幾何学、位相幾何学および物理学(第2版)<br>Geometry, Topology and Physics (2ND)

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幾何学、位相幾何学および物理学(第2版)
Geometry, Topology and Physics (2ND)

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  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 650 p.
  • 言語 ENG
  • 商品コード 9780750306065
  • DDC分類 516.36

Full Description


Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.

Table of Contents

Preface to the First Edition                       xvii
Preface to the Second Edition xix
How to Read this Book xxi
Notation and Conventions xxii
Quantum Physics 1 (66)
Analytical mechanics 1 (8)
Newtonian mechanics 1 (1)
Lagrangian formalism 2 (3)
Hamiltonian formalism 5 (4)
Canonical quantization 9 (10)
Hilbert space, bras and kets 9 (1)
Axioms of canonical quantization 10 (3)
Heisenberg equation, Heisenberg picture 13 (1)
and Schrodinger picture
Wavefunction 13 (4)
Harmonic oscillator 17 (2)
Path integral quantization of a Bose 19 (12)
particle
Path integral quantization 19 (7)
Imaginary time and partition function 26 (2)
Time-ordered product and generating 28 (3)
functional
Harmonic oscillator 31 (7)
Transition amplitude 31 (4)
Partition function 35 (3)
Path integral quantization of a Fermi 38 (13)
particle
Fermionic harmonic oscillator 39 (1)
Calculus of Grassmann numbers 40 (1)
Differentiation 41 (1)
Integration 42 (1)
Delta-function 43 (1)
Gaussian integral 44 (1)
Functional derivative 45 (1)
Complex conjugation 45 (1)
Coherent states and completeness 46 (1)
relation
Partition function of a fermionic 47 (4)
oscillator
Quantization of a scalar field 51 (4)
Free scalar field 51 (3)
Interacting scalar field 54 (1)
Quantization of a Dirac field 55 (1)
Gauge theories 56 (4)
Abelian gauge theories 56 (2)
Non-Abelian gauge theories 58 (2)
Higgs fields 60 (1)
Magnetic monopoles 60 (3)
Dirac monopole 61 (1)
The Wu-Yang monopole 62 (1)
Charge quantization 62 (1)
Instantons 63 (4)
Introduction 63 (1)
The (anti-) self-dual solution 64 (2)
Problems 66 (1)
Mathematical Preliminaries 67 (26)
Maps 67 (8)
Definitions 67 (3)
Equivalence relation and equivalence 70 (5)
class
Vector spaces 75 (6)
Vectors and vector spaces 75 (1)
Linear maps, images and kernels 76 (1)
Dual vector space 77 (1)
Inner product and adjoint 78 (2)
Tensors 80 (1)
Topological spaces 81 (4)
Definitions 81 (1)
Continuous maps 82 (1)
Neighbourhoods and Hausdorff spaces 82 (1)
Closed set 83 (1)
Compactness 83 (2)
Connectedness 85 (1)
Homeomorphisms and topological invariants 85 (8)
Homeomorphisms 85 (1)
Topological invariants 86 (2)
Homotopy type 88 (1)
Euler characteristic: an example 88 (3)
Problems 91 (2)
Homology Groups 93 (28)
Abelian groups 93 (5)
Elementary group theory 93 (3)
Finitely generated Abelian groups and 96 (1)
free Abelian groups
Cyclic groups 96 (2)
Simplexes and simplicial complexes 98 (2)
Simplexes 98 (1)
Simplicial complexes and polyhedra 99 (1)
Homology groups of simplicial complexes 100 (17)
Oriented simplexes 100 (2)
Chain group, cycle group and boundary 102 (4)
group
Homology groups 106 (4)
Computation of H0(K) 110 (1)
More homology computations 111 (6)
General properties of homology groups 117 (4)
Connectedness and homology groups 117 (1)
Structure of homology groups 118 (1)
Betti numbers and the Euler--Poincare 118 (2)
theorem
Problems 120 (1)
Homotopy Groups 121 (48)
Fundamental groups 121 (6)
Basic ideas 121 (1)
Paths and loops 122 (1)
Homotopy 123 (2)
Fundamental groups 125 (2)
General properties of fundamental groups 127 (4)
