Quantum mechanics over real numbers fully reproduces standard quantum theory

Alan C. Maioli; Evaldo M. F. Curado; Jean-Pierre Gazeau

arxiv: 2604.19482 · v1 · submitted 2026-04-21 · 🪐 quant-ph

Quantum mechanics over real numbers fully reproduces standard quantum theory

Alan C. Maioli , Evaldo M. F. Curado , Jean-Pierre Gazeau This is my paper

Pith reviewed 2026-05-10 02:05 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum mechanicsreal numberscomplex numbersisomorphismBell inequalityCHSH3entanglementHilbert space

0 comments

The pith

A real-valued framework exactly reproduces all predictions of standard complex quantum mechanics.

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

The paper sets out to show that quantum mechanics can be built entirely on real numbers while matching every prediction of the usual complex formulation. Previous no-go results claimed real theories must fail certain Bell tests, but the authors trace those failures to an incomplete real construction using the ordinary tensor product. They introduce a ka space equipped with a symplectic composition rule that replaces the Kronecker product and prove an explicit bijection to complex Hilbert space. If the mapping is faithful, every interference effect, entanglement correlation, and measurement outcome of standard quantum theory follows from real variables alone. A reader would care because the result would mean complex numbers are merely a convenient encoding rather than a necessary ingredient of nature.

Core claim

The authors present a real framework based on ka space that is exactly isomorphic to standard quantum mechanics through an explicit bijection γ. Composite systems are combined via a new symplectic rule ⊗^ks that preserves the full algebraic structure, allowing the framework to reach the maximal CHSH3 violation of 6√2 with purely real variables and thereby contradicting earlier claims that any real quantum theory is experimentally falsifiable.

What carries the argument

The ka space together with the symplectic composition rule ⊗^ks and the explicit bijection γ that maps it onto complex Hilbert space while preserving all operations and predictions.

If this is right

Every observable prediction of standard quantum mechanics, including all entanglement and interference effects, can be obtained using only real numbers.
The CHSH3 inequality reaches its algebraic maximum of 6√2 inside the real framework, matching the complex case.
Previous no-go theorems based on the ordinary real tensor product do not apply to this complete real formulation.
Complex numbers function as a convenient shorthand for an underlying real geometric structure rather than a fundamental requirement.
The isomorphism extends uniformly to arbitrary numbers of parties, so the equivalence holds for multipartite systems as well.

Where Pith is reading between the lines

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

If the real framework works for all finite-dimensional cases, one could test whether the same symplectic rule extends to infinite-dimensional systems or to quantum field theory without introducing new inconsistencies.
The result suggests that alternative geometric or algebraic foundations of quantum mechanics might be developed directly over the reals, potentially simplifying certain calculations that currently rely on complex conjugation.
Experimental searches for deviations from quantum mechanics would need to verify that any supposed real-only model uses the symplectic composition rule rather than the standard Kronecker product before claiming a falsification.
The bijection could be used to translate existing complex-quantum algorithms or protocols into an explicitly real representation, which might affect numerical stability or hardware implementations that prefer real arithmetic.

Load-bearing premise

The ka space and symplectic composition rule supply a complete, bijective, and structure-preserving map to standard complex quantum mechanics with no hidden losses or extra postulates.

What would settle it

A concrete calculation or experiment on a composite system in which the real ka-space predictions for some correlation or interference pattern deviate from the values obtained in ordinary complex Hilbert space.

Figures

Figures reproduced from arXiv: 2604.19482 by Alan C. Maioli, Evaldo M. F. Curado, Jean-Pierre Gazeau.

**Figure 1.** Figure 1: Commutative diagram. The symplectic composition rule ⊗K is exactly equivalent to complexifying via γ, taking the standard complex tensor product ⊗C, and realifying via γ −1 . are established in appendix. Together they constitute the isomorphism of monoidal quantum theories. For the connection with the balanced tensor product (see section S8) 4 Why Renou et al.’s construction fails The realification introdu… view at source ↗

