We analyze the relations between the standard formulation of quantum mechanics in complex Hilbert space and its counterpart in real Kähler space. Specifically, we demonstrate that the conventional quantum framework in complex Hilbert space can be faithfully reformulated within a real Kähler space without loss of structure or predictive power. A key result is the explicit isomorphism between Hermitian operators in the complex Hilbert space and Kähler-Hermitian operators in the real formulation. A remarkable property of the double degeneracy of observable spectra in Kähler quantum mechanics will be discussed. Furthermore, we also address the implications of this reformulation for composite quantum systems, emphasizing how entanglement and tensor-product structures manifest in the Kähler-space framework.
The talk is based on
I. Aref'eva and I. Volovich, Notes on Real Quantum Mechanics in a Kähler Space, in preparation.