Arcwise connectedness and fundamental 127 (1)
groups
Homotopic invariance of fundamental 128 (3)
groups
Examples of fundamental groups 131 (3)
Fundamental group of torus 133 (1)
Fundamental groups of polyhedra 134 (11)
Free groups and relations 134 (2)
Calculating fundamental groups of 136 (8)
polyhedra
Relations between H1(K) and π1(|K|) 144 (1)
Higher homotopy groups 145 (3)
Definitions 146 (2)
General properties of higher homotopy 148 (2)
groups
Abelian nature of higher homotopy groups 148 (1)
Arcwise connectedness and higher 148 (1)
homotopy groups
Homotopy invariance of higher homotopy 148 (1)
groups
Higher homotopy groups of a product 148 (1)
space
Universal covering spaces and higher 148 (2)
homotopy groups
Examples of higher homotopy groups 150 (3)
Orders in condensed matter systems 153 (6)
Order parameter 153 (1)
Superfluid 4He and superconductors 154 (3)
General consideration 157 (2)
Defects in nematic liquid crystals 159 (4)
Order parameter of nematic liquid 159 (1)
crystals
Line defects in nematic liquid crystals 160 (1)
Point defects in nematic liquid crystals 161 (1)
Higher dimensional texture 162 (1)
Textures in superfluid 3He-A 163 (6)
Superfluid 3He-A 163 (2)
Line defects and non-singular vortices 165 (1)
in 3He-A
Shankar monopole in 3He-A 166 (1)
Problems 167 (2)
Manifolds 169 (57)
Manifolds 169 (9)
Heuristic introduction 169 (2)
Definitions 171 (2)
Examples 173 (5)
The calculus on manifolds 178 (10)
Differentiable maps 179 (2)
Vectors 181 (3)
One-forms 184 (1)
Tensors 185 (1)
Tensor fields 185 (1)
Induced maps 186 (2)
Submanifolds 188 (1)
Flows and Lie derivatives 188 (8)
One-parameter group of transformations 190 (1)
Lie derivatives 191 (5)
Differential forms 196 (8)
Definitions 196 (2)
Exterior derivatives 198 (3)
Interior product and Lie derivative of 201 (3)
forms
Integration of differential forms 204 (3)
Orientation 204 (1)
Integration of forms 205 (2)
Lie groups and Lie algebras 207 (9)
Lie groups 207 (2)
Lie algebras 209 (3)
The one-parameter subgroup 212 (3)
Frames and structure equation 215 (1)
The action of Lie groups on manifolds 216 (10)
Definitions 216 (3)
Orbits and isotropy groups 219 (4)
Induced vector fields 223 (1)
The adjoint representation 224 (1)
Problems 224 (2)
de Rham Cohomology Groups 226 (18)
Stokes' theorem 226 (4)
Preliminary consideration 226 (2)
Stokes' theorem 228 (2)
de Rham cohomology groups 230 (5)
Definitions 230 (3)
Duality of Hr(M) and Hr(M); de Rham's 233 (2)
theorem
Poincare's lemma 235 (2)
Structure of de Rham cohomology groups 237 (7)
Poincare duality 237 (1)
Cohomology rings 238 (1)
The Kunneth formula 238 (2)
Pullback of de Rham cohomology groups 240 (1)
Homotopy and H1(M) 240 (4)
Riemannian Geometry 244 (64)
Riemannian manifolds and 244 (3)
pseudo-Riemannian manifolds
Metric tensors 244 (2)
Induced metric 246 (1)
Parallel transport, connection and 247 (7)
covariant derivative
Heuristic introduction 247 (2)
Affine connections 249 (1)
Parallel transport and geodesics 250 (1)
The covariant derivative of tensor 251 (1)
fields
The transformation properties of 252 (1)
connection coefficients
The metric connection 253 (1)
Curvature and torsion 254 (7)
Definitions 254 (2)
Geometrical meaning of the Riemann 256 (4)
tensor and the torsion tensor
The Ricci tensor and the scalar 260 (1)
curvature
Levi-Civita connections 261 (10)
The fundamental theorem 261 (1)
The Levi-Civita connection in the 262 (1)
classical geometry of surfaces
Geodesics 263 (3)
The normal coordinate system 266 (2)
