The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or nonlocal boundary conditions. The theory is introduced at a level suitable for graduate students. Numerous examples are then given to illustrate the peculiar properties of the Dirac operator, and the role of boundary conditions in heat-kernel asymptotics and quantum field theory. Topics covered include the introduction of spin-structures in Riemannian and Lorentzian manifolds; applications of index theory; heat-kernel asymptotics for operators of Laplace type; quark boundary conditions; one-loop quantum cosmology; conformally covariant operators; and the role of the Dirac operator in some recent investigations of four-manifolds. This volume provides graduate students with a rigorous introduction and researchers with a invaluable reference to the Dirac operator and its applications in theoretical physics.
Author(s): Giampiero Esposito
Year: 1998
Language: English
Pages: 224
Contents......Page 10
Preface......Page 12
Acknowledgments......Page 14
1.1 The Dirac equation......Page 16
1.2 Spinor fields in Lorentzian manifolds......Page 21
1.3 Spin-structures: Riemannian case......Page 24
1.4 Global theory of the Dirac operator......Page 27
Appendix 1.A......Page 30
2.1 Operators of Laplace type......Page 37
2.2 Clifford algebras......Page 39
2.3 Index of elliptic operators and signature operator......Page 40
2.4 Dirac operator......Page 43
2.5 Some outstanding problems......Page 48
2.6 Pseudo-differential operators......Page 50
3.1 Index problem for manifolds with boundary......Page 56
3.2 Elliptic boundary conditions......Page 59
3.3 Index theorems and gauge fields......Page 60
3.4 Index of two-parameter families......Page 67
3.5 Determinants of Dirac operators......Page 68
3.6 Bott periodicity......Page 70
3.7 K-theory......Page 80
Appendix 3.A......Page 83
Appendix 3.B......Page 85
4 Spectral asymmetry......Page 89
4.1 Spectral asymmetry and Riemannian geometry......Page 90
4.2 7](0) calculation......Page 92
4.3 A further example......Page 97
4.4 Massless spin-l/2 fields......Page 99
4.5 ((0) value with non-local boundary conditions......Page 101
4.6 Essential self-adjoint ness......Page 105
Appendix 4.A......Page 110
Appendix 4.B......Page 116
5 Spectral geometry with operators ofLaplace type......Page 121
5.1 On hearing the shape of a drum......Page 122
5.2 The Laplacian on manifolds with boundary......Page 126
5.3 Functorial method......Page 130
5.4 Mixed boundary conditions......Page 141
5.5 Heat equation and index theorem......Page 143
5.6 Heat kernel and (-function......Page 145
5.7 Majorizations of the heat kernel......Page 146
Appendix 5.A......Page 151
6.1 Introduction......Page 163
6.2 Quark boundary conditions......Page 165
6.3 Quantum cosmology......Page 167
6.4 Conformal gauges and manifolds with boundary......Page 169
6.5 Conformally covariant operators......Page 177
6.6 Euclidean quantum gravity......Page 180
6.7 Dirac operator and the theory of four-manifolds......Page 189
Appendix 6.A......Page 196
Appendix 6.B......Page 198
Appendix 6.C......Page 201
Appendix 6.D......Page 202
References......Page 206
Index......Page 222
