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Geometry, Topology and Physics 2nd edition [Hardback]

(Kinki University, Osaka, Japan)
  • Formāts: Hardback, 596 pages, height x width: 234x156 mm, weight: 1260 g
  • Izdošanas datums: 27-Jul-2017
  • Izdevniecība: CRC Press
  • ISBN-10: 1138413364
  • ISBN-13: 9781138413368
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Geometry, Topology and Physics 2nd editionPalielināt
  • Formāts: Hardback, 596 pages, height x width: 234x156 mm, weight: 1260 g
  • Izdošanas datums: 27-Jul-2017
  • Izdevniecība: CRC Press
  • ISBN-10: 1138413364
  • ISBN-13: 9781138413368
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781138413368.html
Offers an introduction to the ideas and techniques of differential geometry and topology. This work starts with a brief survey of quantum field theory, gauge theory, general relativity, vector spaces, and topology. It also demonstrates elaborate concepts of topology and geometry, including fiber bundles, characteristic classes, and index theorems.

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.

Recenzijas

"a very impressive book." -Australian and New Zealand Physicists

"The clarity of the presentation is enhanced by explicit calculations and diagrams; the proof of a theorem is given only when it is instructive and not very technical. There is also a large number of exercises and problems, and last but not least, an index superb layout" - Zentralblatt fur Mathematick un ihre Grenzgebiete

"I believe that the book will not only boost modernization of the traditional courses of theoretical physics but will prompt the specialist in topology and differential geometry to have a closer look at the applications. So I welcome this second edition." -Christopher Gilmour

Preface to the First Edition xvii
Preface to the Second Edition xix
How to Read this Book xxi
Notation and Conventions xxii
1 Quantum Physics 1(66)
1.1 Analytical mechanics
1(8)
1.1.1 Newtonian mechanics
1(1)
1.1.2 Lagrangian formalism
2(3)
1.1.3 Hamiltonian formalism
5(4)
1.2 Canonical quantization
9(10)
1.2.1 Hilbert space, bras and kets
9(1)
1.2.2 Axioms of canonical quantization
10(3)
1.2.3 Heisenberg equation, Heisenberg picture and Schrodinger picture
13(1)
1.2.4 Wavefunction
13(4)
1.2.5 Harmonic oscillator
17(2)
1.3 Path integral quantization of a Bose particle
19(12)
1.3.1 Path integral quantization
19(7)
1.3.2 Imaginary time and partition function
26(2)
1.3.3 Time-ordered product and generating functional
28(3)
1.4 Harmonic oscillator
31(7)
1.4.1 Transition amplitude
31(4)
1.4.2 Partition function
35(3)
1.5 Path integral quantization of a Fermi particle
38(13)
1.5.1 Fermionic harmonic oscillator
39(1)
1.5.2 Calculus of Grassmann numbers
40(1)
1.5.3 Differentiation
41(1)
1.5.4 Integration
42(1)
1.5.5 Delta-function
43(1)
1.5.6 Gaussian integral
44(1)
1.5.7 Functional derivative
45(1)
1.5.8 Complex conjugation
45(1)
1.5.9 Coherent states and completeness relation
46(1)
1.5.10 Partition function of a fermionic oscillator
47(4)
1.6 Quantization of a scalar field
51(4)
1.6.1 Free scalar field
51(3)
1.6.2 Interacting scalar field
54(1)
1.7 Quantization of a Dirac field
55(1)
1.8 Gauge theories
56(4)
1.8.1 Abelian gauge theories
56(2)
1.8.2 Non-Abelian gauge theories
58(2)
1.8.3 Higgs fields
60(1)
1.9 Magnetic monopoles
60(3)
1.9.1 Dirac monopole
61(1)
1.9.2 The Wu-Yang monopole
62(1)
1.9.3 Charge quantization