Riemann curvature tensor with 268 (3)
Levi-Civita connection
Holonomy 271 (2)
Isometries and conformal transformations 273 (6)
Isometries 273 (1)
Conformal transformations 274 (5)
Killing vector fields and conformal 279 (4)
Killing vector fields
Killing vector fields 279 (3)
Conformal Killing vector fields 282 (1)
Non-coordinate bases 283 (6)
Definitions 283 (1)
Cartan's structure equations 284 (1)
The local frame 285 (2)
The Levi-Civita connection in a 287 (2)
non-coordinate basis
Differential forms and Hodge theory 289 (8)
Invariant volume elements 289 (1)
Duality transformations (Hodge star) 290 (1)
Inner products of r-forms 291 (2)
Adjoints of exterior derivatives 293 (1)
The Laplacian, harmonic forms and the 294 (2)
Hodge decomposition theorem
Harmonic forms and de Rham cohomology 296 (1)
groups
Aspects of general relativity 297 (5)
Introduction to general relativity 297 (1)
Einstein--Hilbert action 298 (2)
Spinors in curved spacetime 300 (2)
Bosonic string theory 302 (6)
The string action 303 (2)
Symmetries of the Polyakov strings 305 (2)
Problems 307 (1)
Complex Manifolds 308 (40)
Complex manifolds 308 (7)
Definitions 308 (1)
Examples 309 (6)
Calculus on complex manifolds 315 (5)
Holomorphic maps 315 (1)
Complexifications 316 (1)
Almost complex structure 317 (3)
Complex differential forms 320 (4)
Complexification of real differential 320 (1)
forms
Differential forms on complex manifolds 321 (1)
Dolbeault operators 322 (2)
Hermitian manifolds and Hermitian 324 (6)
differential geometry
The Hermitian metric 325 (1)
Kahler form 326 (1)
Covariant derivatives 327 (2)
Torsion and curvature 329 (1)
Kahler manifolds and Kahler differential 330 (6)
geometry
Definitions 330 (4)
Kahler geometry 334 (1)
The holonomy group of Kahler manifolds 335 (1)
Harmonic forms and ∂-cohomology 336 (5)
groups
The adjoint operators ∂† 337 (1)
and ∂†
Laplacians and the Hodge theorem 338 (1)
Laplacians on a Kahler manifold 339 (1)
The Hodge numbers of Kahler manifolds 339 (2)
Almost complex manifolds 341 (3)
Definitions 342 (2)
Orbifolds 344 (4)
One-dimensional examples 344 (2)
Three-dimensional examples 346 (2)
Fibre Bundles 348 (26)
Tangent bundles 348 (2)
Fibre bundles 350 (7)
Definitions 350 (3)
Reconstruction of fibre bundles 353 (1)
Bundle maps 354 (1)
Equivalent bundles 355 (1)
Pullback bundles 355 (2)
Homotopy axiom 357 (1)
Vector bundles 357 (6)
Definitions and examples 357 (2)
Frames 359 (1)
Cotangent bundles and dual bundles 360 (1)
Sections of vector bundles 361 (1)
The product bundle and Whitney sum 361 (2)
bundle
Tensor product bundles 363 (1)
Principal bundles 363 (11)
Definitions 363 (7)
Associated bundles 370 (2)
Triviality of bundles 372 (1)
Problems 372 (2)
Connections on Fibre Bundles 374 (45)
Connections on principal bundles 374 (10)
Definitions 375 (1)
The connection one-form 376 (1)
The local connection form and gauge 377 (4)
potential
Horizontal lift and parallel transport 381 (3)
Holonomy 384 (1)
Definitions 384 (1)
Curvature 385 (6)
Covariant derivatives in principal 385 (1)
bundles
Curvature 386 (2)
Geometrical meaning of the curvature 388 (1)
and the Ambrose--Singer theorem
Local form of the curvature 389 (1)
The Bianchi identity 390 (1)
The covariant derivative on associated 391 (8)
vector bundles
The covariant derivative on associated 391 (2)
bundles
A local expression for the covariant 393 (3)
derivative
Curvature rederived 396 (1)
A connection which preserves the inner 396 (1)
product
Holomorphic vector bundles and 397 (2)
