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Natural and Gauge Natural Formalism for Classical Field Theorie : A Geometric Perspective including Spinors and Gauge Theories - L. Fatibene

Natural and Gauge Natural Formalism for Classical Field Theorie

A Geometric Perspective including Spinors and Gauge Theories

Hardcover

Published: 1st November 2003
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The mathematical theory of gauge natural bundles has recently been developed to provide a natural mathematical framework for gauge covariant fields (principal connections and associated objects). For this reason the language and the techniques of gauge natural bundles provide a privileged unifying scheme to treat the dynamics of all classical field theories. In this volume the authors develop and work out applications to gravity and gauge theories and their interactions with generic matter fields, including spinors in full detail. Spinor fields in particular appear to be the prototypes of truly gauge-natural objects, which are not purely gauge nor purely natural, so that they are a paradigmatic example of the intriguing relations between gauge natural geometry and physical phenomenology. In particular, the gauge natural framework for spinors is developed in this book in full detail, and it is shown to be fundamentally related to the interaction between fermions and dynamical tetrad gravity. The gauge natural framework also provides an extended area to deal with conservation laws and it encompasses all apparently unrelated covariant prescriptions commonly used in relativistic theories to generate other currents and conserved integrals. This book should be of interest to researchers and graduate students in the fields of theoretical physics, mathematical physics as well as differential geometry.

Forewordp. xi
Introductionp. xv
Acknowledgmentsp. xxi
The Geometric Setting
Introductionp. 3
Fiber Bundlesp. 9
Definition of Fiber Bundlesp. 10
Fibered Coordinatesp. 22
Particular Classes of Fiber Bundlesp. 23
Fiber Bundles with Structure Groupp. 34
Canonical Constructions of Bundlesp. 39
Vector Fields on a Bundlep. 46
Exercisesp. 50
Jet Bundlesp. 53
The Jet Bundlesp. 53
Prolongation of Morphisms and Sectionsp. 57
Contact Formsp. 60
The Infinite Jet Bundlep. 63
Generalized Vector Fieldsp. 67
Lie Derivative of Sectionsp. 70
Examplesp. 72
Differential Equationsp. 74
Exercisesp. 77
Principal Bundles and Connectionsp. 79
Principal Bundles and Their Right Actionp. 80
Invariant Vector Fieldsp. 81
Local Expression of Principal Automorphismsp. 82
Infinitesimal Generators of Automorphismsp. 85
Bundle Connectionp. 88
Induced Connections on Associated Bundlesp. 98
Examplesp. 99
Structures Induced by a Connectionp. 105
Exercisesp. 112
Natural Bundlesp. 113
Natural Bundlesp. 113
Natural Morphismsp. 118
Lie Derivatives of Sectionsp. 120
Examplesp. 122
Exercisesp. 124
Gauge Natural Bundlesp. 125
Gauge Natural Bundlesp. 125
Gauge Natural Prolongations of a Principal Bundlep. 128
The Infinitesimal Generators of Automorphisms (IGA)p. 135
Lie Derivatives of Sectionsp. 138
Examplesp. 139
Exercisesp. 142
The Variational Structure of Field Theories
Introductionp. 145
The Lagrangian Formalismp. 151
The Lagrangian and the Action Functionalp. 152
The Variational Morphismsp. 155
Euler-Lagrange Equationsp. 169
Poincare-Cartan Formp. 173
Symmetries and Nother's Theoremp. 180
Examplesp. 187
Conserved Quantities in Field Theoryp. 204
BRST Transformationsp. 213
Exercisesp. 217
Natural Theoriesp. 219
Definition of Natural Theoriesp. 220
Bianchi Identities and Superpotentialsp. 224
Conserved Quantitiesp. 226
Geodesicsp. 233
General Relativityp. 234
On Maxwell and Yang-Mills Naturalityp. 247
Gravity and Bosonic Matterp. 251
Metric-Affine Formalism for Gravityp. 265
Exercisesp. 268
Gauge Natural Theoriesp. 269
Motivationsp. 270
Gauge Natural Field Theoriesp. 272
Generalized Bianchi Identities and Superpotentialsp. 273
Conserved Quantitiesp. 275
Yang-Mills Theoriesp. 277
Gauge Natural Formulation of Hilbert Lagrangianp. 280
Vielbein Formulation of Gravitational Theoriesp. 281
Kosmann Liftp. 288
Exercisesp. 291
Spinor Fields
Introductionp. 295
Spin Structures and Spin Framesp. 299
Clifford Algebras and Spin Groupsp. 299
Spin Structuresp. 304
Example: Spin Structures on Spheresp. 307
Spin Framesp. 314
The Bundle of Spin Framesp. 317
Other Formulations of General Relativityp. 319
Kosmann Liftp. 321
Exercisesp. 323
Spinor Theoriesp. 325
Dirac Spinor Fieldp. 325
Neutrinosp. 329
(Reduced) Electroweak Modelp. 332
Anticommuting Majorana Spinor Fieldsp. 339
Wess-Zumino Modelp. 343
Rarita-Schwinger Modelp. 345
Exercisesp. 347
Final Wordp. 349
Referencesp. 353
Bibliographyp. 356
Analitic Indexp. 359
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781402017032
ISBN-10: 1402017030
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 365
Published: 1st November 2003
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 1.6