Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition

by Choquet-Bruhat; DeWitt-Morette

• ISBN13: 9780444504739
• ISBN10: 0444504737
• Edition: Revised
• Format: Hardcover
• Copyright: 2000-11-08
• Publisher: Elsevier Science

Summary

Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in volume 1 of the 2-Volume set Analysis, Manifolds and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number of parenthesis following the title.

Table of Contents

Preface to the second edition	v
Preface	vii
Contents	ix
Conventions	xv
Review of Fundamental Notions of Analysis	1	(50)
Graded algebras	1	(2)
Berezinian	3	(2)
Tensor product of algebras	5	(1)
Clifford algebras	6	(8)
Clifford algebra as a coset of the tensor algebra	14	(1)
Fierz identity	15	(2)
Pin and Spin groups	17	(10)
Weyl spinors, helicity operator; Majorana pinors, charge conjugation	27	(6)
Representations of Spin(n, m), n + m odd	33	(3)
Dirac adjoint	36	(1)
Lie algebra of Pin(n, m) and Spin(n, m)	37	(2)
Compact spaces	39	(1)
Compactness in weak star topology	40	(2)
Homotopy groups, general properties	42	(4)
Homotopy of topological groups	46	(1)
Spectrum of closed and self-adjoint linear operators	47	(4)
Differential Calculus on Banach Spaces	51	(40)
Supersmooth mappings	51	(6)
Berezin integration; Gaussian integrals	57	(7)
Noether's theorems I	64	(7)
Noether's theorems II	71	(8)
Invariance of the equations of motion	79	(3)
String action	82	(1)
Stress-energy tensor; energy with respect to a timelike vector field	83	(8)
Differentiable Manifolds	91	(36)
Sheaves	91	(1)
Differentiable submanifolds	91	(1)
Subgroups of Lie groups. When are they Lie subgroups?	92	(1)
Cartan-Killing form on the Lie algebra g of a Lie group G	93	(2)
Direct and semidirect products of Lie groups and their Lie algebra	95	(7)
Homomorphisms and antihomomorphisms of a Lie algebra into spaces of vector fields	102	(1)
Homogeneous spaces; symmetric spaces	103	(5)
Examples of homogeneous spaces, Stiefel and Grassmann manifolds	108	(2)
Abelian representations of nonabelian groups	110	(1)
Irreducibility and reducibility	111	(3)
Characters	114	(1)
Solvable Lie groups	114	(1)
Lie algebras of linear groups	115	(3)
Graded bundles	118	(9)
Integration on Manifolds	127	(108)
Cohomology. Definitions and exercises	127	(7)
Obstruction to the construction of Spin and Pin bundles; Stiefel-Whitney classes	134	(16)
Inequivalent spin structures	150	(8)
Cohomology of groups	158	(3)
Lifting a group action	161	(2)
Short exact sequence; Weyl Heisenberg group	163	(4)
Cohomology of Lie algebras	167	(4)
Quasi-linear first-order partial differential equation	171	(2)
Exterior differential systems	173	(8)
B. Kent Harrison
Backlund transformations for evolution equations	181	(3)
N.H. Ibragimov
Poisson manifolds I	184	(16)
Poisson manifolds II	200	(19)
C. Moreno
Completely integrable systems	219	(16)
C. Moreno
Riemannian Manifolds. Kahlerian Manifolds	235	(68)
Necessary and sufficient conditions for Lorentizian signature	235	(3)
First fundamental form (induced metric)	238	(1)
Killing vector fields	239	(1)
Sphere Sn	240	(4)
Curvature of Einstein cylinder	244	(1)
Conformal transformation of Yang-Mills, Dirac and Higgs operators in d dimensions	244	(5)
Conformal system for Einstein equations	249	(7)
Conformal transformation of nonlinear wave equations	256	(6)
Masses of ``homothetic'' space-time	262	(1)
Invariant geometries on the squashed seven spheres	263	(11)
Harmonic maps	274	(7)
Composition of maps	281	(5)
Kaluza-Klein theories	286	(8)
Kahler manifolds; Calabi-Yau spaces	294	(9)
Bis. Connections on a Principal Fibre Bundle	303	(70)
An explicit proof of the existence of infinitely many connections on a principal bundle with paracompact base	303	(2)
Gauge transformations	305	(2)
Hopf fibering S3 → S3	307	(1)
Subbundles and reducible bundles	308	(2)
Broken symmetry and bundle reduction, Higgs mechanism	310	(11)
The Euler-Poincare characteristic	321	(13)
Equivalent bundles	334	(1)
Universal bundles. Bundle classification	335	(5)
Generalized Bianchi identity	340	(1)
Chern-Simons classes	340	(9)
Cocycles on the Lie algebra of a gauge group; Anomalies	349	(14)
Virasoro representation of l (Diff S1). Ghosts. BRST operator	363	(10)
Distributions	373	(60)
Elementary solution of the wave equation in d-dimensional spacetime	373	(4)
Sobolev embedding theorem	377	(9)
Multiplication properties of Sobolev spaces	386	(3)
The best possible constant for a Sobolev inequality on ≥ 3	389	(2)
H. Grosse
Hardy-Littlewood-Sobolev inequality	391	(2)
H. Grosse
Spaces Hs, δ(Rn)	393	(3)
Spaces Hs(Sn) and Hs, δ(Rn)	396	(2)
Completeness of a ball on Wsp in Wsp-1	398	(1)
Distribution with laplacian in L2(Rn)	399	(1)
Nonlinear wave equation in curved spacetime	400	(5)
Harmonic coordinates in general relativity	405	(2)
Leray theory of hyperbolic systems. Temporal gauge in general relativity	407	(6)
Einstein equations with sources as a hyperbolic system	413	(1)
Distributions and analyticity: Wightman distributions and Schwinger functions	414	(11)
C. Doering
Bounds on the number of bound states of the Schrodinger operator	425	(3)
Sobolev spaces on Riemannian manifolds	428	(5)
Supplements and Additional Problems	433	(92)
The isomorphism H ⊗ ≃ M4(R). A supplement to Problem I.4 (I.17)	435	(2)
Lie derivative of spinor fields (III.15)	437	(6)
Poisson-Lie groups, Lie bialgebras, and the generalized classical Yang-Baxter equation (IV.14)	443	(33)
Carlos Moreno
Luis Valero
Volume of the sphere Sn. A supplement to Problem V.4 (V.15)	476	(2)
Teichmuller spaces (V.16)	478	(5)
Yamabe property on compact manifolds (V.17)	483	(12)
The Euler class. A supplement to Problem Vbis.6 (Vbis.13)	495	(1)
Formula for lapacians at a point of the frame bundle (Vbis.14)	496	(4)
The Berry and Aharanov-Anandan phases (Vbis.15)	500	(12)
A density theorem. A supplement to Problem VI.6 ``Spaces Hs,δ(Rn)'' (VI.17)	512	(1)
Tensor distributions on submanifolds, multiple layers, and shocks (VI.18)	513	(8)
Discrete Boltzmann equation (VI.19)	521	(4)
Subject Index	525	(6)
Errata to Part I	531