Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.19482

Quantum mechanics over real numbers fully reproduces standard quantum theory

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

quant-phSubmitted Apr 21, 2026• Announced Apr 22, 2026• 0 citations
v1 · Apr 21, 2026v2 · May 26, 2026v3 · Jun 23, 2026
arXivHTMLPDF

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 $γ$. 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.

Related papers

Experimental Equivalence of Real and Complex Quantum Mechanics: Resolving the CHSH$_3$ No-Go Result

Pushpendra Singh

A recent study by Renou et al. (Nature 600, 625, 2021) reported that quantum theory formulated over real numbers can be experimentally falsified via a gap in the maximal Clauser–Horne–Shimony–Holt CHSH$_3$ violation. Here, we show that this apparent distinction is not fundamental but arises from an incomplete composition rule for composite systems. Using a structure‑preserving real representation of quantum mechanics (Singh, Ann. Phys. 487, 170368, 2026), we demonstrate that the maximal violation $6\sqrt{2}$ is achieved using only real arithmetic. Our explicit construction for the tripartite entanglement‑swapping scenario reproduces exactly the complex quantum prediction. We further establish an algebraic isomorphism between real and complex quantum mechanics that extends to a structure-preserving strict monoidal functor. This establishes an operational equivalence between real and complex quantum mechanics, demonstrating that all quantum correlations can be realized without invoking complex numbers.Keywords: Quantum foundations, Real quantum mechanics, Symplectic geometry, Tensor product structure, Real Hilbert space, Real-complex isomorphism, Bell nonlocality, Entanglement, Quantum correlations.

•Apr 30, 2026•0 citations

Notes on Real Quantum Mechanics in a Kahler Space

I. Ya. Aref’eva, Igor Vasil'evich Volovich

The necessity of complex numbers in quantum mechanics has long been debated. This paper develops a real Kahler space formulation of quantum mechanics [19], asserting equivalence to the standard complex Hilbert space framework. By mapping the complex Hilbert space ${\mathbb C}^n$ to a real Kahler space $K ^{2n}$, i.e. ${\mathbb R}^{2n}$ equipped with a metric, a symplectic structure and an automorphism, we establish a correspondence between Hermitian operators in ${\mathbb C}^n$ and real operators in $K^{2n}$. While the isomorphism appears straightforward some subtleties emerge: (i) the overcounting of composite system states under tensor products in ${\mathbb R}^{2n}$, and (ii) the double degeneracy of operator spectra in the real formulation. Through a systematic investigation of these challenges, we clarify the structural relationship between real and complex formulations, resolve ambiguities in composite system representations, and analyze spectral consequences. Our results reaffirm the equivalence of the two frameworks while highlighting nuanced distinctions with implications for foundational debates on locality, phase invariance, and the role of complex numbers in quantum theory.

arXiv:2506.07632•Jun 9, 2025•0 citations

Quantum theory based on real numbers cannot be experimentally falsified

Timothée Hoffreumon, Mischa P. Woods

Whether the complex numbers of standard quantum theory are experimentally indispensable has remained open for decades. Real quantum theory (RQT), obtained by replacing complex amplitudes with real ones while retaining the usual Kronecker-product composition rule, reproduces all single-party and bipartite Bell correlations of quantum theory (QT), but its lack of local tomography suggested that the two theories might diverge in more general local experiments. This possibility appeared to be confirmed by Renou et al., who argued that a bilocal network experiment can falsify RQT without falsifying QT. Here we show that this conclusion relies on an experimentally untestable assumption. The key distinction is between product-state independence, which constrains the mathematical form of source states, and operational independence, which is defined entirely by the absence of observable cross-source correlations. We prove that, once source independence is imposed operationally, every finite network correlation achievable in QT is also achievable in RQT with the same locality structure of the measurements. We then extend this equivalence to arbitrary finite sequential multipartite protocols involving channels and measurements with prescribed locality structure. Thus, as long as no violation of QT is observed, RQT cannot be experimentally falsified. Our results restore the empirical indistinguishability of QT and RQT, while showing that they support markedly different pictures of the correlation structure underlying the same observed world.

quant-pharXiv:2603.19208•Mar 20, 2026•1 citation

Real Quantum Mechanics in a Kahler Space

Igor Vasil'evich Volovich

In this paper, we demonstrate the equivalence between the complex Hilbert space and real Kahler space formulations of quantum mechanics. Complex numbers play an important role in the traditional formulation of quantum mechanics in complex Hilbert spaces. However, the necessity of complex numbers--as opposed to their mere convenience--remains a subject of debate. Several alternative formulations of quantum mechanics using real numbers have been proposed. In this paper, we demonstrate that standard quantum mechanics, formulated in a complex Hilbert space, admits an equivalent reformulation in a real Kahler space. By establishing a natural isomorphism between the operator theories of the complex Hilbert space and the real Kahler space, we prove the equivalence of the two formulations including composite system. This Kahler-space framework preserves all essential features of quantum mechanics while offering a key advantage: it inherently incorporates a Hamiltonian symplectic structure analogous to classical mechanics. This structural alignment provides a unified geometric perspective for both classical and quantum dynamics. Additionally, we show that the ergodicity of finite-dimensional quantum systems becomes manifest in this framework, resolving interpretational ambiguities present in conventional complex formulations.

