Geometry, topology and physics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Taylor & Francis
2003
|
| Edition: | 2. ed. |
| Series: | Graduate student series in physics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XXII, 573 S. graph. Darst. |
| ISBN: | 0750306068 9780750306065 |
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| 100 | 1 | |a Nakahara, Mikio |e Verfasser |0 (DE-588)1028332297 |4 aut | |
| 245 | 1 | 0 | |a Geometry, topology and physics |c Mikio Nakahara |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Taylor & Francis |c 2003 | |
| 300 | |a XXII, 573 S. |b graph. Darst. | ||
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| 490 | 0 | |a Graduate student series in physics | |
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Record in the Search Index
| _version_ | 1804130330394755072 |
|---|---|
| adam_text | CONTENTS
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
1.1
Analytical mechanics
1
1.1.1
Newtonian mechanics
1
1.1.2
Lagrangian formalism
2
1.1.3
Hamiltonian formalism
5
1.2
Canonical quantization
9
1.2.1
Hubert space, bras and
kets
9
1.2.2
Axioms of canonical quantization
10
1.2.3 Heisenberg
equation,
Heisenberg
picture and
Schrödinger
picture
13
1.2.4
Wavefunction
13
1.2.5
Harmonic oscillator
17
1.3
Path integral quantization of
a Bose
particle
19
1.3.1
Path integral quantization
19
1.3.2
Imaginary time and partition function
26
1.3.3
Time-ordered product and generating functional
28
1.4
Harmonic oscillator
31
1.4.1
Transition amplitude
31
1.4.2
Partition function
35
1.5
Path integral quantization of a Fermi particle
38
1.5.1
Fermionic harmonic oscillator
39
1.5.2
Calculus of
Grassmann
numbers
40
1.5.3
Differentiation
41
1.5.4
Integration
42
1.5.5
Delta-function
43
1.5.6
Gaussian integral
44
1.5.7
Functional derivative
45
1.5.8
Complex conjugation
45
1.5.9
Coherent states and completeness relation
46
1.5.10
Partition
function of a fermionic oscillator
1.6
Quantization of a scalar field
1.6.1
Free scalar field
1.6.2
Interacting scalar field
1.7
Quantization of a Dirac field
1.8
Gauge theories
1.8.1
Abelian gauge theories
1.8.2
Non-
Abelian gauge theories
1.8.3
Higgs fields
1.9
Magnetic
monopoles
1.9.1
Dirac
monopole
1.9.2
The Wu-Yang
monopole
1.9.3
Charge quantization
1.10
Instantons
1.10.1
Introduction
1.10.2
The (anti-)self-dual solution
Problems
2
Mathematical Preliminaries
2.1
Maps
2.1.1
Definitions
2.1.2
Equivalence relation and equivalence class
2.2
Vector spaces
2.2.1
Vectors and vector spaces
2.2.2
Linear maps, images and kernels
2.2.3
Dual vector space
2.2.4
Inner product and adjoint
2.2.5
Tensors
2.3
Topoiogical spaces
2.3.1
Definitions
2.3.2
Continuous maps
2.3.3
Neighbourhoods and Hausdorff spaces
2.3.4
Closed set
2.3.5
Compactness
2.3.6
Connectedness
2.4
Homeomorphisms and topological invariants
2.4.1
Homeomorphisms
2.4.2
Topological invariants
2.4.3
Homotopy type
2.4.4
Euler
characteristic: an example
Problems
47
51
51
54
55
56
56
58
60
60
61
62
62
63
63
64
66
67
67
67
70
75
75
76
77
78
80
81
81
82
82
83
83
85
85
85
91
3
Homology Groups
93
3.1
Abelian groups
93
3.1.1
Elementary group theory
93
3.1.2
Finitely generated Abelian groups and free Abelian groups
96
3.1.3
Cyclic groups
96
3.2 Simplexes
and simplicial complexes
98
3.2.1 Simplexes 98
3.2.2
Simplicial complexes and polyhedra
99
3.3
Homology groups of simplicial complexes
100
3.3.1
Oriented
Simplexes 100
3.3.2
Chain group, cycle group and boundary group
102
3.3.3
Homology groups
106
3.3.4
Computation of Ho
(
К
) 110
3.3.5
More homology computations 111
3.4
General properties of homology groups
117
3.4.1
Connectedness and homology groups
117
3.4.2
Structure of homology groups
118
3.4.3
Betti
numbers and the
