Hidden Complex Structure in Quotient-Space Real Quantum Mechanics

• URL: http://arxiv.org/abs/2607.05865v1
• Date: Tue, 07 Jul 2026 05:47:59 GMT
• Title: Hidden Complex Structure in Quotient-Space Real Quantum Mechanics
• Abstract summary: We argue that quantum mechanics can be formulated over the real numbers by replacing the tensor-product postulate with a quotient-space construction.<n>We show that the operational content of this construction is not that of a generic real Hilbert-space theory.<n>This clarifies what is, and is not, excluded by experiments testing the necessity of complex numbers.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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