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Studies in Geometric Quantization INIS-mf--11287 STUDIES IN GEOMETRIC QUANTIZATION G.M. TUYNMAN I STUDIES IN GEOMETRIC QUANTIZATION ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam, op gezag van de Rector Magnificus Prof. dr. S.K. Thoden van Velzen in het openbaar te verdedigen in de Aula der Universiteit (Oude Lutherse Kerk, ingang Singel 411, hoek Spui), op woensdag 3 februari 1988 te 15 uur door 1 GIJSBERT MAARTEN TUYNMAN | geboren te Amsterdam s* druk: Quick Service drukkerij Enschede Stellingen 1. De fasefactor exp( i/fi ƒ L dt ) is een wezenlijk onderdeel van de Feynman-pad- integraal-quantisatie methode. In het artikel van Dirac [Di] dat voor Feynman het uitgangspunt was voor zijn pad-integraal-quantisatie, wordt uitgegaan van het Lagrange formalisme om deze factor af te leiden. Gezien recente resultaten aangaande niet integreerbare fasefactoren [Ho] en gezien de prequantisatie constructie [Tu] is het veel bevredigender om deze factor af te leiden binnen het Hamilton formalisme. [Di] P.A.M. Dirac : The Lagrangian in quantum mechanics. - PhysilcZ.Sowjetunion 3 (1933) p64-72. [Ho] P.A. Horvathy : Etude géométrique du monopole magnétique. - J.Geom. & Physics 1 (1984) p39-78. [Tu] G.M. Tuynman : What is prequantization, and what is geometrie quantization? - Dit proefschrift 2. Wanneer men het proces van geometrische quantisatie toepast op een Kahler variëteit M, dan vindt men voor bijna alle observabelen f:M-»R een zelfgeadjungeerde operator O(f) waarvan het contravariante symbool in de zin van Berezin [Be] gegeven wordt door : . 'contravar v * • •" "dR . waarbij A^ de Laplace - de Rham operator is op M die hoort bij de Kahler structuur. - Dit proefschrift hoofdstuk 5. [Be] F.A. Berezin : Quantization. - Malh. USSR Izvestija 8 (1974) pi 109-1165. 3. De algemene definitie van een (Banach) variëteit zoals die in het boek "Manifolds, Tensor Analysis, and Applications" van Abraham, Marsden & Ratiu [Ab...] gegeven wordt is niet de gebruikelijke, doch definieert iets dat staat tussen een gebruikelijke variëteit en een S-atlas zoals gedefinieerd door W.T. van Est [vE]. [Ab...] R. Abraham, J.E. Manden & T. Ratiu : Manifolds, Tensor Analysis, and Applications. - Addison-Wesley, Reading, Massachuseu. 1983. [vE] W.T. van Est : Rapport sur les S-atlas. - Astérisque 116 (1984) p235-292. Stellingen bij het proefschrift "Studies in geometrie quantization" van G.M. Tuynman. 4. In tegenstelling tot wat algemeen wordt aangenomen, bestaan er drie, niet twee, verschillende varianten van geometrische quantisatie : met half-dichtheden, met half-vormen en met polarisatoren. 5. Het proces van geometrische quantisatie is een links-inverse van het proces dat quantummechanica beschrijft in het symplectisch formalisme. Dit proefschrift hoofdstuk 2. 6. De alternatieve beschrijving van centrale extensies door middel van commutator afbeeldingen in plaats van door 2-cohomologie klassen [Fr&Ka], [Le] werkt in het algemeen alleen voor vrije abelse groepen. [Fr&Ka] LB. Frenkel & V.G. Kac : Basic representations of affine Lie algebras and dual resonance models. - Proc.Natl.Acad.Sci. USA 82 (1985) p8295-8299. n n 7. De sytnplectische variëteit T*0S (dc cotangentiaal bundel van de eenheidsbol S in Rn+1 zonder de nul-sectie) bezit een familie van inequivalcnte complexe structuren die van n n+1 2 T*0S een Kahler variëteit maken die isomorf is met de ruimte {z e C \{0) I £. z = 0}. Het verdient aanbeveling deze opmerking in voorkomende gevallen op te nemen ter voorkoming van misverstanden bij onzorgvuldig vergelijken van resultaten. 8. Definities van tensoren in termen van grootheden met veel indices dragen niet bij tot een beter begrip van deze materie. Stellingen bij het proefschrift "Studie» in geometrie quantization'' van G.M. Tuynman. STUDIES IN GEOMETRIC QUANTIZATION STUDIES IN GEOMETRIC QUANTIZATION ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam, op gezag van de Rector Magnificus Prof. dr. S.K. Thoden van Velzen in het openbaar te verdedigen in de Aula der Universiteit (Oude Luthersc Kerk, ingang Singel 411, hoek Spui), op woensdag 3 februari 1988 te IS uur door GUSBERT MAARTEN TUYNMAN geboren te Amsterdam druk: Quick Service drukkerij Enschede Promotores: Prof, dr W. T. van Est Prof, dr J.-M. Souriau De promotie vindt plaats binnen de faculteit Wiskunde en Informatica Dedicated to the memory of my grandmother H.J. Tuijnman - Verhoeven I. i I J 1 Preface This thesis contains five chapters, of which the first, entitled "What is prequantization, and what is geometric quantization?", is meant as an introduction to geometric quantization for the non-specialist. The second chapter, entitled "Central extensions and physics" is written in collaboration with W. Wiegcrinck (student theoretical physics) and is submitted for publication. It deals with the notion of central extensions of manifolds and elaborates and proves the statements made in the first chapter. Central extensions of manifolds occur in physics as the freedom of a phase factor in the quantum mechanical state vector, as the phase factor in the prequantization