Euler-Poincarć
theorem
118
Problems
120
4
Homotopy Groups
121
4.1
Fundamental groups
121
4.1.1
Basic ideas
121
4.1.2
Paths and loops
122
4.1.3
Homotopy
123
4.1.4
Fundamental groups
125
4.2
General properties of fundamental groups
127
4.2.1
Arcwise connectedness and fundamental groups
127
4.2.2
Homotopic
invariance
of fundamental groups
128
4.3
Examples of fundamental groups
131
4.3.1
Fundamental group of torus
133
4.4
Fundamental groups of polyhedra
134
4.4.1
Free groups and relations
134
4.4.2
Calculating fundamental groups of polyhedra
136
4.4.3
Relations between Hi{K) and
π ( Κ )
144
4.5
Higher homotopy groups
145
4.5.1
Definitions
146
4.6
General properties of higher homotopy groups
148
4.6.1
Abelian nature of higher homotopy groups
148
4.6.2
Arcwise connectedness and higher homotopy groups
148
4.6.3
Homotopy
invariance
of higher homotopy groups
148
4.6.4
Higher homotopy groups of a product space
148
4.6.5
Universal covering spaces and higher homotopy groups
148
4.7
Examples of higher homotopy groups
150
4.8
Orders
in condensed matter systems
4.8.1
Order parameter
4.8.2
Superfluid 4He and superconductors
4.8.3
General consideration
4.9
Defects in nematic liquid crystals
4.9.1
Order parameter of nematic liquid crystals
4.9.2
Line defects in nematic liquid crystals
4.9.3
Point defects in nematic liquid crystals
4.9.4
Higher dimensional texture
4.10
Textures in superfluid 3He-A
4.10.1
Superfluid 3He-A
4.10.2
Line defects and non-singular vortices in 3He-A
4.10.3
Shankar
monopolein 3He-A
Problems
5
Manifolds
5.1
Manifolds
5.1.1
Heuristic introduction
5.1.2
Definitions
5.1.3
Examples
5.2
The calculus on manifolds
5.2.1
Differentiable maps
5.2.2
Vectors
5.2.3
One-forms
5.2.4
Tensors
5.2.5
Tensor fields
5.2.6
Induced maps
5.2.7
Submanifolds
5.3
Flows and Lie derivatives
5.3.1
One-parameter group of transformations
5.3.2
Lie derivatives
5.4
Differential forms
5.4.1
Definitions
5.4.2
Exterior derivatives
5.4.3
Interior product and Lie derivative of forms
5.5
Integration of differential forms
5.5.1
Orientation
5.5.2
Integration of forms
5.6
Lie groups and Lie algebras
5.6.1
Lie groups
5.6.2
Lie algebras
5.6.3
The one-parameter subgroup
5.6.4
Frames and structure equation
5.7
The action of Lie groups on manifolds
153
153
154
157
159
159
160
161
162
163
163
165
166
167
169
169
169
171
173
178
179
181
184
185
185
186
188
188
190
191
196
196
198
201
204
204
205
207
207
209
212
215
216
5.7.1
Definitions
5.7.2
Orbits and isotropy groups
5.7.3
Induced vector fields
5.7.4
The adjoint representation
Problems
de Rham
Cohomology Groups
6.1
Stokes theorem
6.1.1
Preliminary consideration
6.1.2
Stokes theorem
6.2 de Rham
cohomology groups
6.2.1
Definitions
6.2.2
Duality of Hr{M) and Hr(M);
de Rham s
theorem
6.3
Poincaré s
lemma
6.4
Structure of
de Rham
cohomology groups
6.4.1
Poincaré
duality
Cohomology rings
The Kiinneth formula
Pullback of
de Rham
cohomology groups
Homotopy and
H
l (M)
6.4.2
6.4.3
6.4.4
6.4.5
7.2.3
7.2.4
7.2.5
7.2.6
7
Riemannian Geometry
7.1
Riemannian manifolds and pseudo-Riemannian manifolds
7.1.1
Metric tensors
7.1.2
Induced metric
7.2
Parallel transport, connection and covariant derivative
7.2.1
Heuristic introduction
7.2.2
Affine
connections
Parallel transport and geodesies
The covariant derivative of tensor fields
The transformation properties of connection coefficients
The metric connection
7.3
Curvature and torsion
7.3.1
Definitions
7.3.2
Geometrical meaning of the Riemann tensor and the
torsion tensor
7.3.3
The
Ricci
tensor and the scalar curvature
7.4
Levi-Civita connections
7.4.1
The fundamental theorem
7.4.2
The Levi-Civita connection in the classical geometry of
surfaces
7.4.3
Geodesies
7.4.4
The normal coordinate system
7.4.5
