A real-valued framework based on Kähler space with a symplectic composition rule exactly reproduces all standard quantum mechanics predictions, including maximal CHSH violation.
Real quantum mechanics in a K¨ ahler space.arXiv preprint, arXiv:2504.16838 (2025)
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abstract
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.
fields
quant-ph 4years
2026 4representative citing papers
The quotient-space real formulation of quantum mechanics is shown to be standard complex QM in real notation, not an independent real-amplitude theory, because it requires a hidden complex structure J, a superselection rule, and a balanced tensor product isomorphic to the complex tensor product.
A symplectic framework links quantum evolution to classical Hamiltonian dynamics on Kähler manifolds, yielding exponentially compressed quantum representations for integrable systems and approximate versions for others via perturbation theory.
The proposed postulate for choosing between real-number versions of quantum theory does not hold in Fermionic Information Theory and therefore is not general.
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