Hermitian inner products
Gauge theories 399 (10)
U(1) gauge theory 399 (1)
The Dirac magnetic monopole 400 (2)
The Aharonov--Bohm effect 402 (2)
Yang--Mills theory 404 (1)
Instantons 405 (4)
Berry's phase 409 (10)
Derivation of Berry's phase 410 (1)
Berry's phase, Berry's connection and 411 (7)
Berry's curvature
Problems 418 (1)
Characteristic Classes 419 (34)
Invariant polynomials and the Chern--Weil 419 (7)
homomorphism
Invariant polynomials 420 (6)
Chern classes 426 (5)
Definitions 426 (2)
Properties of Chern classes 428 (1)
Splitting principle 429 (1)
Universal bundles and classifying spaces 430 (1)
Chern characters 431 (5)
Definitions 431 (3)
Properties of the Chern characters 434 (1)
Todd classes 435 (1)
Pontrjagin and Euler classes 436 (7)
Pontrjagin classes 436 (3)
Euler classes 439 (3)
Hirzebruch L-polynomial and A-genus 442 (1)
Chern-Simons forms 443 (5)
Definition 443 (1)
The Chern-Simons form of the Chern 444 (1)
character
Cartan's homotopy operator and 445 (3)
applications
Stiefel--Whitney classes 448 (5)
Spin bundles 449 (1)
Cech cohomology groups 449 (1)
Stiefel--Whitney classes 450 (3)
Index Theorems 453 (48)
Elliptic operators and Fredholm operators 453 (6)
Elliptic operators 454 (2)
Fredholm operators 456 (1)
Elliptic complexes 457 (2)
The Atiyah--Singer index theorem 459 (1)
Statement of the theorem 459 (1)
The de Rham complex 460 (2)
The Dolbeault complex 462 (2)
The twisted Dolbeault complex and the 463 (1)
Hirzebruch--Riemann--Roch theorem
The signature complex 464 (3)
The Hirzebruch signature 464 (1)
The signature complex and the 465 (2)
Hirzebruch signature theorem
Spin complexes 467 (5)
Dirac operator 468 (3)
Twisted spin complexes 471 (1)
The heat kernel and generalized 472 (5)
ζ-functions
The heat kernel and index theorem 472 (3)
Spectral ζ -functions 475 (2)
The Atiyah--Patodi--Singer index theorem 477 (4)
η-invariant and spectral flow 477 (1)
The Atiyah--Patodi--Singer (APS) index 478 (3)
theorem
Supersymmetric quantum mechanics 481 (6)
Clifford algebra and fermions 481 (1)
Supersymmetric quantum mechanics in 482 (3)
flat space
Supersymmetric quantum mechanics in a 485 (2)
general manifold
Supersymmetric proof of index theorem 487 (14)
The index 487 (3)
Path integral and index theorem 490 (10)
Problems 500 (1)
Anomalies in Gauge Field Theories 501 (27)
Introduction 501 (2)
Abelian anomalies 503 (5)
Fujikawa's method 503 (5)
Non-Abelian anomalies 508 (4)
The Wess--Zumino consistency conditions 512 (6)
The Becchi--Rouet--Stora operator and 512 (1)
the Faddeev--Popov ghost
The BRS operator, FP ghost and moduli 513 (2)
space
The Wess--Zumino conditions 515 (1)
Descent equations and solutions of WZ 515 (3)
conditions
Abelian anomalies versus non-Abelian 518 (5)
anomalies
m dimensions versus m + 2 dimensions 520 (3)
The parity anomaly in odd-dimensional 523 (5)
spaces
The parity anomaly 524 (1)
The dimensional ladder: 4--3--2 525 (3)
Bosonic String Theory 528 (32)
Differential geometry on Riemann surfaces 528 (7)
Metric and complex structure 528 (1)
Vectors, forms and tensors 529 (2)
Covariant derivatives 531 (2)
The Riemann-Roch theorem 533 (2)
Quantum theory of bosonic strings 535 (20)
Vacuum amplitude of Polyakov strings 535 (3)
Measures of integration 538 (12)
Complex tensor calculus and string 550 (4)
measure
Moduli spaces of Riemann surfaces 554 (1)
One-loop amplitudes 555 (5)
Moduli spaces, CKV, Beltrami and 555 (2)
quadratic differentials
The evaluation of determinants 557 (3)
References 560 (5)
Index 565