Complex numbers are central to the traditional formulation of quantum mechanics in complex Hilbert spaces. However, whether their use represents a fundamental necessity — as opposed to mere convenience — remains an open question. While several real-number reformulations of quantum mechanics have been proposed, we focus here on a framework proposed in [1] that recasts quantum mechanics within real Kähler spaces.
This Kähler-space formulation retains all defining features of quantum mechanics while introducing a critical advantage: it naturally embeds a Hamiltonian symplectic structure akin to classical mechanics. This alignment unifies classical and quantum dynamics under a shared geometric perspective. Furthermore, we demonstrate that finite-dimensional quantum systems exhibit manifest ergodicity within this framework, resolving long-standing interpretive ambiguities inherent to conventional complex formulations. Relations with p-dic and adelic quantum mechanics will be also discussed.
References
[1] I.~Volovich, Real Quantum Mechanics in a Kahler Space, arXiv:2504.16838.