This open access book explains geometric quantization from a physicist’s perspective. After presenting the general formalism, it delves into several examples reflecting current research interests in high-energy physics and condensed matter physics. Applications explore Chern-Simons theory, theta vacuum, the Hall effect, fluid dynamics, and elements of noncommutative geometry.The content is tailored to appeal to researchers, graduate students, and advanced undergraduates in high-energy physics, particle physics, and mathematical physics. A background in differential geometry and group theory is beneficial for a comprehensive understanding of the discussions. Preface Contents 1 Introduction 2 Symplectic Form and Poisson Brackets 2.1 Symplectic Structure 2.2 Poisson Brackets 2.3 Phase Volume 2.4 Darboux's Theorem 3 Classical Dynamics 4 Geometric Quantization 4.1 Pre-Quantization 4.2 Polarization 4.3 Measure of Integration 4.4 Representation of Operators 4.5 Comments on the Measure of Integration, Corrected Operators, Etc 5 Topological Features of Quantization 5.1 The Case of Nontrivial script upper H Superscript 1 Baseline left parenthesis upper M comma double struck upper R right parenthesismathcalH1(M, mathbbR) 5.2 The Case of Nontrivial script upper H squared left parenthesis upper M comma double struck upper R right parenthesismathcalH2(M, mathbbR) 5.3 Summary of Holomorphic Polarization and Quantization 6 Coherent States, the Two-Sphere and upper G divided by upper HG/H Spaces 6.1 Coherent States 6.2 Quantizing the Two-Sphere 6.2.1 Quantization Using Local Coordinates 6.2.2 Quantization Using Homogeneous Coordinates 6.2.3 Group Theoretic Version 6.3 Kähler Spaces of the upper G divided by upper HG/H-Type 6.3.1 Quantizing double struck upper C upper P squaredmathbbCP2 6.3.2 Quantizing General upper G divided by upper HG/H Spaces 6.3.3 A Note on an Index Theorem 6.3.4 A Short Historical Note 7 The Chern-Simons Theory in 2+1 Dimensions 7.1 Analysis on upper S squared times double struck upper RS2 timesmathbbR 7.2 Argument for Quantization of kk 7.3 The Ground State Wave Function 7.4 Abelian Theory on the Torus 8 thetaθ-Vacua in a Nonabelian Gauge Theory 9 Fractional Statistics in Quantum Hall Effect 9.1 Quantum Hall Effect and the Landau Problem 9.2 Excitations in Fractional QHE 10 Fluid Dynamics 10.1 The Lagrange Formulation 10.2 Clebsch Variables and the General Form of Action 10.3 Assorted Comments 10.4 Examples 10.4.1 Nonabelian Magnetohydrodynamics 10.4.2 Spin and Fluids 10.5 Anomalies in Fluid Dynamics 10.5.1 Anomalous Electrodynamics 10.5.2 Anomalies in the Fluid Phase of the Standard Model 10.5.3 The Chiral Magnetic Effect 11 Quantization Rules 12 A Comment on the Metaplectic Correction Solutions to Problems References Index