Riemann curvature tensor with Levi-Civita connection
7.5
Holonomy
216
219
223
224
224
226
226
226
228
230
230
233
235
237
237
238
238
240
240
244
244
244
246
247
247
249
250
251
252
253
254
254
256
260
261
261
262
263
266
268
271
7.6
Isometries and
conformai transformations
273
7.6.1
Isometries
273
7.6.2
Conformai transformations
274
7.7
Killing vector fields and
conformai
Killing vector fields
279
7.7.1
Killing vector fields
279
7.7.2
Conformai
Killing vector fields
282
7.8
Non-coordinate bases
283
7.8.1
Definitions
283
7.8.2
Cartan s structure equations
284
7.8.3
The local frame
285
7.8.4
The Levi-Civita connection in a non-coordinate basis
287
7.9
Differential forms and Hodge theory
289
7.9.1
Invariant volume elements
289
7.9.2
Duality transformations (Hodge star)
290
7.9.3
Inner products of /--forms
291
7.9.4
Adjoints
of exterior derivatives
293
7.9.5
The Laplacian, harmonic forms and the Hodge
decomposition theorem
294
7.9.6
Harmonic forms and
de Rham
cohomology groups
296
7.10
Aspects of
generał
relativity
297
7.10.1
Introduction to general relativity
297
7.10.2
Einstein-Hilbert action
298
7.10.3
Spinors in curved spacetime
300
7.11
Bosonic string theory
302
7.11.1
The string action
303
1
.W.I Symmetries of the Polyakov strings
305
Problems
307
8
Complex Manifolds
308
8.1
Complex manifolds
308
8.1.1
Definitions
308
8.1.2
Examples
309
8.2
Calculus on complex manifolds
315
8.2.1
Holomorphic maps
315
8.2.2
Complexifications
316
8.2.3
Almost complex structure
317
8.3
Complex differential forms
320
8.3.1
Complexification of real differential forms
320
8.3.2
Differential forms on complex manifolds
321
8.3.3
Dolbeault operators
322
8.4
Hermitian manifolds and Hermitian differential geometry
324
8.4.1
The Hermitian metric
325
8.4.2 Kahler form 326
8.4.3
Covariant derivatives
327
8.4.4
Torsion and curvature
8.5 Kahler
manifolds and
Kahler
differential geometry
8.5.1
Definitions
8.5.2 Kahler
geometry
8.5.3
The holonomy group of
Kahler
manifolds
8.6
Harmonic forms and
Э
-cohomology
groups
8.6.1
The adjoint operators
9^
and
9
8.6.2
Laplacians and the Hodge theorem
8.6.3
Laplacians on
a
Kahler
manifold
8.6.4
The Hodge numbers of
Kahler
manifolds
8.7
Almost complex manifolds
8.7.1
Definitions
8.8
Orbifolds
8.8.1
One-dimensional examples
8.8.2
Three-dimensional examples
9
Fibre Bundles
9.1
Tangent bundles
9.2
9.3
9.4
Fibre
1
Dundles
9.2.1
Definitions
9.2.2
Reconstruction of fibre bundles
9.2.3
Bundle maps
9.2.4
Equivalent bundles
9.2.5
Pullback bundles
9.2.6
Homotopy axiom
Vector bundles
9.3.1
Definitions and examples
9.3.2
Frames
9.3.3
Cotangent bundles and dual bundles
9.3.4
Sections of vector bundles
9.3.5
The product bundle and Whitney sum bundle
9.3.6
Tensor product bundles
Principal bundles
9.4.1
Definitions
9.4.2
Associated bundles
9.4.3
Triviality of bundles
Problems
10
Connections on Fibre Bundles
10.1
Connections on principal bundles
10.1.1
Definitions
10.1.2
The connection one-form
10.1.3
The local connection form and gauge potential
10.1.4
Horizontal lift and parallel transport
10.2
Holonomy
329
330
330
334
335
336
337
338
339
339
341
342
344
344
346
348
348
350
350
353
354
355
355
357
357
357
359
360
361
361
363
363
363
370
372
372
374
374
375
376
377
381
384
10.2.1
Definitions
384
10.3
Curvature
385
10.3.1
Covariant derivatives in principal bundles
385
10.3.2
Curvature
386
10.3.3
Geometrical meaning of the curvature and the Ambrose-
Singer theorem
388
10.3.4
Local form of the curvature
389
10.3.5
The
Bianchi
identity
390
10.4
The covariant derivative on associated vector bundles
391
10.4.1
The covariant derivative on associated bundles
391
10.4:2
A local expression for the covariant derivative
393
10.4.3
Curvature
rederived
396
