Dans ce document, 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. Cependant, 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. Dans ce document, 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. Ce
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.

Cet article explore les excursions dans le temps et leurs implications.

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