The Role of Topology in Classical and Quantum Physics

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Springer Science & Business Media, Jan 1, 1992 - Electronic books - 239 pages
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In solid-state physics especially topological techniques have turned out to be extremely useful for modelling and explaining physical properties of matter. This book illustrates various applications of algebraic topology in classical field theory (non-linear sigma-models) and in quantizationsin multiply connected spaces (anyons). It treats Chern-Simon Lagrangians, Berry's phase, the polarization of light and the fractional quantum Hall effect.
  

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Contents

AN ELEMENTARY INTRODUCTION TO ALGEBRAIC TOPOLOGY
1
12 Some examples
2
13 Topology of Defects the case of planar spins
4
14 The Fundamental Group
8
15 The Abstract Fundamental Group Conjugacy Classes and freely Homotopic Loops
18
16 The OrderParameter Space as a Coset Space
28
17 Preliminary Theorems Concerning Fundamental Groups
42
18 Higher Homotopy Groups
46
34 Dynamical Implementation of Braid Statistics
135
TOPICS IN CHERNSIMONS PHYSICS
145
42 ChernSimons Lagrangians
146
43 Charged Particles Interacting with a CS Field
153
44 Gauge Fixing and an Explicit Solution for A𝜇
159
45 Effective Lafrangian for Statistical Particle Interaction
167
A SHORT INTRODUCTION TO CONNECTIONS ON UI BUNDLES AND BERRYS PHASE
172
52 Light Polarization the Hopf Bundle and Pancharatnams Phase
173

19 Relative Homotopy and Relative Homotopy Groups
57
110 The Exact Homotopy Sequence
58
TOPOLOGICAL METHODS IN CLASSICAL FIELD THEORY
67
22 The Pontrjagin Index and the Hopf Invariant
74
23 The SO3 Nonlinear SigmaModels in One and Two Space Dimensions
85
24 The d2 SO3 Nonlinear SigmaModel as a Gauge Theory
95
25 The GinzburgLandau Theory of Superconductivity
104
INEQUIVALENT QUANTIZATIONS IN MULTIPLY CONNECTED SPACES BRAID GROUPS AND ANYONS
114
32 Quantum Mechanics in Nonsimplyconnected Spaces
117
33 The Case of Identical Particles
124
53 The Quantum Adiabatic Phase
181
ELECTRONS IN A MAGNETIC FIELD AND A CURSORY LOOK AT THE QUANTUM HALL EFFECT
191
62 Preliminaries
194
63 The Classical Hall Effect
201
64 Bloch Electrons in a Magnetic Field
203
65 The Integer Quantum Hall Effect
210
66 The Fractional Quantum Hall Effect
217
67 FractionallyCharged Quasiparticles and the Hierarchy of Quantum Hall States
222
REFERENCES
231
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