62(1)
1.10 Instantons
63(3)
1.10.1 Introduction
63(1)
1.10.2 The (anti-)self-dual solution
64(2)
Problems
66(1)
2 Mathematical Preliminaries 67(26)
2.1 Maps
67(8)
2.1.1 Definitions
67(3)
2.1.2 Equivalence relation and equivalence class
70(5)
2.2 Vector spaces
75(6)
2.2.1 Vectors and vector spaces
75(1)
2.2.2 Linear maps, images and kernels
76(1)
2.2.3 Dual vector space
77(1)
2.2.4 Inner product and adjoint
78(2)
2.2.5 Tensors
80(1)
2.3 Topological spaces
81(4)
2.3.1 Definitions
81(1)
2.3.2 Continuous maps
82(1)
2.3.3 Neighbourhoods and Hausdorff spaces
82(1)
2.3.4 Closed set
83(1)
2.3.5 Compactness
83(2)
2.3.6 Connectedness
85(1)
2.4 Homeomorphisms and topological invariants
85(6)
2.4.1 Homeomorphisms
85(1)
2.4.2 Topological invariants
86(2)
2.4.3 Homotopy type
88(1)
2.4.4 Euler characteristic: an example
88(3)
Problems
91(2)
3 Homology Groups 93(28)
3.1 Abelian groups
93(5)
3.1.1 Elementary group theory
93(3)
3.1.2 Finitely generated Abelian groups and free Abelian groups
96(1)
3.1.3 Cyclic groups
96(2)
3.2 Simplexes and simplicial complexes
98(2)
3.2.1 Simplexes
98(1)
3.2.2 Simplicial complexes and polyhedra
99(1)
3.3 Homology groups of simplicial complexes
100(17)
3.3.1 Oriented simplexes
100(2)
3.3.2 Chain group, cycle group and boundary group
102(4)
3.3.3 Homology groups
106(4)
3.3.4 Computation of H0(K)
110(1)
3.3.5 More homology computations
111(6)
3.4 General properties of homology groups
117(3)
3.4.1 Connectedness and homology groups
117(1)
3.4.2 Structure of homology groups
118(1)
3.4.3 Betti numbers and the Euler-Poincare theorem
118(2)
Problems
120(1)
4 Homotopy Groups 121(48)
4.1 Fundamental groups
121(6)
4.1.1 Basic ideas
121(1)
4.1.2 Paths and loops
122(1)
4.1.3 Homotopy
123(2)
4.1.4 Fundamental groups
125(2)
4.2 General properties of fundamental groups
127(4)
4.2.1 Arcwise connectedness and fundamental groups
127(1)
4.2.2 Homotopic invariance of fundamental groups
128(3)
4.3 Examples of fundamental groups
131(3)
4.3.1 Fundamental group of torus
133(1)
4.4 Fundamental groups of polyhedra
134(11)
4.4.1 Free groups and relations
134(2)
4.4.2 Calculating fundamental groups of polyhedra
136(8)
4.4.3 Relations between H1(K) and π1(|K|)
144(1)
4.5 Higher homotopy groups
145(3)
4.5.1 Definitions
146(2)
4.6 General properties of higher homotopy groups
148(2)
4.6.1 Abelian nature of higher homotopy groups
148(1)
4.6.2 Arcwise connectedness and higher homotopy groups
148(1)
4.6.3 Homotopy invariance of higher homotopy groups
148(1)
4.6.4 Higher homotopy groups of a product space
148(1)
4.6.5 Universal covering spaces and higher homotopy groups
148(2)
4.7 Examples of higher homotopy groups
150(3)
4.8 Orders in condensed matter systems
153(6)
4.8.1 Order parameter
153(1)
4.8.2 Superfluid 4He and superconductors
154(3)
4.8.3 General consideration
157(2)
4.9 Defects in nematic liquid crystals
159(4)
4.9.1 Order parameter of nematic liquid crystals
159(1)
4.9.2 Line defects in nematic liquid crystals
160(1)
4.9.3 Point defects in nematic liquid crystals
161(1)
4.9.4 Higher dimensional texture
162(1)
4.10 Textures in superfluid 3He-A
163(4)
4.10.1 Superfluid 3He-A
163(2)
4.10.2 Line defects and non-singular vortices in 3He-A
165(1)
4.10.3 Shankar monopole in 3He-A
166(1)
Problems
167(2)
5 Manifolds 169(57)
5.1 Manifolds
169(9)
5.1.1 Heuristic introduction
169(2)
5.1.2 Definitions
171(2)
5.1.3 Examples
173(5)
5.2 The calculus on manifolds
178(10)
5.2.1 Differentiable maps