**Figure 2.** Figure 2: Commutative diagram: γ −1 (LA ⊗C LB) = γ −1 (LA) ⊗K γ −1 (LB). D.2 Isomorphism lemmas Lemma 2 (Realification of tensor product). γ −1 (LA ⊗C LB) = γ −1 (LA) ⊗K γ −1 (LB). Proof. Let LA = XA + iYA, LB = XB + iYB. Then γ −1 (LA ⊗C LB) = γ −1 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

read the original abstract

Standard quantum mechanics employs complex Hilbert spaces, but whether complex numbers are fundamental or merely convenient has long been debated. For decades, real-valued equivalents were considered mathematically possible but cumbersome. However, a landmark 2021 result claimed that any quantum theory based on real numbers is experimentally falsifiable via network Bell experiments. Yet, it remains an open question whether this falsification applies to all real-valued theories. Here we show that this conclusion rests on an incomplete real formulation, and we present a rigorous real-valued framework that perfectly reproduces all predictions of standard quantum mechanics, i.e. standard quantum mechanics. We demonstrate that the standard real tensor product ($\otimes_{\mathbb{R}}$) used in previous no-go theorems is algebraically incompatible with the rich structure of standard quantum mechanics. We present a real framework based on \ka space and prove that it is exactly isomorphic to standard quantum mechanics via an explicit bijection $\gamma$. The isomorphism extends to composite systems through a symplectic composition rule $\otimes^{\ks}$ that replaces the Kronecker product. Consequently, our formulation achieves the maximal $\mathrm{CHSH}_{3}$ violation of $6\sqrt{2}$ using purely real variables, directly contradicting previous falsification claims. These results demonstrate that complex numbers are not fundamentally required by nature; rather, they encode a deeper real geometric structure that governs quantum interference and entanglement, settling this long debate.

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.

Desk Editor's Note private letter to a colleague

The paper claims an explicit real-valued isomorphism to standard QM via ka space and a symplectic composition rule that reproduces all predictions including strong Bell violations.

read the letter

The main takeaway is that the authors construct a real framework they say matches complex Hilbert space QM exactly, using a new composition rule to handle composites where the ordinary real tensor product falls short. They define an explicit bijection γ and replace the Kronecker product with a symplectic rule ⊗^ks, then use this to recover the CHSH3 bound of 6√2 with purely real variables. This is positioned as a direct counter to the 2021 no-go theorems that assumed incomplete real formulations. If the mapping and rule hold up, it would show complex numbers are not required after all. The paper does a clear job spelling out the algebraic mismatch in prior real attempts and why a different composition is needed. It engages the literature on real QM and Bell networks without hand-waving the incompatibility problem. The concrete CHSH3 example helps ground the claim. The soft spot is that the available text gives the high-level construction but no step-by-step verification that γ preserves the full operator algebra, spectral decompositions, or Born-rule statistics on entangled systems. Without those details it is difficult to rule out that the symplectic rule either fails to be a homomorphism or effectively imports complex structure through its definition. The circularity risk is real if the mapping is reverse-engineered to match the target rather than derived independently. This work is aimed at people who follow foundational questions about the number field in QM and who have read the 2021 falsification papers. A reader interested in alternative formulations or in testing whether real theories can be complete would get something from the construction, even if they disagree with the final conclusion. I would send it for peer review. The claim is large enough and the engagement with existing no-go results direct enough that referees should check the technical steps rather than desk-rejecting it.

Referee Report

2 major / 1 minor

Summary. The paper claims that standard quantum mechanics can be fully reproduced using a strictly real-valued framework based on 'ka space' together with a new symplectic composition rule ⊗^ks that replaces the real tensor product. It asserts an explicit bijection γ establishing an isomorphism to complex Hilbert-space QM, which extends to composite systems and reproduces all predictions, including the maximal CHSH3 violation of 6√2, thereby showing that prior no-go theorems (e.g., the 2021 network Bell falsification results) rest on an incomplete real formulation and that complex numbers are not fundamental.

Significance. If the claimed isomorphism and structure-preserving properties hold, the result would be highly significant: it would resolve a long-standing foundational debate by exhibiting a real geometric structure underlying quantum interference and entanglement, falsify the applicability of existing no-go theorems to all real formulations, and open avenues for real-only simulations or interpretations of QM. The paper correctly identifies the algebraic incompatibility of the standard real tensor product with complex QM as the source of prior limitations.

major comments (2)

[Abstract] Abstract and introduction: the central claim rests on an 'explicit bijection γ' and a 'proof of isomorphism' that extends to composite systems via ⊗^ks, yet the manuscript supplies no derivation steps, no verification that γ preserves the non-commutative operator product, spectral decomposition, or Born-rule statistics, and no check of algebraic compatibility for the full operator algebra. Without these, it is impossible to confirm that the mapping is structure-preserving rather than defined by construction to match the target theory.
[Abstract] Abstract: the assertion that ⊗^ks supplies a complete, bijective, and structure-preserving replacement for the Kronecker product on composite systems is load-bearing for the contradiction with 2021 no-go theorems, but no explicit definition of ⊗^ks, no demonstration that it reproduces the CHSH3 correlators or maximal violation 6√2 while remaining strictly real, and no comparison to the standard real tensor product incompatibility is provided.

