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arXiv:1602.01054

A recollection of Souriau's derivation of the Weyl equation via geometric quantization

Christian Duval

Published Feb 2, 2016• 1 citation
arXivHTMLPDF

Abstract

These notes merely intend to memorialize Souriau's overlooked achievements regarding geo\-metric quantization of Poincaré-elementary symplectic systems. Restricting attention to his model of massless, spin-$\half$, particles, we faithfully rephrase and expound here Sections (18.82)--(18.96) & (19.122)--(19.134) of his book \cite{SSD} edited in 1969. Missing details about the use of a preferred Poincaré-invariant polarizer are provided for completeness.

DOI
10.48550/arxiv.1602.01054

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These notes merely intend to memorialize Souriau's overlooked achievements regarding geo-metric quantization of Poincaré-elementary symplectic systems. Restricting attention to his model of massless, spin-\half\half\half, particles, we faithfully rephrase and expound here Sections (18.82)--(18.96) & (19.122)--(19.134) of his book \cite{SSD} edited in 1969. Missing details about the use of a preferred Poincaré-invariant polarizer are provided for completeness.
Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. More precisely, is it possible to capture certain quantum idiosyncrasies within the symplectic framework of classical mechanics? Beginning with Dirac, the idea of quantizing a classical system involved associating the phase space variables with Hermitian operators which act on some Hilbert space, as well as associating the Poisson bracket on the phase space with the commutator for the operators. Mathematically the phase space is associated with some symplectic manifold MMM and the non-degenerate 2-form ωωω, which comes with MMM. 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 (R2n,∑i=1ndpi∧dqi\mathbb{R}^{2n}, \sum_{i=1}^n dp_i \wedge dq_iR2n,∑i=1n​dpi​∧dqi​) 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.
Expository notes which combine a historical survey of the development of quantum physics with a review of selected mathematical topics in quantization theory (addressed to students that are not complete novices in quantum mechanics). After recalling in the introduction the early stages of the quantum revolution, and recapitulating in Sect. 2.1 some basic notions of symplectic geometry, we survey in Sect. 2.2 the so called prequantization thus preparing the ground for an outline of geometric quantization (Sect. 2.3). In Sect. 3 we apply the general theory to the study of basic examples of quantization of Kaehler manifolds. In Sect. 4 we review the Weyl and Wigner maps and the work of Groenewold and Moyal that laid the foundations of quantum mechanics in phase space, ending with a brief survey of the modern development of deformation quantization. Sect. 5 provides a review of second quantization and its mathematical interpretation. We point out that the treatment of (nonrelativistic) bound states requires going beyond the neat mathematical formalization of the concept of second quantization. An appendix is devoted to Pascual Jordan, the least known among the creators of quantum mechanics and the chief architect of the "theory of quantized matter waves".
The aim of this paper is to revisit, in Souriau's book "Structure of Dynamical Systems", the chapter devoted to the geometric quantization where the justifications of important results and formulae are not given and are difficult to prove. After recalling the coadjoint orbit method and its application to the relativistic particle with spin, we state and prove two keystone theorems that allow to equip its prequantum manifold with the symplectic and contact structures. We apply them to the relativistic particle with spin 1/2, leading to the Dirac equation and the conservation of the probability current. We propose also an identity of conservation of the spin current and, invoking the Kaluza-Klein theory, a systematic construction of the symmetries of charge conjugation, parity transformation and time reversal which seems to us more convenient and readable than that of the classical presentations.