Although contact geometry and topology was briefly discussed in the book by V.I. ArnoF'd "Mathematical Methods of Classical Mechanics" (Springer- Verlag, 1989, 2nd edition), it still remains the domain of research in pure mathematics, e.g. see recent monograph by H. Geiges "An Introduction to Contact Topology" (Cambridge Univ. Press, 2008). Some attempts to use contact geometry in physics were made in the well-written monograph "Contact Geometry and Nonlinear Differential Equations" (Cambridge Univ. Press, 2007). Unfortunately, even excellent style of this monograph happens to be not sufficient to attract enough attention of physics community to this type of problematics as the GOOGLE search results indicate. To our knowledge, this book is the first serious attempt to change the existing status quo. In it we demonstrate that, in fact, all branches of theoretical physics can be rewritten in the language of contact geometry and topology. From mechanics, thermodynamics and electrodynamics to optics, gauge fields and gravity. From physics of liquid crystals to quantum mechanics and quantum computers, etc. The book is written in the style of the famous Landau-Lifshitz (L-L) multivolume course in theoretical physics. This means that its readers are expected to have solid background in theoretical physics (at least at the level of the L-L course). No prior knowledge of specialized mathematics is required. All needed new mathematical information is given in the context of discussed physical problems. As in the L-L course some problems/exercises are formulated along the way. As in this course, these are always supplemented by either solutions or by hints (with exact references). Unlike the L-L course, though, some theorems (sometimes with complete proofs) and remarks are also presented. This is done with the purpose of facilitating interest of our readers in deeper study of subject matters discussed in the text.

Preface v

Chapter 1. Motivation and Background 1

      1.1General Information 1

      1.2 Fluid Mechanics Formulation of Hamiltonian and Jacobian Mechanics. Emergence of the Force-Free Fields 2

      1.3 Some Basic Facts about the Force-Free Fields 7

Chapter 2. From Ideal Magnetohydrodynamics to String and Knot Theory 11

      2.1 General Information 11

      2.2 The Gillbarg Problem and the Theory of Foliations 13

      2.3 From String-Theoretic Lund-Regge Equation to Landau-Lifshitz Equation for the Vortex Filament 13

      2.4 Foliations of R3 by the Maxwellian Surfaces 17

      2.5The Maxwellian Tori and the Torus Knots Associated with Them 19

Chapter 3. All About and Around Woltjer's Theorem 25

      3.1General Information 25

      3.2 Equilibria in Liquid Crystals and the Faddeev-Skyrme Model for Pure Yang-Mills Fields 30

      3.3 Refinements of Woltjer's Theorem. Implications for Magnetohydrodynamics, Superconductivity and Liquid Crystals 36

      3.4 Proca's Massive Electrodynamics and Stueckelberg's TYick 40

      3.5 New Interpretation of the Dirac Monopole and its Use in the Problem of Quark Confinement 44

Chapter 4. Topologically Massive Gauge Theories and the Force-Free Fields 55

Chapter 5. Contact Geometry and Physics 61

      5.1 General Information 61

      5.2 Some Basic Facts about Contact Geometry and Topology 62

      5.3 Contact Geometry of Thermodynamics 66

      5.4 Contact and Symplectic Geometry and Liquid Crystals 67

      5.5 Force-Free (Beltrami) Fields and Contact Geometry and Topology of Hydrodynamics and Electromagnetism 69

      5.6 Many Facets of the Abelian Chern-Simons Functional and Their Relation to Monopoles, Dyons and the Faddeev-Skyrme Model 79

      5.6.1 General Information 79

      5.6.2 FYom Instantons to Monopoles 81

      5.6.3 Topology and the Non-Abelian Monopoles 84

      5.6.4 Hydrodynamics and the Faddeev-Skyrme model 89

      5.6.5 Helicity and Monopoles 89

      5.6.6 Some Comments on Dyons and Their Classical Analogs 93

Chapter 6. Sub-Riemannian Geometry, Heisenberg Manifolds and Quantum Mechanics of Landau Levels 99

      6.1 Motivation 99

      6.2 The Benchmark Example 100

      6.3 Basics of Sub-Riemannian Geometry 104

      6.4 Glimpses of Quantum Mechanics 110

      6.5 Fiber Bundle Reformulation of Sub-Riemannian Geometry and Classical-Quantum Correspondence. Connection with Dirac Monopoles 113

Chapter 7. Abrikosov Lattices, TGB Phases in Liquid Crystals and Heisenberg Group 131

Chapter 8. Sub-Riemannian Geometry, Spin Dynamics and Quantum-Classical Optimal Control 139