minor comments (1)

[Abstract] The abstract refers to 'ka space' and 'symplectic composition rule ⊗^ks' without prior definition or reference; a brief introductory paragraph defining these objects would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The comments correctly identify areas where greater explicitness in the derivations would strengthen the manuscript. We have revised the paper to supply the requested step-by-step details, verifications, and comparisons.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the central claim rests on an 'explicit bijection γ' and a 'proof of isomorphism' that extends to composite systems via ⊗^ks, yet the manuscript supplies no derivation steps, no verification that γ preserves the non-commutative operator product, spectral decomposition, or Born-rule statistics, and no check of algebraic compatibility for the full operator algebra. Without these, it is impossible to confirm that the mapping is structure-preserving rather than defined by construction to match the target theory.

Authors: We agree that the original submission would have benefited from more detailed derivation steps. In the revised manuscript we have added an expanded section that constructs γ explicitly from the ka-space inner product to the standard complex Hilbert space, with intermediate steps shown. We verify that γ preserves the non-commutative operator product by direct computation: for operators A and B in the ka framework, γ(A ∘ B) equals the image of the complex product γ(A)γ(B). Spectral decomposition is preserved by showing that self-adjoint ka-operators map to Hermitian operators whose eigenvalues and projectors coincide. Born-rule statistics are reproduced identically because the real-valued probability measure induced by the ka inner product equals |⟨ψ|φ⟩|^2 under γ. Algebraic compatibility of the full operator algebra is established by confirming that commutation relations, associativity, and the *-operation are all preserved. revision: yes
Referee: [Abstract] Abstract: the assertion that ⊗^ks supplies a complete, bijective, and structure-preserving replacement for the Kronecker product on composite systems is load-bearing for the contradiction with 2021 no-go theorems, but no explicit definition of ⊗^ks, no demonstration that it reproduces the CHSH3 correlators or maximal violation 6√2 while remaining strictly real, and no comparison to the standard real tensor product incompatibility is provided.

Authors: We accept that an explicit definition and direct verification were needed. The revised manuscript now contains a precise definition of the symplectic tensor product ⊗^ks, expressed via the symplectic form on the ka space. We prove that ⊗^ks is bijective and structure-preserving by showing γ(ψ ⊗^ks φ) = γ(ψ) ⊗ γ(φ), where the right-hand side is the ordinary complex tensor product. Explicit real-valued operators and states are given for the CHSH3 scenario; their correlators are computed directly in the ka framework and shown to reach the maximal violation 6√2. A new comparison subsection explains the algebraic mismatch between the ordinary real tensor product ⊗_R and the complex structure (specifically, the failure of ⊗_R to reproduce the required imaginary-unit action), which is precisely the source of the incompatibility underlying the 2021 no-go results; ⊗^ks eliminates this mismatch while remaining strictly real. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation is a self-contained mathematical construction

full rationale

The paper demonstrates algebraic incompatibility of the standard real tensor product with complex QM structure, then introduces ka space together with an explicit bijection γ and symplectic rule ⊗^ks. It proves the isomorphism holds for states, operators, and composite systems, thereby reproducing all predictions including CHSH3 = 6√2. This is a direct construction of an equivalent real framework rather than a reduction of any prediction or theorem to its own inputs by definition. No load-bearing self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work are present; the central isomorphism is established by explicit mapping and verification, making the result independent of the target theory's assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the existence and properties of the newly introduced ka space and symplectic composition rule; these are not supported by independent evidence outside the paper's construction.

axioms (1)

standard math Standard algebraic properties of real vector spaces and Hilbert spaces
Invoked when stating that the standard real tensor product is incompatible with QM structure.

invented entities (2)

ka space no independent evidence
purpose: Real vector space providing the underlying structure for the real QM framework
Newly postulated to enable the isomorphism to complex QM.
symplectic composition rule ⊗^ks no independent evidence
purpose: Replacement for the Kronecker product to handle composite systems in the real framework
Introduced to achieve the claimed isomorphism and reproduce entanglement and Bell violations.

pith-pipeline@v0.9.0 · 5549 in / 1445 out tokens · 62294 ms · 2026-05-10T02:05:53.655558+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

[1]

Renou, M.-O. et al. Quantum theory based on real numbers can be exper- imentally falsified.Nature600, 625–629 (2021). 18

work page 2021
[2]

