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REVIEW 3 major objections 4 minor 9 references

A comment argues that the target paper's 'real' quantum mechanics is complex quantum mechanics in disguise, making its unfalsifiability claim flawed.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 00:42 UTC pith:J75NVUSE

load-bearing objection Makes one good point about the flag construction hiding a complex structure, but the central 'CQM in disguise' claim rests on treating inaccessible flags as physical; still worth refereeing. the 3 major comments →

arxiv 2607.28668 v1 pith:J75NVUSE submitted 2026-07-25 physics.gen-ph

Comment on "Quantum mechanics based on real numbers: a consistent description"

S C Tiwari This is my paper
classification physics.gen-ph PACS 03.65
keywords real quantum mechanicsflag statescomplex numbersHilbert spacefalsifiabilityquantum foundationsU(1) phase ambiguity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This comment takes aim at a recent proposal for quantum mechanics based on real numbers, which uses auxiliary 'flag states' to convert complex probability amplitudes into real ones. The comment argues that the flag degrees of freedom carry the complex structure back in: the matrix J_F that appears in the operator map is a real substitute for i, since J_F^2 = -I_F. Therefore the construction is not a genuinely real-valued theory but a replication of ordinary complex quantum mechanics on an enlarged real space. As a result, the original claim that real-valued QM cannot be falsified is conceptually flawed, because this particular version is only complex QM in disguise. The comment also notes that other, genuinely real formulations remain possible and are not excluded by this construction.

Core claim

The central claim is that the flag-state mapping S:|Ψ>→Re|Ψ>⊗|0>_F + Im|Ψ>⊗|1>_F embeds the complex number i into a real degree of freedom. Under the operator transform T(Π)=ReΠ⊗I_F + ImΠ⊗J_F, the matrix J_F satisfies J_F^2=-I_F and thus behaves exactly like i. The expanded real Hilbert space R^d⊗R^2_F therefore contains two real components that play the role of real and imaginary parts, so the theory still has a complex structure; it is CQM written in a larger real space. The comment concludes that the target paper's ReQM is a trivial replication of CQM, and that its claim that real-valued quantum mechanics cannot be falsified is conceptually flawed.

What carries the argument

The key object is the matrix J_F (the real 2x2 antisymmetric matrix with J_F^2 = -I_F), used in the operator map as a real representative of i. The flag state map S uses two flag basis vectors to store the real and imaginary parts of a complex vector separately, and the operator map T uses I_F and J_F to mimic complex multiplication. Together they show that the flag degree of freedom is doing exactly the work of the imaginary unit, so the theory retains a complex structure despite being written in real numbers.

Load-bearing premise

The argument that this construction is CQM in disguise assumes that the flag states, despite being declared 'not directly accessible,' are legitimate physical dimensions of the Hilbert space; if they are merely auxiliary bookkeeping devices, the replication conclusion collapses.

What would settle it

Find a concrete calculation in which the flag-state real QM and ordinary complex QM give different predictions for an observable (e.g., a multipartite correlation or an interference pattern), or demonstrate that the map S can be inverted to a real theory with no residual complex structure. Either result would break the claimed trivial equivalence.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The target paper's real-number QM is equivalent to ordinary complex QM; it introduces no new physical content.
  • The conclusion that real-valued quantum mechanics cannot be falsified does not follow; only this specific construction is unfalsifiable because it is already complex.
  • The construction depends on an inaccessible flag degree of freedom, which functions as hidden entanglement.
  • Any claim that complex numbers are optional in quantum theory needs a formulation that does not hide i in an auxiliary space.

