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Cambridge University Press eBooks

Geometric quantization and algebraic Lagrangian geometry

Nikolai Andreevich Tyurin

Published Jan 18, 2007• 15 citations
Publisher

Abstract

Introduction The main theme of this survey is the quantization of classical mechanical systems in terms of algebraic geometry. The aim is thus to relate questions of mathematics and theoretical physics. We first recall briefly the main problems and methods which turn us to study the new subject. Quantization itself is the main problem of theoretical physics. The need to introduce and develop it was dictated by the creators of quantum theory. According to the ‘Copenhagen philosophy’, the physical predictions of a quantum theory must be formulated in terms of classical concepts (the first sentence of Woodhouse; here we quote the beginning of this survey). Thus in addition to the usual structures (Hilbert space, unitary transformations, selfadjoint operators …, see, for example, Landau and Lifschitz) any reasonable quantum theory has to admit an appropriate passage to a classical limit under which quantum observables are transferred to their classical analogues. However, as Dirac pointed out at the beginning of the quantum age, the correspondence between quantum theory and classical theory should be based not just on numerical coincidences taking place in the limit h → ∞, but on an analogy between their mathematical structures. Classical theory does approximate the quantum theory, but it does more – it supplies a framework for some interpretation of quantum theory. Using this idea, we can in general understand a quantization procedure as a correspondence between classical and quantum theories. In this sense, quantization of the classical mechanical systems is a movement in one direction, while taking the quasiclassical limit goes in the opposite direction.

DOI
10.1017/cbo9780511721472.009

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Introduction The main theme of this survey is the quantization of classical mechanical systems in terms of algebraic geometry. The aim is thus to relate questions of mathematics and theoretical physics. We first recall briefly the main problems and methods which turn us to study the new subject. Quantization itself is the main problem of theoretical physics. The need to introduce and develop it was dictated by the creators of quantum theory. According to the ‘Copenhagen philosophy’, the physical predictions of a quantum theory must be formulated in terms of classical concepts (the first sentence of Woodhouse; here we quote the beginning of this survey). Thus in addition to the usual structures (Hilbert space, unitary transformations, selfadjoint operators …, see, for example, Landau and Lifschitz) any reasonable quantum theory has to admit an appropriate passage to a classical limit under which quantum observables are transferred to their classical analogues. However, as Dirac pointed out at the beginning of the quantum age, the correspondence between quantum theory and classical theory should be based not just on numerical coincidences taking place in the limit h → ∞, but on an analogy between their mathematical structures. Classical theory does approximate the quantum theory, but it does more – it supplies a framework for some interpretation of quantum theory. Using this idea, we can in general understand a quantization procedure as a correspondence between classical and quantum theories. In this sense, quantization of the classical mechanical systems is a movement in one direction, while taking the quasiclassical limit goes in the opposite direction.
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.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
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".
These notes collect the lectures given by the author to the "XXIII International Workshop on Geometry and Physics" held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoretical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by "Feynman Path Integral Technique", is the method which is most widely used in quantum field theory. Finally, we will describe the "Weyl Quantization Approach" which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.
Usually one uses completely different mathematical descriptions to formulate classical and quantum mechanics. Classical theory may be nicely formulated in terms of symplectic geometry, and the quantum one in terms of algebraic objects related to a complex Hilbert space. However, it turns out that standard, nonrelativistic quantum mechanics possesses natural geometric structure that is even richer than that found in classical mechanics. This section reveals the beauty of the geometric approach to quantum theory and stands as a basis for the elegant geometrical ideas of Pancharatnam and, later on, of Aharonov and Anandan. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract It is the case that quantum mechanics has a deep geometric structure and can be presented accordingly. Quantum mechanics is to a certain degree foreshadowed by the geometry inherent in the geometric structure of classical mechanics. The purpose here is to present some new results and proofs which impact the mathematical structure of quantum mechanics. The relationship between integrability and quantum physics is investigated in terms of geometric ideas and structures. Two physical examples are drawn from these mathematical ideas which directly relate to physics. An introduction as to how these ideas can be extended to infinite degrees of freedom is also described.