arXiv:2504.16838•Apr 23, 2025•0 citations

Hidden Complex Structure in Quotient-Space Real Quantum Mechanics

Jeongho Bang, Kyoungho Cho, Kyunghyun Baek

Barrios Hita et al. [Phys. Rev. Lett. $\bf{136}$, 240202 (2026)] argued that quantum mechanics can be formulated over the real numbers by replacing the tensor-product postulate with a quotient-space construction, and concluded that complex numbers are therefore a matter of convenience. We show that the operational content of this construction is not that of a generic real Hilbert-space theory. Empirical equivalence requires a distinguished real linear operator $J$ with $J^2 = -\mathbb{1}$, and all physical effects, instruments, and dynamics must preserve the corresponding $SO(2)$ gauge. Moreover, the composite-system rule is a balanced tensor product over this hidden complex structure, not the ordinary tensor product over $\mathbb{R}$. In multipartite network scenarios, this changes the meaning of source independence: canonical real representatives are not source-factorizable in the usual tensor-product sense. Thus, the construction is best understood as standard complex quantum mechanics written in real notation, not as an independent real-amplitude theory. This clarifies what is, and is not, excluded by experiments testing the necessity of complex numbers.

quant-pharXiv:2607.05865•Jul 8, 2026•0 citations

Application of Real valued quantum mechanical framework on quantum mechanics, quantum field theory and quantum computing

Stotra Goswami

The historical debate surrounding the necessity of complex numbers in quantum mechanics has been revitalized by recent experimental falsifications of standard real-amplitude models. However, Barrios Hita et al. (2026) demonstrated that a physically consistent real-number quantum theory can be constructed by mapping complex Hilbert spaces onto real quotient Hilbert spaces with local flags. In this work, we extensively expand this framework to full quantum field theory and numerous applications. We present explicit, index-level, and operator-level derivations of non-relativistic dynamics (the Schr¨odinger equation, Ehrenfest’s theorem, uncertainty principle), relativistic quantum mechanics (Klein-Gordon and Dirac equations), and quantum field theory (canonical quantization, Fock space, propagators, path integrals). We further apply the real formulation to harmonic oscillators, spin systems, perturbation theory, scattering theory, quantum entanglement, gauge theories and Quantum computing. We show that U(1) gauge transformations are rigorously isomorphic to SO(2) rotations in the real flag space, and that all physical predictions—including Bell violations and scattering amplitudes—are exactly reproduced. All derivations are presented at the explicit index level with full algebraic details, using a symplectic structure J that replaces the complex unit i.