10.4.4
A connection which preserves the inner product
396
10.4.5
Holomorphic vector bundles and Hermitian inner
products
397
10.5
Gauge theories
399
10.5.1
U(l) gauge theory
399
10.5.2
The Dirac magnetic
monopole
400
10.5.3
The Aharonov-Bohm effect
402
10.5.4
Yang-Mills theory
404
10.5.5
Instantons
405
10.6
Berry s phase
409
10.6.1
Derivation of Berry s phase
410
10.6.2
Berry s phase, Berry s connection and Berry s curvature
411
Problems
418
11
Characteristic Classes
419
11.1
Invariant polynomials and the Chern-Weil homomorphism
419
11.1.1
Invariant polynomials
420
11.2
Chern classes
426
11.2.1
Definitions
426
11.2.2
Properties of Chern classes
428
11.2.3
Splitting principle
429
11.2.4
Universal bundles and classifying spaces
430
11.3
Chern characters
431
11.3.1
Definitions
431
11.3.2
Properties of the Chern characters
434
11.3.3
Todd classes
435
11.4
Pontrjagin and
Euler
classes
436
11.4.1
Pontrjagin classes
436
11.4.2
Euler
classes ^
439
11.4.3
Hirzebruch L-polynomial and
Â-genus
442
11.5
Chern-Simons forms
443
11.5.1
Definition
443
11.5.2
The Chern-Simons form of the Chern character
444
11.5.3
Cartan s homotopy operator and applications
445
11.6 Stiefel-Whitney
classes
448
11.6.1
Spin bundles
449
11.6.2
Cech
cohomology groups
449
11.6.3 Stiefel-Whitney
classes
450
12
Index Theorems
453
12.1
Elliptic operators and
Fredholm
operators
453
12.1.1
Elliptic operators
454
12.1.2
Fredholm
operators
456
12.1.3
Elliptic complexes
457
12.2
The Atiyah-Singer index theorem
459
12.2.1
Statement of the theorem
459
12.3
The
de
Rham complex
460
12.4
The Dolbeault complex
462
12.4.1
The twisted Dolbeault complex and the Hirzebruch-
Riemann-Roch theorem
463
12.5
The signature complex
464
12.5.1
The Hirzebruch signature
464
12.5.2
The signature complex and the Hirzebruch signature
theorem
465
12.6
Spin complexes
467
12.6.1
Dirac operator
468
12.6.2
Twisted spin complexes
471
12.7
The heat kernel and generalized
f
-functions
472
12.7.1
The heat kernel and index theorem
472
12.7.2
Spectral
f
-functions
475
12.8
The Atiyah-Patodi-Singer index theorem
477
12.8.1
^-invariant and spectral flow
477
12.8.2
The Atiyah-Patodi-Singer (APS) index theorem
478
12.9
Supersymmetric quantum mechanics
481
12.9.1
Clifford algebra and
fermions
481
12.9.2
Supersymmetric quantum mechanics in flat space
482
12.9.3
Supersymmetric quantum mechanics in a general
manifold
485
12.10
Supersymmetric proof of index theorem
487
12.10.1
The index
487
12.10.2
Path integral and index theorem
490
Problems
500
13
Anomalies
in Gauge Field Theories
501
13.1
Introduction
501
13.2
Abelian anomalies
503
13.2.1
Fujikawa s method
503
13.3
Non-Abelian anomalies
508
13.4
The Wess-Zumino consistency conditions
512
13.4.1
The Becchi-Rouet-Stora operator and the Faddeev-
Popov ghost
512
13.4.2
The BRS operator, FP ghost and moduli space
513
13.4.3
The Wess-Zumino conditions
515
13.4.4
Descent equations and solutions of WZ conditions
515
13.5
Abelian anomalies versus non-Abelian anomalies
518
13.5.1
m
dimensions versus
m
+ 2
dimensions
520
13.6
The parity anomaly in odd-dimensional spaces
523
13.6.1
The parity anomaly
524
13.6.2
The dimensional ladder:
4-3-2 525
14
Bosonic String Theory
528
14.1
Differential geometry on Riemann surfaces
528
14.1.1
Metric and complex structure
528
14.1.2
Vectors, forms and tensors
529
14.1.3
Covariant derivatives
531
14.1.4
The Riemann-Roch theorem
533
14.2
Quantum theory of bosonic strings
535
14.2.1
Vacuum amplitude of Polyakov strings
535
14.2.2
Measures of integration
538
14.2.3
Complex tensor calculus and string measure
550
14.2.4
Moduli spaces of Riemann surfaces
554
14.3
One-loop amplitudes
555
14.3.1
Moduli spaces, CKV, Beltrami and quadratic differentials
555