179(2)
5.2.2 Vectors
181(3)
5.2.3 One-forms
184(1)
5.2.4 Tensors
185(1)
5.2.5 Tensor fields
185(1)
5.2.6 Induced maps
186(2)
5.2.7 Submanifolds
188(1)
5.3 Flows and Lie derivatives
188(8)
5.3.1 One-parameter group of transformations
190(1)
5.3.2 Lie derivatives
191(5)
5.4 Differential forms
196(8)
5.4.1 Definitions
196(2)
5.4.2 Exterior derivatives
198(3)
5.4.3 Interior product and Lie derivative of forms
201(3)
5.5 Integration of differential forms
204(3)
5.5.1 Orientation
204(1)
5.5.2 Integration of forms
205(2)
5.6 Lie groups and Lie algebras
207(9)
5.6.1 Lie groups
207(2)
5.6.2 Lie algebras
209(3)
5.6.3 The one-parameter subgroup
212(3)
5.6.4 Frames and structure equation
215(1)
5.7 The action of Lie groups on manifolds
216(8)
5.7.1 Definitions
216(3)
5.7.2 Orbits and isotropy groups
219(4)
5.7.3 Induced vector fields
223(1)
5.7.4 The adjoint representation
224(1)
Problems
224(2)
6 de Rham Cohomology Groups 226(18)
6.1 Stokes' theorem
226(4)
6.1.1 Preliminary consideration
226(2)
6.1.2 Stokes' theorem
228(2)
6.2 de Rham cohomology groups
230(5)
6.2.1 Definitions
230(3)
6.2.2 Duality of Hr(M) and Hr(M); de Rham's theorem
233(2)
6.3 Poincare's lemma
235(2)
6.4 Structure of de Rham cohomology groups
237(7)
6.4.1 Poincare duality
237(1)
6.4.2 Cohomology rings
238(1)
6.4.3 The Kunneth formula
238(2)
6.4.4 Pullback of de Rham cohomology groups
240(1)
6.4.5 Homotopy and H1(M)
240(4)
7 Riemannian Geometry 244(64)
7.1 Riemannian manifolds and pseudo-Riemannian manifolds
244(3)
7.1.1 Metric tensors
244(2)
7.1.2 Induced metric
246(1)
7.2 Parallel transport, connection and covariant derivative
247(7)
7.2.1 Heuristic introduction
247(2)
7.2.2 Affine connections
249(1)
7.2.3 Parallel transport and geodesics
250(1)
7.2.4 The covariant derivative of tensor fields
251(1)
7.2.5 The transformation properties of connection coefficients
252(1)
7.2.6 The metric connection
253(1)
7.3 Curvature and torsion
254(7)
7.3.1 Definitions
254(2)
7.3.2 Geometrical meaning of the Riemann tensor and the torsion tensor
256(4)
7.3.3 The Ricci tensor and the scalar curvature
260(1)
7.4 Levi-Civita connections
261(10)
7.4.1 The fundamental theorem
261(1)
7.4.2 The Levi-Civita connection in the classical geometry of surfaces
262(1)
7.4.3 Geodesics
263(3)
7.4.4 The normal coordinate system
266(2)
7.4.5 Riemann curvature tensor with Levi-Civita connection
268(3)
7.5 Holonomy
271(2)
7.6 Isometries and conformal transformations
273(6)
7.6.1 Isometries
273(1)
7.6.2 Conformal transformations
274(5)
7.7 Killing vector fields and conformal Killing vector fields
279(4)
7.7.1 Killing vector fields
279(3)
7.7.2 Conformal Killing vector fields
282(1)
7.8 Non-coordinate bases
283(6)
7.8.1 Definitions
283(1)
7.8.2 Cartan's structure equations
284(1)
7.8.3 The local frame
285(2)
7.8.4 The Levi-Civita connection in a non-coordinate basis
287(2)
7.9 Differential forms and Hodge theory
289(8)
7.9.1 Invariant volume elements
289(1)
7.9.2 Duality transformations (Hodge star)
290(1)
7.9.3 Inner products of r-forms
291(2)
7.9.4 Adjoints of exterior derivatives
293(1)
7.9.5 The Laplacian, harmonic forms and the Hodge decomposition theorem
294(2)
7.9.6 Harmonic forms and de Rham cohomology groups
296(1)
7.10 Aspects of general relativity
297(5)
7.10.1 Introduction to general relativity
297(1)
7.10.2 Einstein-Hilbert action
298(2)
7.10.3 Spinors in curved spacetime
300(2)
7.11 Bosonic string theory
302(5)
