Barrios Hita et al. [Phys. Rev. Lett. 136, 240202 (2026)] proposed a quotient-space formulation of quantum mechanics over the real numbers and concluded that complex numbers are only a convenience. A preceding analysis [arXiv:2607.05865] showed that, when the construction is empirically equivalent to complex quantum mechanics, its real flag is protected by a hidden complex structure $\hat{J}$ and its composition rule is the balanced tensor product over that structure. Here we ask what would follow from the stronger operational reading that the flag is an accessible real degree of freedom. The answer is a direct clash with overlap-determinability (OD), introduced in [arXiv:2601.14638], and with standard no-go theorems. A readable or coherently controllable flag fixes a representative of each ray, and hence supplies precisely the phase convention that OD identifies as the missing resource for universal superposition of unknown states. Once promoted to a generic phase-access primitive, this resource enables probabilistic cloning of a linearly dependent set, steering-based signaling, and logarithmic-query unstructured search. Thus the quotient construction has a sharp follow-up interpretation: as a protected gauge representation it is standard complex quantum mechanics in real notation; as an accessible real theory it leaves the quantum operational framework.
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This work is a follow-up to our previous study, arXiv:2607.05865
Barrios Hita et al. [Phys. Rev. Lett. 136, 240202 (2026)] proposed a quotient-space formulation of quantum mechanics over the real numbers and concluded that complex numbers are only a convenience. A preceding analysis [arXiv:2607.05865] showed that, when the construction is empirically equivalent to complex quantum mechanics, its real flag is protected by a hidden complex structure J^ and its composition rule is the balanced tensor product over that structure. Here we ask what would follow from the stronger operational reading that the flag is an accessible real degree of freedom. The answer is a direct clash with overlap-determinability (OD), introduced in [arXiv:2601.14638], and with standard no-go theorems. A readable or coherently controllable flag fixes a representative of each ray, and hence supplies precisely the phase convention that OD identifies as the missing resource for universal superposition of unknown states. Once promoted to a generic phase-access primitive, this resource enables probabilistic cloning of a linearly dependent set, steering-based signaling, and logarithmic-query unstructured search. Thus the quotient construction has a sharp follow-up interpretation: as a protected gauge representation it is standard complex quantum mechanics in real notation; as an accessible real theory it leaves the quantum operational framework.
Barrios Hita et al. [Phys. Rev. Lett. 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 J2=−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 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.
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 highly cited 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. We demonstrate that the standard real tensor product (⊗R) used in previous no-go theorems is algebraically incompatible with the rich structure of conventional quantum mechanics. We present a real framework based on Kähler space and prove that it is exactly isomorphic to established 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 CHSH3 violation of 62 using purely real variables, demonstrating that the no-go theorem is specific to a particular real representation of states and operators and to the composition rule ⊗R built upon it, neither of which extends to the present Kähler framework. 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 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 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 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.
> In standard quantum mechanics, the imaginary part of the wavefunction is treated as a mathematical convenience. This work shows that it corresponds to a real internal coordinate in a strictly real two-component representation Ψ = (R, I). The quantum phase is identified as a rotation angle in this internal plane, and Schrödinger dynamics becomes a continuous rotation between R and I. Crucially, the I-component — though not directly measurable in a single projective measurement — can be fully reconstructed via controlled phase shifts and interferometry against a known reference. This explains why real-only models fail: they lack the second degree of freedom required to produce the sinθ term in interference patterns (P(θ) = A + B cosθ + C sinθ, with C ≠ 0). The model is empirically equivalent to complex quantum mechanics but provides a clear physical interpretation of the “imaginary” part as an internal information coordinate. > This document is self-contained, uses no complex notation, and is fully compatible with established quantum theory. It includes a testable experimental signature and connects to quantum tomography, interferometry, and foundational questions about the role of complex numbers in physics. This work is conceptually related to recent discussions on the necessity of complex numbers in quantum theory (Renou et al. 2021; Chen et al. 2022; Weilenmann et al. 2025), but takes a different approach by giving an explicitly real two-component interpretation of the imaginary part.”