Real Quantum Mechanics in a Kahler Space

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.