7.11.1 The string action
303(2)
7.11.2 Symmetries of the Polyakov strings
305(2)
Problems
307(1)
8 Complex Manifolds 308(40)
8.1 Complex manifolds
308(7)
8.1.1 Definitions
308(1)
8.1.2 Examples
309(6)
8.2 Calculus on complex manifolds
315(5)
8.2.1 Holomorphic maps
315(1)
8.2.2 Complexifications
316(1)
8.2.3 Almost complex structure
317(3)
8.3 Complex differential forms
320(4)
8.3.1 Complexification of real differential forms
320(1)
8.3.2 Differential forms on complex manifolds
321(1)
8.3.3 Dolbeault operators
322(2)
8.4 Hermitian manifolds and Hermitian differential geometry
324(6)
8.4.1 The Hermitian metric
325(1)
8.4.2 Kahler form
326(1)
8.4.3 Covariant derivatives
327(2)
8.4.4 Torsion and curvature
329(1)
8.5 Kahler manifolds and Kahler differential geometry
330(6)
8.5.1 Definitions
330(4)
8.5.2 Kahler geometry
334(1)
8.5.3 The holonomy group of Kahler manifolds
335(1)
8.6 Harmonic forms and partial derivative-cohomology groups
336(5)
8.6.1 The adjoint operators partial derivative+ and partial derivative-+
337(1)
8.6.2 Laplacians and the Hodge theorem
338(1)
8.6.3 Laplacians on a Kahler manifold
339(1)
8.6.4 The Hodge numbers of Kahler manifolds
339(2)
8.7 Almost complex manifolds
341(3)
8.7.1 Definitions
342(2)
8.8 Orbifolds
344(4)
8.8.1 One-dimensional examples
344(2)
8.8.2 Three-dimensional examples
346(2)
9 Fibre Bundles 348(26)
9.1 Tangent bundles
348(2)
9.2 Fibre bundles
350(7)
9.2.1 Definitions
350(3)
9.2.2 Reconstruction of fibre bundles
353(1)
9.2.3 Bundle maps
354(1)
9.2.4 Equivalent bundles
355(1)
9.2.5 Pullback bundles
355(2)
9.2.6 Homotopy axiom
357(1)
9.3 Vector bundles
357(6)
9.3.1 Definitions and examples
357(2)
9.3.2 Frames
359(1)
9.3.3 Cotangent bundles and dual bundles
360(1)
9.3.4 Sections of vector bundles
361(1)
9.3.5 The product bundle and Whitney sum bundle
361(2)
9.3.6 Tensor product bundles
363(1)
9.4 Principal bundles
363(9)
9.4.1 Definitions
363(7)
9.4.2 Associated bundles
370(2)
9.4.3 Triviality of bundles
372(1)
Problems
372(2)
10 Connections on Fibre Bundles 374(45)
10.1 Connections on principal bundles
374(10)
10.1.1 Definitions
375(1)
10.1.2 The connection one-form
376(1)
10.1.3 The local connection form and gauge potential
377(4)
10.1.4 Horizontal lift and parallel transport
381(3)
10.2 Holonomy
384(1)
10.2.1 Definitions
384(1)
10.3 Curvature
385(6)
10.3.1 Covariant derivatives in principal bundles
385(1)
10.3.2 Curvature
386(2)
10.3.3 Geometrical meaning of the curvature and the Ambrose-Singer theorem
388(1)
10.3.4 Local form of the curvature
389(1)
10.3.5 The Bianchi identity
390(1)
10.4 The covariant derivative on associated vector bundles
391(8)
10.4.1 The covariant derivative on associated bundles
391(2)
10.4.2 A local expression for the covariant derivative
393(3)
10.4.3 Curvature rederived
396(1)
10.4.4 A connection which preserves the inner product
396(1)
10.4.5 Holomorphic vector bundles and Hermitian inner products
397(2)
10.5 Gauge theories
399(10)
10.5.1 U(1) gauge theory
399(1)
10.5.2 The Dirac magnetic monopole
400(2)
10.5.3 The Aharonov-Bohm effect
402(2)
10.5.4 Yang-Mills theory
404(1)
10.5.5 Instantons
405(4)
10.6 Berry's phase
409(9)
10.6.1 Derivation of Berry's phase
410(1)
10.6.2 Berry's phase, Berry's connection and Berry's curvature
411(7)
Problems
418(1)
11 Characteristic Classes 419(34)
11.1 Invariant polynomials and the Chern-Weil homomorphism
419(7)
11.1.1 Invariant polynomials
420(6)
11.2 Chern classes
426(5)