•Aug 9, 2026•0 citations
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 (⊗R\otimes_{\mathbb{R}}⊗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 γγγ. The isomorphism extends to composite systems through a symplectic composition rule ⊗\ks\otimes^{\ks}⊗\ks that replaces the Kronecker product. Consequently, our formulation achieves the maximal CHSH3\mathrm{CHSH}_{3}CHSH3​ violation of 626\sqrt{2}62​ 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.
Experimental Equivalence of Real and Complex Quantum Mechanics: Resolving the CHSH3_33​ No-Go Result
A recent study by Renou et al. (Nature 600, 625, 2021) reported that quantum theory formulated over real numbers can be experimentally falsified via a gap in the maximal Clauser–Horne–Shimony–Holt CHSH3_33​ violation. Here, we show that this apparent distinction is not fundamental but arises from an incomplete composition rule for composite systems. Using a structure‑preserving real representation of quantum mechanics (Singh, Ann. Phys. 487, 170368, 2026), we demonstrate that the maximal violation 626\sqrt{2}62​ is achieved using only real arithmetic. Our explicit construction for the tripartite entanglement‑swapping scenario reproduces exactly the complex quantum prediction. We further establish an algebraic isomorphism between real and complex quantum mechanics that extends to a structure-preserving strict monoidal functor. This establishes an operational equivalence between real and complex quantum mechanics, demonstrating that all quantum correlations can be realized without invoking complex numbers.Keywords: Quantum foundations, Real quantum mechanics, Symplectic geometry, Tensor product structure, Real Hilbert space, Real-complex isomorphism, Bell nonlocality, Entanglement, Quantum correlations.
The necessity of complex numbers in quantum mechanics has long been debated. This paper develops a real Kahler space formulation of quantum mechanics [19], asserting equivalence to the standard complex Hilbert space framework. By mapping the complex Hilbert space Cn{\mathbb C}^nCn to a real Kahler space K2nK ^{2n}K2n, i.e. R2n{\mathbb R}^{2n}R2n equipped with a metric, a symplectic structure and an automorphism, we establish a correspondence between Hermitian operators in Cn{\mathbb C}^nCn and real operators in K2nK^{2n}K2n. While the isomorphism appears straightforward some subtleties emerge: (i) the overcounting of composite system states under tensor products in R2n{\mathbb R}^{2n}R2n, and (ii) the double degeneracy of operator spectra in the real formulation. Through a systematic investigation of these challenges, we clarify the structural relationship between real and complex formulations, resolve ambiguities in composite system representations, and analyze spectral consequences. Our results reaffirm the equivalence of the two frameworks while highlighting nuanced distinctions with implications for foundational debates on locality, phase invariance, and the role of complex numbers in quantum theory.
Whether the complex numbers of standard quantum theory are experimentally indispensable has remained open for decades. Real quantum theory (RQT), obtained by replacing complex amplitudes with real ones while retaining the usual Kronecker-product composition rule, reproduces all single-party and bipartite Bell correlations of quantum theory (QT), but its lack of local tomography suggested that the two theories might diverge in more general local experiments. This possibility appeared to be confirmed by Renou et al., who argued that a bilocal network experiment can falsify RQT without falsifying QT. Here we show that this conclusion relies on an experimentally untestable assumption. The key distinction is between product-state independence, which constrains the mathematical form of source states, and operational independence, which is defined entirely by the absence of observable cross-source correlations. We prove that, once source independence is imposed operationally, every finite network correlation achievable in QT is also achievable in RQT with the same locality structure of the measurements. We then extend this equivalence to arbitrary finite sequential multipartite protocols involving channels and measurements with prescribed locality structure. Thus, as long as no violation of QT is observed, RQT cannot be experimentally falsified. Our results restore the empirical indistinguishability of QT and RQT, while showing that they support markedly different pictures of the correlation structure underlying the same observed world.
In this paper, we demonstrate the equivalence between the complex Hilbert space and real Kahler space formulations of quantum mechanics. Complex numbers play an important role in the traditional formulation of quantum mechanics in complex Hilbert spaces. However, the necessity of complex numbers--as opposed to their mere convenience--remains a subject of debate. Several alternative formulations of quantum mechanics using real numbers have been proposed. In this paper, we demonstrate that standard quantum mechanics, formulated in a complex Hilbert space, admits an equivalent reformulation in a real Kahler space. By establishing a natural isomorphism between the operator theories of the complex Hilbert space and the real Kahler space, we prove the equivalence of the two formulations including composite system. This Kahler-space framework preserves all essential features of quantum mechanics while offering a key advantage: it inherently incorporates a Hamiltonian symplectic structure analogous to classical mechanics. This structural alignment provides a unified geometric perspective for both classical and quantum dynamics. Additionally, we show that the ergodicity of finite-dimensional quantum systems becomes manifest in this framework, resolving interpretational ambiguities present in conventional complex formulations.
Barrios Hita et al. [Phys. Rev. Lett. 136\bf{136}136, 240202 (2026)] argued that quantum mechanics can be formulated over the real numbers by replacing the tensor-product postulate with a quotient-space construction, and concluded that complex numbers are therefore a matter of convenience. We show that the operational content of this construction is not that of a generic real Hilbert-space theory. Empirical equivalence requires a distinguished real linear operator JJJ with J2=−1J^2 = -\mathbb{1}J2=−1, and all physical effects, instruments, and dynamics must preserve the corresponding SO(2)SO(2)SO(2) gauge. Moreover, the composite-system rule is a balanced tensor product over this hidden complex structure, not the ordinary tensor product over R\mathbb{R}R. In multipartite network scenarios, this changes the meaning of source independence: canonical real representatives are not source-factorizable in the usual tensor-product sense. Thus, the construction is best understood as standard complex quantum mechanics written in real notation, not as an independent real-amplitude theory. This clarifies what is, and is not, excluded by experiments testing the necessity of complex numbers.
The historical debate surrounding the necessity of complex numbers in quantum mechanics has been revitalized by recent experimental falsifications of standard real-amplitude models. However, Barrios Hita et al. (2026) demonstrated that a physically consistent real-number quantum theory can be constructed by mapping complex Hilbert spaces onto real quotient Hilbert spaces with local flags. In this work, we extensively expand this framework to full quantum field theory and numerous applications. We present explicit, index-level, and operator-level derivations of non-relativistic dynamics (the Schr¨odinger equation, Ehrenfest’s theorem, uncertainty principle), relativistic quantum mechanics (Klein-Gordon and Dirac equations), and quantum field theory (canonical quantization, Fock space, propagators, path integrals). We further apply the real formulation to harmonic oscillators, spin systems, perturbation theory, scattering theory, quantum entanglement, gauge theories and Quantum computing. We show that U(1) gauge transformations are rigorously isomorphic to SO(2) rotations in the real flag space, and that all physical predictions—including Bell violations and scattering amplitudes—are exactly reproduced. All derivations are presented at the explicit index level with full algebraic details, using a symplectic structure J that replaces the complex unit i.