Notes on Real Quantum Mechanics in a Kahler Space

TL;DR

This paper presents a real Kahler space formulation of quantum mechanics, demonstrating its equivalence to the standard complex Hilbert space approach while exploring subtle structural differences and their implications.

Contribution

It develops a systematic real Kahler space framework for quantum mechanics, clarifying the relationship with complex Hilbert spaces and addressing subtleties in composite systems and spectra.

Findings

Establishes a correspondence between Hermitian operators and real operators in Kahler space

Identifies and analyzes subtleties like overcounting and spectral degeneracy

Reaffirms equivalence while highlighting nuanced distinctions with foundational implications

Abstract

The necessity of complex numbers in quantum mechanics has long been debated. This paper develops a real Kahler space formulation of quantum mechanics [19], asserting equivalence to the standard complex Hilbert space framework. By mapping the complex Hilbert space ${\mathbb C}^n$ to a real Kahler space $K ^{2n}$, i.e. ${\mathbb R}^{2n}$ equipped with a metric, a symplectic structure and an automorphism, we establish a correspondence between Hermitian operators in ${\mathbb C}^n$ and real operators in $K^{2n}$. While the isomorphism appears straightforward some subtleties emerge: (i) the overcounting of composite system states under tensor products in ${\mathbb R}^{2n}$, and (ii) the double degeneracy of operator spectra in the real formulation. Through a systematic investigation of these challenges, we clarify the structural relationship between real and complex formulations, resolve ambiguities in composite system representations, and analyze spectral consequences. Our results reaffirm the equivalence of the two frameworks while highlighting nuanced distinctions with implications for foundational debates on locality, phase invariance, and the role of complex numbers in quantum theory.