11.2.1 Definitions
426(2)
11.2.2 Properties of Chern classes
428(1)
11.2.3 Splitting principle
429(1)
11.2.4 Universal bundles and classifying spaces
430(1)
11.3 Chern characters
431(5)
11.3.1 Definitions
431(3)
11.3.2 Properties of the Chern characters
434(1)
11.3.3 Todd classes
435(1)
11.4 Pontrjagin and Euler classes
436(7)
11.4.1 Pontrjagin classes
436(3)
11.4.2 Euler classes
439(3)
11.4.3 Hirzebruch L-polynomial and A-genus
442(1)
11.5 Chern-Simons forms
443(5)
11.5.1 Definition
443(1)
11.5.2 The Chern-Simons form of the Chern character
444(1)
11.5.3 Cartan's homotopy operator and applications
445(3)
11.6 Stiefel-Whitney classes
448(5)
11.6.1 Spin bundles
449(1)
11.6.2 Cech cohomology groups
449(1)
11.6.3 Stiefel-Whitney classes
450(3)
12 Index Theorems 453(48)
12.1 Elliptic operators and Fredholm operators
453(6)
12.1.1 Elliptic operators
454(2)
12.1.2 Fredholm operators
456(1)
12.1.3 Elliptic complexes
457(2)
12.2 The Atiyah-Singer index theorem
459(1)
12.2.1 Statement of the theorem
459(1)
12.3 The de Rham complex
460(2)
12.4 The Dolbeault complex
462(2)
12.4.1 The twisted Dolbeault complex and the Hirzebruch-Riemann-Roch theorem
463(1)
12.5 The signature complex
464(3)
12.5.1 The Hirzebruch signature
464(1)
12.5.2 The signature complex and the Hirzebruch signature theorem
465(2)
12.6 Spin complexes
467(5)
12.6.1 Dirac operator
468(3)
12.6.2 Twisted spin complexes
471(1)
12.7 The heat kernel and generalized ζ-functions
472(5)
12.7.1 The heat kernel and index theorem
472(3)
12.7.2 Spectral ζ-functions
475(2)
12.8 The Atiyah-Patodi-Singer index theorem
477(4)
12.8.1 η-invariant and spectral flow
477(1)
12.8.2 The Atiyah-Patodi-Singer (APS) index theorem
478(3)
12.9 Supersymmetric quantum mechanics
481(6)
12.9.1 Clifford algebra and fermions
481(1)
12.9.2 Supersymmetric quantum mechanics in flat space
482(3)
12.9.3 Supersymmetric quantum mechanics in a general manifold
485(2)
12.10 Supersymmetric proof of index theorem
487(13)
12.10.1 The index
487(3)
12.10.2 Path integral and index theorem
490(10)
Problems
500(1)
13 Anomalies in Gauge Field Theories 501(27)
13.1 Introduction
501(2)
13.2 Abelian anomalies
503(5)
13.2.1 Fujikawa's method
503(5)
13.3 Non-Abelian anomalies
508(4)
13.4 The Wess-Zumino consistency conditions
512(6)
13.4.1 The Becchi-Rouet-Stora operator and the Faddeev-Popov ghost
512(1)
13.4.2 The BRS operator, FP ghost and moduli space
513(2)
13.4.3 The Wess-Zumino conditions
515(1)
13.4.4 Descent equations and solutions of WZ conditions
515(3)
13.5 Abelian anomalies versus non-Abelian anomalies
518(5)
13.5.1 m dimensions versus m + 2 dimensions
520(3)
13.6 The parity anomaly in odd-dimensional spaces
523(5)
13.6.1 The parity anomaly
524(1)
13.6.2 The dimensional ladder: 4-3-2
525(3)
14 Bosonic String Theory 528(32)
14.1 Differential geometry on Riemann surfaces
528(7)
14.1.1 Metric and complex structure
528(1)
14.1.2 Vectors, forms and tensors
529(2)
14.1.3 Covariant derivatives
531(2)
14.1.4 The Riemann-Roch theorem
533(2)
14.2 Quantum theory of bosonic strings
535(20)
14.2.1 Vacuum amplitude of Polyakov strings
535(3)
14.2.2 Measures of integration
538(12)
14.2.3 Complex tensor calculus and string measure
550(4)
14.2.4 Moduli spaces of Riemann surfaces
554(1)
14.3 One-loop amplitudes
555(5)
14.3.1 Moduli spaces, CKV, Beltrami and quadratic differentials
555(2)
14.3.2 The evaluation of determinants
557(3)
References 560(5)
Index 565
Mikio Nakahara