Where Pith is reading between the lines

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

  • If the flags are interpreted as pure mathematical bookkeeping rather than physical degrees of freedom, the 'disguise' argument loses force; the theory could be genuinely real with gauge-like redundancy, so the comment's conclusion is not airtight.
  • One could test the equivalence by constructing a multipartite scenario where the flag-space structure leaves a residual signature, such as a modified SO(2) phase rotation, that differs from CQM.
  • The comment's observation about J_F points to a broader possibility: any real representation of CQM must either import an effective i (via a matrix like J_F) or adopt a non-trivial spinor/topological structure, which could be explored as a research direction.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This comment claims that the real-valued quantum mechanics proposed in arXiv:2503.17307 (Hita et al., PRL 136, 240202) is not a genuinely real-valued theory but rather a 'trivial replication' of complex quantum mechanics. The author observes that the flag-state map S in Eq. (1) embeds a complex state into R^d ⊗ R^2_F, that the U(1) phase freedom becomes an SO(2) rotation on the flag space as in Eq. (2), and that the operator J_F defined in Eq. (4) satisfies J_F^2 = -I, thereby reproducing the complex unit i. From these algebraic facts, the comment concludes that ReQM in [1] is 'CQM in disguised form' and that the claim that real-valued quantum mechanics cannot be falsified is conceptually flawed. The comment further suggests, referencing the author's own prior work [9], that a more promising real formulation would replace i by a Cartan/symplectic matrix within a spinor formalism.

Significance. If the central conclusion were correct, it would challenge the interpretation of a published PRL result and the broader program of real-number quantum foundations. The technical observations are individually correct: Eq. (2) correctly shows that the global phase becomes an SO(2) rotation in flag space, and J_F^2 = -I is a standard real representation of the imaginary unit. The paper is also honest about the auxiliary role of flags. However, the paper's main claim — that ReQM in [1] is a trivial replication of CQM — is not established. The argument rests on an interpretative leap: it treats the flag space as physical despite [1]'s explicit statement that the flag is not a directly accessible degree of freedom. No rigorous map is given from the enlarged space to the physical Hilbert space, and the multipartite quotient construction of [1] is not analyzed. The paper is a short comment with no new calculations or falsifiable predictions, and its force depends on settling an ontological question that the manuscript does not resolve.

major comments (3)
  1. [Discussion after Eq. (4)] The central claim that ReQM in [1] is 'CQM in disguised form' depends on treating the flag space R^2_F as part of the physical Hilbert space. The author acknowledges that 'flag states are auxiliary and if inaccessible it can be argued that these effect the single state |Ψ̃⟩ via invisible entanglement,' but this 'invisible entanglement' is asserted, not derived. If the flag is only a mathematical bookkeeping device, then J_F is a formal complex structure on an auxiliary space, and the physical theory may be genuinely real with a gauge-like redundancy. The comment does not show that the inaccessible flags contribute to any measurement statistic or that they cannot be quotiented out. Without such a demonstration, the conclusion that the theory is 'CQM in disguise' is an interpretive assertion rather than a proven result.
  2. [Multipartite quotient construction] The comment entirely ignores the second novel postulate of [1]: the replacement of the tensor product for multipartite systems by a quotient with the kernel of the inverse of the map from complex to real. This is a potentially load-bearing omission. If the quotient removes the flag degrees from the physical description, then the global operator J_F acting on R^d ⊗ R^2_F may not descend to a well-defined complex structure on the physical quotient Hilbert space. The argument that 'ReQM in [1] is a trivial replication of CQM' must be checked for composite systems, where [1]'s construction is specifically designed to avoid the flag degrees. The comment provides no analysis of whether a global J with J^2 = -I survives the quotient. Until this is addressed, the central conclusion is incomplete.
  3. [Eqs. (1)–(4) and final paragraph] The paper's conclusion that ReQM in [1] is a 'trivial replication of CQM' is stronger than the mathematics shown. The author demonstrates that the real vector space R^d ⊗ R^2_F carries an SO(2) rotation and a matrix J_F with J_F^2 = -I, but this only shows that a complex structure exists on the enlarged space. It does not establish a bijective, physically meaningful equivalence between the full algebra of complex observables, inner products, tensor products, and measurement outcomes in [1] and those of standard complex QM. The phrase 'trivial replication' implies a systematic correspondence that the comment neither constructs nor proves. The argument would need to show that every complex amplitude and observable expectation value is reproduced uniquely by the real formalism without surplus structure, and that the surplus structure is physically meaningful. This is not done.
minor comments (4)
  1. [Abstract] Typo: 'covenience' should be 'convenience'.
  2. [First paragraph] The analogy with integers/rationals as subsets of reals is a heuristic illustration, not an argument. It would be helpful to state explicitly what formal property of ReQM is being illustrated, since as written it could be read as a mere rhetorical device.
  3. [Reference [5]] The comment cites [5] as a 'well-reasoned critique' but does not explain how the author's perspective differs from or extends [5]. Given that [5] apparently already argues for a hidden complex structure, the novelty of the present comment should be stated more precisely.
  4. [Final paragraph] The statement 'It is not clear as to why authors did not use it' is speculative about the authors' intentions and should be removed or recast as a mathematical observation about the equivalence of J_F and i.

