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Geometric Quantization and The Orbit Method July 12, 2019 Author: Khallil Berrekkal Supervisors: prof. dr. Eric Opdam dhr. dr. Vladimir Gritsev Bachelor Thesis Mathematics and Physics Institute of Physics Korteweg-de Vries Instituut voor Wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam Abstract In this thesis the process of geometric quantization is described. Using geometry, a quantum system is constructed from a classical system, meaning a Hilbert space and a quantum observable are constructed, given a phase space and classical observable. We also look at the orbit method, where unitary irreducible representations of Lie groups are found, using geometrical objects called coadjoint orbits. The orbit method is the mathematical counterpart of geometrical quantization. Here we have looked at two examples, namely the Heisenberg Lie group and SU(2). Title: Geometric Quantization and The Orbit Method Author: Khallil Berrekkal, [email protected], 11293780 Supervisors: prof. dr. Eric Opdam, dhr. dr. Vladimir Gritsev Second reviewers: dhr. dr. H.B. Posthuma and dr. W.J. Waalewijn End date: July 12, 2019 Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math Institute of Physics Universiteit van Amsterdam Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl Contents 1 Introduction ...................................................5 I Part One: Mathematical Framework 7 2 Symplectic Geometry and Classical Mechanics .................9 2.1 Hamilton Formalism9 2.2 Symmetries and Lie Groups 12 3 Mathematical Notions ......................................... 15 3.1 Connections and Bundles 15 3.1.1 Bundles........................................................ 15 3.1.2 Connections.................................................... 16 3.2 Integrality 19 3.2.1 Parallel Transport................................................. 19 3.2.2 The Condition................................................... 20 II Part Two: Quantization 23 4 Geometric Quantization ....................................... 25 4.1 Prequantization 25 4.2 Polarization 27 4.2.1 The real case.................................................... 28 4.2.2 The complex case................................................ 28 4.2.3 The reduced Hilbert space.......................................... 29 4.2.4 Kähler......................................................... 29 4.3 Quantization of R2 30 4.3.1 Integrality condition and the prequantum Hilbert space..................... 30 4.3.2 The correct Hilbert space - Real polarization............................. 30 4.3.3 Quantizing observables............................................ 31 4.4 Quantization of S2 31 4.4.1 Integrality condition............................................... 31 4.4.2 Constructing the prequantum Hilbert space of S2 .......................... 31 4.4.3 The correct Hilbert space of S2 ....................................... 32 4.4.4 Quantizing the observables......................................... 33 4 III Part Three: Orbits 35 5 The Orbit Method ............................................. 37 5.1 Coadjoint Orbits 37 5.1.1 The symplectic structure on coadjoint orbits.............................. 38 5.2 Quantization and Representations 39 5.2.1 The Heisenberg algebra and group.................................... 39 5.2.2 Quantizing the coadjoint orbits of the Heisenberg Lie group.................. 40 5.2.3 Heisenberg Lie group/algebra representations............................ 41 5.2.4 SU(2) and S2 .................................................... 42 5.2.5 Quantization.................................................... 43 Conclusion 45 Popular Summary 46 Bibliography ................................................. 47 Books 47 Articles 47 Index ........................................................ 47 1. Introduction On the scale of everything around us that we encounter in our daily lives, everything can be described pretty well with the laws of Newton. The corresponding model is also known as classical mechanics, which fits well into our intuition. Classical mechanics has multiple formulations, of which the Hamilton formulation. Hamiltonian mechanics is simply a reformulation of classical mechanics using a geometrical language. This formalism provides a more abstract understanding of the theory. A classical state is described as being an element of a symplectic manifold and the physical quantities that can be measured are smooth real-valued functions from the manifold. Since the beginning of the 20th century it is well known that when we zoom into the world of atoms, electrons and other microscopical particles, that the laws of Newton clearly fall apart. Quantum mechanics was a new theory that described the