Geomatric Quantization of Complex Phase Space with Kahler Metric

道夫西岡

Published Oct 1, 1979• 0 citations

Abstract

By introducing a complex phase into classical mechanical system, we study geomatic quantization of complex phase space woth Kahler metric to give a method of deriving Hilbert space representig the corresponding quantum mechanical system. We present two …

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Geomatric Quantization of Complex Phase Space with Kahler Metric

道夫西岡

Published Oct 1, 1979• 0 citations

OpenAlex

Abstract

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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.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.

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{\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.

M

and the non-degenerate 2-form

ω

, which comes with

M

. Geometric prequantization is a process by which one does this in a mathematically "rigorous" manner and we shall attempt to just introduce the methods here. We do this by exploring this contruction for (

\mathbb{R}^{2n}, \sum_{i=1}^n dp_i \wedge dq_i

) which avoids using complex line bundles in any non-trivial way. One should note however that the Hilbert space one obtains is in fact too "big", in the sense that it has too many functions in order to correspond with actual physically significant Hilbert spaces. Geometric quantization remedies this situation but it should be noted that not all manifolds are prequantizable. We shall not discuss either of these issues however.