantum mechanics is true by construction (axiom Q7), so its contribution is a reformulation, not a predictive derivation. With appropriate caveats, it is a valuable foundational contribution.

major comments (4)
  1. [Section 7, Eq. (7.8)] The master bi-trajectory measure Q[η+,η−][Dη+][Dη−] is introduced as a supposition, not derived from the phenomenology or from the previously established properties. The text explicitly states that its existence cannot be proved with the method used for (Q6) and defers to reference [7]. The supporting argument is conditional: Eq. (7.11) defines the superoperator Λ_t only 'assuming that the master measure exists', and the identification of Λ_t with the CPTP map in Eq. (7.14) is exactly the existence statement at issue. No Kolmogorov-consistency or regularity verification for the family Q_{τ_n|τ_0} on the uncountable index set R × S^{d²−1} is supplied. Consequently, the concluding identification in Section 8 of the quantum system with its bi-trajectory measure is unsupported. This is load-bearing for the paper's central claim, and the authors should either provide a proof or clearly reformulate the master-object statement as a conjecture.
  2. [Section 4.2, Eq. (4.12)] The mapping of the coarse-grained device's projectors and metric into H(K) via 'projectors correspond to projectors, and metrics correspond to metrics' is an unforced simplicity choice. The text acknowledges that 'there is no purely logical reason' for the Hilbert spaces H(Kbar) and H(K) to be related. This choice is necessary to obtain the measurement–projector link (4.15), the common metric (4.14), and hence the system Hilbert space H_S used throughout Sections 5 and 6. Since the paper claims to deduce the formalism solely from experimental observations, an extra-empirical economy postulate needs explicit justification, or the argument must be reframed as a derivation conditional on that postulate. The non-uniqueness of the embedding also affects the subsequent derivation of a single unitary U for all observables in Sec. 5.1.
  3. [Section 4.4, Eqs. (4.28)–(4.31)] The derivation of the off-diagonal representation of the bi-probability is underdetermined. Equation (4.30) equates the sum over off-diagonal pairs at position j with the corresponding sum of trace expressions, but the individual off-diagonal elements Q(f_n^+,...,f_j^+,...,f_1^+; f_n^-,...,f_j^-,...,f_1^-) are not shown to equal the individual trace terms. The conclusion in Eq. (4.31) therefore does not follow from the displayed sum identities alone. To justify the claimed representation, the authors need to show uniqueness, for example by using bi-consistency at all levels together with positive semi-definiteness, or to provide a different argument that fixes each off-diagonal element.
  4. [Section 8] The assertion that the deduced formalism 'is necessarily consistent with the empirical observations it was derived from' is true by construction, because axiom (Q7) identifies the diagonal of the bi-probability with the empirical probability. Thus the agreement between the bi-trajectory formalism and standard quantum mechanics on multi-time correlations (Eq. (8.5)) is not a new prediction but a restatement of the input in a new representation. This observational adequacy should be presented as a consistency property of the reconstruction, not as an independent confirmation of the formalism; the current phrasing overstates the epistemic status of the result.
minor comments (3)
  1. [Section 3.1, Eq. (3.7)] Some glyphs in Eq. (3.7) and the surrounding text (e.g., the strikethrough F_j and f_j) are corrupted in the manuscript; please fix the typesetting.
  2. [Section 5.2, Eq. (5.11)] The conditions in Eq. (5.11) mix empirical constraints with mathematical identities. For example, the anti-Hermiticity of dU/dt U† follows from unitarity, while the inequality involving the quadratic term is automatically satisfied by the unitary expansion. Please clarify which of these conditions are actually imposed by the Zeno phenomenology and which are consequences of the assumed unitarity.
  3. [Section 7] The status of reference [7] should be stated explicitly: does it prove existence of the master bi-trajectory measure, or does it only provide partial results or a different construction? Currently the reader cannot tell from this paper alone whether the missing Kolmogorov extension is a settled issue or an open problem.

Circularity Check

1 steps flagged · score 6.0 of 10

Master bi-trajectory measure Q is stipulated in Eq. (7.8), not deduced; the CPTP supporting argument assumes the measure it aims to justify, so the Section 8 identification of the system with Q is an assumption, not a derivation.

  1. self definitional [Section 7, Eqs. (7.8)-(7.14); Section 8, 'Quantum System' definition]
    "we suppose that Qτn|τ0(η+n,η−n|η+0,η−0) = ∫∫(∏nj=0 δη+j,η+(τj)δη−j,η−(τj)) Q[η+,η−][Dη+][Dη−] ... Unfortunately, for several non-trivial reasons (see [7] for a detailed explanation), we cannot prove the existence of the master bi-trajectory measure Q with the same method used to prove (Q6)."

