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arXiv:2607.05865

Hidden Complex Structure in Quotient-Space Real Quantum Mechanics

Jeongho Bang, Kyoungho Cho, Kyunghyun Baek

quant-phSubmitted Jul 7, 2026• Announced Jul 8, 2026• 0 citations
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Abstract

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.

Comments
17 pages (main text + supplemental material)

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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.
Complex numbers play a crucial role in quantum mechanics. However, their necessity remains debated: whether they are fundamental or merely convenient. Recently, it was shown that any real-number quantum theory satisfying certain postulates can be falsified with multipartite experiments. In this Letter we show that a physically motivated postulate about composite quantum systems allows us to construct quantum mechanics based on real numbers that reproduces predictions for all multipartite quantum experiments. Thus, we argue that real-valued quantum mechanics cannot be falsified, and therefore the use of complex numbers is a matter of convenience.
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.