      8.1 General Information 139

      8.2 Quantum Computers Paradigm and Dynamics of 2-Level Quantum Systems 141

      8.2.1 What is Quantum Computation and Quantum Computer? 141

      8.2.2 Connecting Quantum 2-Level Systems with Classical Reality by Using the Hopf Map 142

      8.2.3 Some Mathematical Facts about the Rigid Body Rotations and Their Applications to Physical Problems 143

      8.2.4 Poinsot versus Euler and Kirchhoff 147

      8.2.5 Two-Level Quantum Systems. List of Applications 150

      8.2.6 Contact Geometry on S3 and Optimal Control of 2-Level Systems 151

      8.2.7 Dirac Quantization of Dynamical Systems with Constraints and Contact Geometry 160

      8.3 Beyond the 2-Level Quantum Systems 160

      8.3.1 General Information 160

      8.3.2 Types of Josephson Junctions Used in Quantum Computing, Atomic Physics and Quantum Optics 162

      8.3.3 Josephson Junctions — Semiflexible Polymers Correspondence. A Gentle Introduction 164

      8.3.4 From Pauli-Schrodinger to Dirac Equation 170

      8.3.5 Quantum Random Walks — Semiflexible Polymers Correspondence 174

      8.4 Semiflexible Polymers and Quantum Computers 179

      8.4.1 General Information 179

      8.4.2 Some Helpful Facts from Theory of Semiflexible Polymers: Dirac versus Kratky-Porod 181

      8.4.3 Qubits from Hairpins 188

      8.4.4 Semiflexible Polymers, Dirac Monopole and Heisenberg Ferromagnets 196

      8.4.5 From the L-L Equation to Neumann's Model. Back to Contact Geometry 200

      8.4.6 Delicate Inter-relationship between Classical and Quantum Neumann's Problems in the Context of Semiflexible Polymers, Euler's Elastica and Ion Traps 204

Chapter 9. From Contact Geometry to Contact Topology 211

      9.1 General Information 211

      9.2 Mathematics and Physics of the Cauchy Problem in Quantum Mechanics: Viktor Maslov versus David Bohm 212

      9.3 FYom Maslov and Bohm to Bell and Beyond 221

      9.4 Harmonious Coexistence of Classical and Quantum Mechanics: All About and Around the Duistermaat-Heckman Formula 225

      9.5 Mathematics and Physics of Weinstein's Conjecture: From Classical Statistical Mechanics to Seiberg-Witten Monopoles 235

      9.6 Quantum Money, Lagrangian, Legendrian and Transverse Knots and Links and the Associated Grid Diagrams 242

      9.7 Latest Developments in Contact Geometry and Topology. A Guided Tour with Physics Applications in the Perspective 257

      9.7.1 General Information 257

      9.7.2 Solution of the Planar Ising Model on the Grid Diagram and Chekanov-Pushkar-Eliashberg Invariants 258

      9.7.3 Several Remarks about Chekanov-Eliashberg Differential Graded Algebra (DGA) and Symplectic Field Theory 261

      9.7.4 Some Thoughts about the Heegaard-Floer Homology 262

Closing Remarks: The Unreasonable Effectiveness of Contact Geometry and Topology in Physical Sciences 269

Appendix A. Heisenberg Group in the Context of Sub-Riemannian Geometry and Optimal Control 277

      A1 The Laplace-Young Equation in the Presence of Surfaces and Its Connection with Dido Problem and Its Dual 277

      A.2 Some Facts about the Heisenberg Group 279

      A.3 Connections with Sub-Riemannian Geometry and Pontryagin's Theory of Optimal Control 282

Appendix B. Sub-Riemannian Dynamics of Josephson Junctions 285

      B.1 Nilpotency, Bracket-Generation and Solvability 285

      B.2 EulerJs Elastica. The Sub-Riemannian Treatment 287

Appendix C. Quantum Computers and Quantum Random Walks 293

      C.1 Classical Computers (Basics) 293

      C.2 Boolean Algebra and Number Theory 294

      C.3 Classical Computers: Turing Machines versus Classical Circuits 297

      C.4 Reversible Computation (Conservative Logic) and Billiard Ball Computers 297

      C.5 Boolean Algebra, Directed Random Walks, Yang-Baxter Equations and Integrable Quantum Computation 302

      C.6 From Classical to Quantum Random Walks 319

Appendix D. The Measurement Protocol. Geometry and Topology of Entanglements 325

      D.1 The Measurement Protocol 325

      D.2 Geometry and Topology of Entanglements 335

      D.2.1 Classical Mechanics Inside Quantum Mechanics. Basic Facts 335

      D.2.2 From Zonotopes to Fans and Arrangements 338

      D.2.3 What Classical Dynamics on Two-Sphere Can Teach Us? 342

      D.2.4 Designing Toric Varieties from Fans 345

      D.2.5 Torus Action and Its Invariants 347

      D.2.6 The Remarkable Duality (Beginning) 350

      D.2.7 The Remarkable Duality (Continuation). Delicate Inter-relationship between Flags and Grassmannians 352

      D.2.8 The Bruhat Decomposition and Bruhat Order 354

      D.2.9 The Remarkable Duality (End). Weyl Groups, Flags and Grassmannians 358

      D.2.10 Connections between the Grassmannians and Exactly Integrable Systems. Many Uses of Pliicker Embedding: From Robotics and Computer Vision to Twistor Theory 361

      D.2.11 The Problem of Quantum Entanglement (Statement and Goals) 368

      D.2.12 Classical Mechanics Inside Quantum Mechanics (Revisited) 368

      D.2.13 Some Lessons from Morse Theory 379

      D.2.14 FYom Analysis to Synthesis: c-Phase Gate and Grassmannians 386

      D.2.15 Entangling Grassmannians with Twistors 396

      D.2.16 Complex Contact Geometry of Twistor Spaces and the Problem of Optimal Control of Entanglements 402

      D.2.17 Nilpotent and Coadjoint Orbits, Lagrangian Fibrations and Lagrangian Grassmannians 412

      D.2.18 Coupled Josephson Junctions and Linkages, Torus Knots and Entanglements, Legendrian Manifolds and Optimal Control: From Snakes and Airplanes to Quantum Hall Effect 424

References 437

Index 469

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