14.3.2
The evaluation of determinants
557
References
560
Index
565
|
| any_adam_object | 1 |
| author | Nakahara, Mikio |
| author_GND | (DE-588)1028332297 |
| author_facet | Nakahara, Mikio |
| author_role | aut |
| author_sort | Nakahara, Mikio |
| author_variant | m n mn |
| building | Verbundindex |
| bvnumber | BV017559024 |
| classification_rvk | SK 950 |
| classification_tum | PHY 014f |
| ctrlnum | (OCoLC)488369729 (DE-599)BVBBV017559024 |
| discipline | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV017559024 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:19:19Z |
| institution | BVB |
| isbn | 0750306068 9780750306065 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010567866 |
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| owner_facet | DE-29T DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-384 DE-20 DE-11 DE-355 DE-BY-UBR |
| physical | XXII, 573 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Taylor & Francis |
| record_format | marc |
| series2 | Graduate student series in physics |
| spelling | Nakahara, Mikio Verfasser (DE-588)1028332297 aut Geometry, topology and physics Mikio Nakahara 2. ed. New York [u.a.] Taylor & Francis 2003 XXII, 573 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate student series in physics Hier auch später erschienene, unveränderte Nachdrucke Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Algebraische Topologie (DE-588)4120861-4 s Mathematische Physik (DE-588)4037952-8 s DE-604 Differentialgeometrie (DE-588)4012248-7 s Topologie (DE-588)4060425-1 s 2\p DE-604 Physik (DE-588)4045956-1 s 3\p DE-604 4\p DE-604 5\p DE-604 Differentialgleichung (DE-588)4012249-9 s 6\p DE-604 Geometrie (DE-588)4020236-7 s 7\p DE-604 Mathematische Methode (DE-588)4155620-3 s 8\p DE-604 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010567866&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 6\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 7\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 8\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Nakahara, Mikio Geometry, topology and physics Differentialgleichung (DE-588)4012249-9 gnd Mathematische Methode (DE-588)4155620-3 gnd Differentialgeometrie (DE-588)4012248-7 gnd Geometrie (DE-588)4020236-7 gnd Algebraische Topologie (DE-588)4120861-4 gnd Mathematische Physik (DE-588)4037952-8 gnd Topologie (DE-588)4060425-1 gnd Physik (DE-588)4045956-1 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4155620-3 (DE-588)4012248-7 (DE-588)4020236-7 (DE-588)4120861-4 (DE-588)4037952-8 (DE-588)4060425-1 (DE-588)4045956-1 (DE-588)4123623-3 |
| title | Geometry, topology and physics |
| title_auth | Geometry, topology and physics |
| title_exact_search | Geometry, topology and physics |
| title_full | Geometry, topology and physics Mikio Nakahara |
| title_fullStr | Geometry, topology and physics Mikio Nakahara |
| title_full_unstemmed | Geometry, topology and physics Mikio Nakahara |
| title_short | Geometry, topology and physics |
| title_sort | geometry topology and physics |
| topic | Differentialgleichung (DE-588)4012249-9 gnd Mathematische Methode (DE-588)4155620-3 gnd Differentialgeometrie (DE-588)4012248-7 gnd Geometrie (DE-588)4020236-7 gnd Algebraische Topologie (DE-588)4120861-4 gnd Mathematische Physik (DE-588)4037952-8 gnd Topologie (DE-588)4060425-1 gnd Physik (DE-588)4045956-1 gnd |
| topic_facet | Differentialgleichung Mathematische Methode Differentialgeometrie Geometrie Algebraische Topologie Mathematische Physik Topologie Physik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010567866&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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