anomalies that were visible on a fundamental scale. A state is now described as a wave function, an element of a Hilbert space, and the physical quantities that can be observed are operators which do not necessarily commute with each other. Quantization is a way to try to understand quantum mechanics through classical mechanics. The aim of geometric quantization is to construct a quantum system out of a classical system, meaning that from a symplectic manifold we construct a Hilbert space and from a classical observable (physical quantity that can be measured) we construct a quantum observable, using a geometric language. At least, this is the aim for a theoretical physicist. In mathematics representation theory is very useful to reduce abstract problems to linear algebra. Finding representations of a group is therefore of great significance. However, finding the represen- tations of a group might be difficult. Kirillov focused on finding the unitary representations of Lie groups through the orbit method. The orbit method constructs a correspondence between unitary representations and geometric objects called coadjoint orbits. This is more a method than a theory, since it has only worked perfectly for certain Lie groups. Moreover, geometric quantization can be thought of as the physical counterpart of the orbit method. So for a mathematician, the aim of geometric quantization could be finding the representations of Lie groups, using a physics intuition as guideline. *** I want to thank my supervisors prof. dr. Eric Opdam and dhr. dr. Vladimir Gritsev for providing me with this amazing subject and for guiding me through my thesis. Whenever I had questions, they were always easily available through email and open to answer my questions. I also want to thank Sliem el Ela for helping me in finding mistakes and for his support. A special thanks to thank David García Zelada who was prepared to help me even when I got stuck around 4 am, when pulling all-nighters. Part One: Mathematical I Framework 2 Symplectic Geometry and Classical Me- chanics ..............................9 2.1 Hamilton Formalism 2.2 Symmetries and Lie Groups 3 Mathematical Notions ................ 15 3.1 Connections and Bundles 3.2 Integrality 2. Symplectic Geometry and Classical Mechanics ymplectic geometry is the geometrical language of classical mechanics. It is a strong tool to give a coordinate free formulation of the equations of motion of some system, considering S phase spaces as symplectic manifolds. First we need to define a symplectic manifold. Definition 2.0.1 — Symplectic manifold. A symplectic manifold (M;w) is a smooth manifold M endowed with a 2-form w such that 1. w is closed, i.e. dw = 0, 2. w is non-degenerate, i.e. wp(v;w) = 0 8w 2 TpM =) v = 0: n In local coordinates we write w = ∑i; j=1 wi jdxi ^ dx j, where wi j is an anti-symmetric, invertible matrix. From this it follows that a symplectic manifold should always be even dimensional, since T n det(wi j) = det(wi j) = det(−(wi j)) = (−1) det(wi j): The determinant is only non-zero if n is even. 2n 2n Example 2.0.2 Let R have global coordinates (q1;:::;qn; p1;:::; pn). We can equip R with n the canonical symplectic form w0 = ∑i=1 dqi ^ dpi. Definition 2.0.3 — Morphism of symplectic manifolds . Let (M;w1) and (N;w2) be sym- plectic manifolds with F : M ! N a smooth map. F is called symplectic or a morhphism of symplectic manifolds if ∗ F w2 = w1: Moreover, if F is a diffeomorphism and F−1 is symplectic as well, we call F a symplectomor- phism. These basic definitions give us a very strong tool to formulate classical mechanis as we will see in the next paragraph. We need one more important result from symplectic geometry which we will not prove. Theorem 2.0.4 — Darboux. Let (M;w) be a 2n−dimensional symplectic manifold. For every m 2 M, there exists an open neighbourhood U, such that (U;wjU ) is symplectomorphic to an 2n open neighbourhood in R endowed with the canonical symplectic form w0. This is a nice result, since it means that every symplectic manifold has locally the same symplectic n structure as R2 . 2.1 Hamilton Formalism First we will formulate the well known coordinate-dependent Hamilton formalism, where we don’t 2n really need a symplectic structure yet. Let M = R with global coordinates (qi; pi), which we call the phase space of some classical system and the smooth functions on M are called the observables of the system. We endow M with an observable H 2 C¥(M), which we call the Hamiltonian , and ¥ we endow C (M) with the Poisson bracket {·;·gp which is (for the coordinate dependent version) defined as n ¶ f ¶g ¶ f ¶g f f ;ggP = ∑ − : i=1 ¶qi ¶ pi ¶ pi ¶qi For the Hamilton formalism we postulate that: 10 Chapter 2.
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