Symplectic Techniques in Physics

Front Cover
Cambridge University Press, May 25, 1990 - Mathematics - 468 pages
Symplectic geometry is very useful for clearly and concisely formulating problems in classical physics and also for understanding the link between classical problems and their quantum counterparts. It is thus a subject of interest to both mathematicians and physicists, though they have approached the subject from different view points. This is the first book that attempts to reconcile these approaches. The authors use the uncluttered, coordinate-free approach to symplectic geometry and classical mechanics that has been developed by mathematicians over the course of the last thirty years, but at the same time apply the apparatus to a great number of concrete problems. In the first chapter, the authors provide an elementary introduction to symplectic geometry and explain the key concepts and results in a way accessible to physicists and mathematicians. The remainder of the book is devoted to the detailed analysis and study of the ideas discussed in Chapter 1. Some of the themes emphasized in the book include the pivotal role of completely integrable systems, the importance of symmetries, analogies between classical dynamics and optics, the importance of symplectic tools in classical variational theory, symplectic features of classical field theories, and the principle of general covariance. This work can be used as a textbook for graduate courses, but the depth of coverage and the wealth of information and application means that it will be of continuing interest to, and of lasting significance for mathematicians and mathematically minded physicists.
 

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Contents

Introduction
1
1 Gaussian optics
7
2 Hamiltons method in Gaussian optics
17
3 Fermats principle
20
4 From Gaussian optics to linear optics
23
5 Geometrical optics Hamiltons method and the theory of geometrical aberrations
34
6 Fermats principle and Hamiltons principle
42
7 Interference and diffraction
47
30 More EulerPoisson equations
233
31 The choice of a collective Hamiltonian
242
32 Convexity properties of toral group actions
249
33 The lemma of stationary phase
260
34 Geometric quantization
265
Motion in a YangMills field and the principle of general covariance
272
36 Curvature
283
37 The energymomentum tensor and the current
296

8 Gaussian integrals
51
9 Examples in Fresnel optics
54
10 The phase factor
60
11 Fresnels formula
71
12 Fresnel optics and quantum mechanics
75
13 Holography
85
14 Poisson brackets
88
15 The Heisenberg group and representation
92
16 The Groenwaldvan Hove theorem
101
17 Other quantizations
104
18 Polarization of light
116
19 The coadjoint orbit structure of a semidirect product
124
20 Electromagnetism and the determination of symplectic structures
130
Why symplectic geometry?
145
The geometry of the moment map
151
22 The DarbouxWeinstein theorem
155
23 Kaehler manifolds
160
24 Leftinvariant forms and Lie algebra cohomology
169
25 Symplectic group actions
172
26 The moment map and some of its properties
183
27 Group actions and foliations
196
28 Collective motion
210
29 Cotangent bundles and the moment map for semidirect products
220
38 The principle of general covariance
304
39 Isotropic and coisotropic embeddings
313
40 Symplectic induction
319
41 Symplectic slices and moment reconstruction
324
42 An alternative approach to the equations of motion
331
43 The moment map and kinetic theory
344
Complete integrability
349
45 Collective complete integrability
359
46 Collective action variables
367
47 The KostantSymes lemma and some of its variants
371
48 Systems of Calogero type
381
49 Solitons and coadjoint structures
391
50 The algebra of formal pseudodifferential operators
397
51 The higherorder calculus of variations in one variable
407
Contractions of symplectic homogeneous spaces
416
52 The Whitehead lemmas
417
53 The HochschildSerre spectral sequence
430
54 Galilean and Poincaré elementary particles
437
55 Coppersmiths theory
446
References
458
Index
467
Copyright

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Page 458 - ATIYAH, MF and SINGER, IM The index of elliptic operators. III. Ann. of Math. 87 (1968), 546^604.

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