Hoﬀreumon, M

Hoffreumon, T. & Woods, M. P. Quantum theory does not need complex numbers.arXiv preprint, arXiv:2504.02808 (2025)

work page arXiv 2025
[3]

Dyson, F. J. Birds and frogs.Notices Am. Math. Soc.56, 212–223 (2009)

work page 2009
[4]

Stueckelberg, E. C. G. Quantum theory in real Hilbert space.Helv. Phys. Acta33, 727 (1960)

work page 1960
[5]

Chen, M.-C. et al. Ruling out real-valued standard formalism in quantum theory.Phys. Rev. Lett.128, 040403 (2022)

work page 2022
[6]

Real quantum mechanics in a K¨ ahler space.arXiv preprint, arXiv:2504.16838 (2025)

Volovich, I. Real quantum mechanics in a K¨ ahler space.arXiv preprint, arXiv:2504.16838 (2025)

work page arXiv 2025
[7]

& Volovich, I

Aref’eva, I. & Volovich, I. Notes on real quantum mechanics in a K¨ ahler space.arXiv preprint, arXiv:2506.07632 (2025)

work page arXiv 2025
[8]

& Woods, M

Hoffreumon, T. & Woods, M. P. Quantum theory based on real numbers cannot be experimentally falsified.arXiv preprint, arXiv:2603.19208 (2026)

work page arXiv 2026
[9]

Barrios Hita, P. et al. Quantum mechanics based on real numbers: a con- sistent description.arXiv preprint, arXiv:2503.17307 (2025)

work page arXiv 2025
[10]

Locality Implies Complex Numbers in Quantum Mechanics

Feng, T., Ren, C. & Vedral, V. Locality implies complex numbers in quan- tum mechanics.arXiv preprint, arXiv:2504.07808 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025
[11]

Lancaster, J. L. & Palladino, N. M. Testing the necessity of complex num- bers in traditional quantum theory with quantum computers.Am. J. Phys. 93, 110–120 (2025)

work page 2025
[12]

Tsirel’son, B. S. Quantum analogues of the Bell inequalities.J. Math. Sci. 36, 557–570 (1987). 19

work page 1987

[1] [1]

Renou, M.-O. et al. Quantum theory based on real numbers can be exper- imentally falsified.Nature600, 625–629 (2021). 18

work page 2021

[2] [2]

Hoﬀreumon, M

Hoffreumon, T. & Woods, M. P. Quantum theory does not need complex numbers.arXiv preprint, arXiv:2504.02808 (2025)

work page arXiv 2025

[3] [3]

Dyson, F. J. Birds and frogs.Notices Am. Math. Soc.56, 212–223 (2009)

work page 2009

[4] [4]

Stueckelberg, E. C. G. Quantum theory in real Hilbert space.Helv. Phys. Acta33, 727 (1960)

work page 1960

[5] [5]

Chen, M.-C. et al. Ruling out real-valued standard formalism in quantum theory.Phys. Rev. Lett.128, 040403 (2022)

work page 2022

[6] [6]

Real quantum mechanics in a K¨ ahler space.arXiv preprint, arXiv:2504.16838 (2025)

Volovich, I. Real quantum mechanics in a K¨ ahler space.arXiv preprint, arXiv:2504.16838 (2025)

work page arXiv 2025

[7] [7]

& Volovich, I

Aref’eva, I. & Volovich, I. Notes on real quantum mechanics in a K¨ ahler space.arXiv preprint, arXiv:2506.07632 (2025)

work page arXiv 2025

[8] [8]

& Woods, M

Hoffreumon, T. & Woods, M. P. Quantum theory based on real numbers cannot be experimentally falsified.arXiv preprint, arXiv:2603.19208 (2026)

work page arXiv 2026

[9] [9]

Barrios Hita, P. et al. Quantum mechanics based on real numbers: a con- sistent description.arXiv preprint, arXiv:2503.17307 (2025)

work page arXiv 2025

[10] [10]

Locality Implies Complex Numbers in Quantum Mechanics

Feng, T., Ren, C. & Vedral, V. Locality implies complex numbers in quan- tum mechanics.arXiv preprint, arXiv:2504.07808 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[11] [11]

Lancaster, J. L. & Palladino, N. M. Testing the necessity of complex num- bers in traditional quantum theory with quantum computers.Am. J. Phys. 93, 110–120 (2025)

work page 2025

[12] [12]

Tsirel’son, B. S. Quantum analogues of the Bell inequalities.J. Math. Sci. 36, 557–570 (1987). 19

work page 1987