process of classical mechanics and it appears in mathematics when studying central extension of Lie groups. In this chapter the connection between these central extensions is investigated and a remarkable similarity between classical and quantum mechanics is shown. Chapter three, entitled "Hyperfine interaction in a classical hydrogen atom and geometric quantization" is written in collaboration with C. Duval and J. Elhadad (University of Marseille) and is published in the Journal of Geometry and Physics (ns 3, 1987). In it is given a classical model for the hydrogen atom including spin-orbit and spin-spin interaction. The method of geometric quantization is applied to this model and the results are discussed. Finally chapters four and five, entitled "Generalised Bergman kernels and geometric quantization" and "Quantization : towards a comparison between methods" are published in the Journal of Mathematical Physics (na 28, 1987). In these chapters an explicit method to calculate the operators corresponding to classical observables is given when the phase space is a Kahler manifold. The obtained formula are then used to quantise symplectic manifolds which are irreducible hermitian symmetric spaces and the results are compared with other quantization procedures applied to these manifolds (in particular to Berezin's quantization). I wish to express my deepest gratitude to my "promotores" Prof. W.T. van Est and Prof. J.-M. Souriau, who have shaped my way of viewing the mathematical world. I would like to thank my coauthors C. Duval and J. Elhadad from the university of Marseille, whose support and friedship is gratefully acknowledged, and also W. Wiegerinck who assisted me in writing chapter two. I also would like to thank all those people who have helped me in one way or another to prepare this thesis, especially R. Brummelhuis, Prof. E.M. de Jager, E.A. de Kerf, A.P.E. ten Kroode, G. Laman, J. Lankelma, H.G.J. Pijls, L.W. Roeland, J. Slothouwer and J.J.O.O. Wiegerinck. Finally I would like to thank the Mathematisch Instituut of the University of Amsterdam for their hospitality and the fact that they provided a Macintosh on which this thesis is written. G.M. Tuynman Amsterdam, 1987 What is prequantization, and what is geometric quantization? by G.M. Tuynman Abstract This paper is intended for the non-specialist (either mathematician or theoretical physicist) with a minimum knowledge of differential geometry, classical mechanics and quantum mechanics. It explains the ideas of prequantization and geometric quantization with emphasis on prequantization. In short, prequantization shows that the mathematical models of classical mechanics and quantum mechanics are much more alike than one should expect from the conventional formulation of these theories. Geometric quantization uses these similarities between classical and quantum mechanics to extend the notion of canonical quantization to general systems in classical mechanics. To appreciate this, one should know that canonical quantization is only applicable to systems which are trivia! in the mathematical sense, e.g. a phase space R2". 1985 AMS subject classification : 70A05,81C99,58F05,81D07,58P06. Keywords: prequantization, geometric quantization. I M §1 Physical systems, states and models Theoretical physics is the science which tries to find rules, called physical laws, according to which a certain class of natural phenomena behave. Moreover, it tries to formulate these rules in terms of mathematical models. It is therefore very important to make a distinction between the natural phenomena, the physical laws and the mathematical models. A mathematical model consists usually of several items, e.g. sets (topological spaces, vector spaces, differentiable manifolds), special objects related to these sets (e.g. a metric tensor, a symplcctic 2-form), variables denoting elements of these sets (local coordinates such as position or pressure) and certain equations in these variables (e.g. "PV = RT" (Boyle - Gay Lussac) or "F = m dtydt2" (Newton)). I will call all these items together the mathematical ingredients of the model. The interpretation (also called the semantics) of such a model relates (some of) the mathematical ingredients to the real world, for instance to the position of the moon in the sky, or to the temperature of the water for your tea (it should be boiling!). Since all things influence each other, it follows that if we want to describe a natural phenomenon, we have to include the whole universe in our description. Clearly this is a task too complex to accomplish, so one has to make some approximations to reality. When studying a phenomenon (motion of the moon, boiling of water) one idealises the situation by neglecting those things which are supposed to be unimportant to the object of study and in fact one neglects as much as possible in order to retain only the essential features of the phenomenon one is studying. The result of neglecting all irrelevant items is called the physical system, hence a physical system is an idealisation of reality.
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