Circularity Check

0 steps flagged

No significant circularity: the comment's argument is a self-contained structural analysis of the flag-state construction.

full rationale

The comment does not fit any circularity pattern. Its central claim that ReQM in [1] is 'CQM in disguised form' is derived directly from the published construction: the map S in Eq. (1), the action in Eq. (2), and the operator T(Π) in Eq. (3) with J_F defined in Eq. (4). The key observation is J_F^2 = -I_F, which provides a complex structure on R^d ⊗ R^2_F. This is a structural inference from the given equations, not an assumption of the conclusion. No parameter is fitted, no prediction is made, and the argument does not rely on the author's own prior work for its load-bearing step. The self-citations to [4], [7], and [9] are contextual or propose an alternative construct; they are not used to justify the main claim. The disputed premise about whether inaccessible flag states are physical dimensions is an interpretive assumption about the formalism, but it is not circular—it is a substantive criticism of [1] that may be right or wrong, but does not reduce the argument to its own inputs. Therefore the paper has no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The comment introduces no free parameters. Its central argument rests on interpretive assumptions about the physical status of flag states and on the equivalence of the mapping S to full CQM. The Cartan-matrix suggestion is an undeveloped alternative, not a working entity.

axioms (3)
  • domain assumption A real-valued theory that uses auxiliary flag states with an inaccessible degree of freedom is still physically complex if it contains a structure isomorphic to i.
    This is the crux of the critique: the presence of J_F with J_F^2=-I is treated as evidence of hidden complex structure rather than as a legitimate real-matrix encoding. It is stated, not derived.
  • ad hoc to paper The flag-state theory is equivalent to CQM if it reproduces CQM amplitudes via the map S.
    The comment assumes equivalence of all predictions follows from the state/operator mapping in Eqs. (1)-(3), without proving that the map is onto the full set of physical states and observables or that the flag states do not change the physical content.
  • domain assumption A physical theory should be free of mysticism/mysteries.
    The author invokes [4] to dismiss QIS-based interpretations; this is a philosophical preference, not a mathematical constraint, yet it supports the rhetorical conclusion.
invented entities (1)
  • Cartan matrix / spinor reformulation no independent evidence
    purpose: Suggested alternative to flag states, claimed to show that i hides spin.
    Mentioned in the final paragraph and referenced to the author's own [9]; no concrete construction, derivation, or prediction is given.

pith-pipeline@v1.3.0-alltime-deepseek · 2195 in / 9315 out tokens · 80021 ms · 2026-08-03T00:42:25.849663+00:00 · methodology

0 comments
read the original abstract

In this comment on arXiv:2503.17307 (PRL, 136, 240202, 2026), a novel idea using flag states to transform complex state vector and operators to real ones is argued to effectively lead to a quantum mechanics that is complex in the disguised form of real. The claim that real-valued quantum mechanics cannot be falsified is conceptually flawed.

discussion (0)

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Reference graph

Works this paper leans on

9 extracted references · 2 linked inside Pith

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    P. B. Hita et al, Quantum mechanics based on real numbers: a consistent description, Phys. Rev. Lett. 136, 240202 (2026); arXiv:2503.17307

    Pith/arXiv arXiv 2026

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    J. Bang, K. Cho and K. Baek, Hidden complex structure in quotient-space real quantum mechanics, arXiv: 2607.05865 v1 [quant-ph]

    Pith/arXiv arXiv

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    J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966)

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    S. C. Tiwari, Comment on ”Experimental test of nonlocal quantum correlation in relativistic configurations”, Phys. Rev. A 65, 016101 (2001)

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    Goldstein, Classical Mechanics (Addison Wesley, 1980)

    S. Goldstein, Classical Mechanics (Addison Wesley, 1980)

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    S. C. Tiwari, Schroedinger equation, spin and topology, Quantum Stud: Math. Found. 10, 293 (2023)

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