    Eq. (7.8) is introduced with 'we suppose that', explicitly not proven. The master object is therefore an input, not an output of the deduction. The supporting argument compounds this: Eq. (7.11) defines Λt only 'Assuming that the master measure exists', and the identification of Λt with the CPTP map (7.14) uses the moment-generating identities whose validity is precisely the content of (7.8). Hence the statement that 'the characteristic function of Q[...] is a well-behaved, regular CPTP map, and thus, the underlying bi-trajectory measure should also be well-behaved' concludes from the assumption back to itself. Section 8 then presents the conditional object as the central result: 'the system is synonymous with its bi-trajectory measure Q(H, Ĥ)' and claims the formalism 'is deduced'.

full rationale

The paper's derivation of the Hilbert-space representation (projectors, density matrices, unitaries, Hamiltonians, tensor products) is largely independent: it uses Gudder's theorem on inner-product representations and the defined bi-probability axioms, so that part is not circular. However, the paper's central claimed achievement—the master bi-trajectory measure Q that unifies all observables—is introduced in Eq. (7.8) as a supposition, not as a theorem, and the paper explicitly states that the existence proof used for the single-observable case (Q6) cannot be extended. The subsequent 'strong supporting argument' via CPTP maps only applies after assuming the measure exists; the equality between the two forms of Λt in Eqs. (7.11) and (7.14) is exactly the existence claim, so it does not provide independent support. Consequently, the identification of the quantum system with Q in Section 8 is an assumption dressed as a result, and the claimed deduction of the master-object picture is partially circular. Additionally, the bi-probability axioms already include the measurement link (Q7) that forces agreement of the deduced formalism with the input multi-time probabilities, so the statement that the formalism is 'necessarily consistent with the empirical observations' is true by construction rather than by novel predictive content.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The paper rests on a small number of strong premises: the bi-probability axioms (which are not raw data but theoretical constructs), Gudder's theorem, an explicit non-unique embedding choice, an assumed stationarity, an assumed existence of devices for arbitrary projectors, and an admitted unproven master measure. There are no numerical free parameters fitted to data.

assumptions (6)
  • domain assumption Bi-probability axioms (Q1)-(Q8), including positivity (Q4), bi-consistency (Q5), the bi-trajectory picture (Q6), the measurement link (Q7), and interference additivity (Q8).
    These are the central inputs, introduced in Section 3.2 and inherited from Part I. They are not raw measurement outcomes but theoretical constructions chosen to encode quantum interference and to make the diagonal equal to observed probabilities, so the deduction is conditional on them.
  • standard math Gudder's theorem on Hilbert space representations of decoherence functionals.
    Used in Section 4.1 to turn the positive semidefinite bi-probability into an inner product. The paper cites [2] but does not reproduce the proof.
  • ad hoc to paper Economy of entities: all observable representations are mapped into a single common Hilbert space by identifying projectors with projectors and metrics with metrics.
    Section 4.2 admits there is no purely logical reason to relate H(K) and H(L); the choice is guided by simplicity and economy of entities, not by observation.
  • domain assumption Stationarity of multi-time probability distributions, Eq. (5.15).
    Used in Section 5.2 to force the Hamiltonian to be time-independent; it is presented as implicit in the initialization procedure, not as a derived fact.
  • domain assumption For any set of orthogonal projectors there exists a measuring device that realizes them.
    Assumed in Section 4.3 (Eq. (4.20)) to eliminate the passage of time and to identify Kt devices; no construction or evidence is given.
  • ad hoc to paper Existence of the master bi-trajectory measure Q[eta+, eta-] over all observables, Eq. (7.8).
    The paper states it cannot prove existence and gives only a strong supporting argument via CPTP maps; the master object and Eq. (8.1) depend on it.
invented entities (1)
  • Master bi-trajectory measure Q(H, H)[eta+, eta-][D eta+][D eta-]
    purpose: Serves as the master object from which all multi-observable bi-probabilities and hence all measurement probabilities are derived.
    The paper postulates its existence in Eq. (7.8) and admits in Section 7 that it cannot be proven; the supporting argument via CPTP maps is not a proof. It makes no new falsifiable predictions beyond standard quantum mechanics.

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Cite this review

Pith. "Pith review of Phenomenological quantum mechanics II: deducing the formalism from experimental observations." pith.science (2026). https://pith.science/paper/F72QAJIQ

@misc{pith2026250704812,
  author       = {Pith},
  title        = {Pith review of: Phenomenological quantum mechanics II: deducing the formalism from experimental observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F72QAJIQ}},
  note         = {Machine review of arXiv:2507.04812}
}
read the original abstract

We propose an exercise in which one attempts to deduce the formalism of quantum mechanics solely from phenomenological observations. The only assumed inputs are the multi-time probability distributions estimated from the results of sequential measurements of quantum observables; no presuppositions about the underlying mathematical structures are permitted. In the concluding Part II of the paper, we carry out the deduction of the formalism from the phenomenological inputs described in Part I. We show that the resulting formalism exhibits an affinity with Hilbert spaces, and we derive an explicit representation in terms of those mathematical structures. Analogues of the obtained elementary building blocks -- such as projection operators -- are readily identifiable within the standard formalism. However, once these building blocks are assembled according to the blueprint of the deduced bi-trajectory formalism, it becomes evident that the new and the standard formalisms differ substantially at the conceptual level. These differences do not negate the fact that both formalisms are in perfect agreement with respect to empirically testable predictions. Rather, the emergence of a novel, non-standard formulation should be seen as a relatively rare opportunity to reassess, from a fresh perspective, some of the long-standing foundational issues in the theory. The hope is that the new approach may prove more successful in addressing problems that have resisted resolution within the established theoretical framework.

Discussion (0). Continue with ORCID to comment.

Reference graph

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