Title Symmetries in fundamental physics / Kurt Sundermeyer

Published Cham : Springer, [2014] ©2014

Table of Contents
1.	Introduction					1
1.1.	Symmetry: Argument, Principle, and Leitmotif					1
1.2.	Operations and Invariants					2
1.3.	"Symmetries" in "Fundamental Physics"					4
1.3.1.	What is Meant by "Fundamental Physics"?					4
1.3.2.	"Physics" on Which Level of Description?					5
1.3.3.	Which Kind of "Symmetry"?					9
1.4.	The Scope of Symmetries					11
1.4.1.	Ontology of Symmetries					11
1.4.2.	Symmetry Groups in Fundamental Physics					13
1.4.3.	The Use of Symmetries					14
1.5.	Bibliographical Notes					16
2.	Classical Mechanics					17
2.1.	Newtonian and Analytical Mechanics					18
2.1.1.	Newtonian Mechanics					18
2.1.2.	Lagrange Form of Mechanics					20
2.1.3.	Hamiltonian Formulation					22
2.1.4.	Principle of Stationary Action					25
2.1.5.	*Classical Mechanics in Geometrical Terms					30
2.2.	Symmetries and Conservation Laws					34
2.2.1.	Conservation Laws					34
2.2.2.	Noether Theorem-A First Glimpse					40
2.2.3.	Symmetry and Canonical Transformations					48
2.2.4.	Conservation Laws and Symmetries					50
2.2.5.	*Noether-Geometrically					60
2.3.	Galilei Group					61
2.3.1.	Transformations and Invariants of Classical Mechanics					61
2.3.2.	Structure of the Galilei Group					62
2.3.3.	Lie Algebra of the Galilei Group					63
2.4.	Concluding Remarks and Bibliographical Notes					64
3.	Electrodynamics and Special Relativity					67
3.1.	Electrodynamics a la Maxwell					67
3.1.1.	Maxwell Equations					67
3.1.2.	Lorentz Boosts					68
3.2.	Special Relativity					69
3.2.1.	"Deriving" Special Relativity					69
3.2.2.	Minkowski Geometry					73
3.2.3.	Relativistic Mechanics					78
3.2.4.	Relativistic Field Theory					81
3.3.	Noether Theorems					87
3.3.1.	Variational Symmetries in Field Theories					88
3.3.2.	Global Symmetries and 1st Noether Theorem					91
3.3.3.	Local Symmetries and 2nd Noether Theorem					96
3.3.4.	Further Topics Relating to Variational Symmetries					101
3.4.	Poincare Transformations					108
3.4.1.	Poincare and Lorentz Groups					108
3.4.2.	Poincare Algebra					110
3.4.3.	Galilei and Bargmann Algebra					112
3.4.4.	Forms of Relativistic Dynamics					113
3.4.5.	Kinematical Groups and Their Mutual Contractions					114
3.5.	*Generalizations of Poincare Symmetry					120
3.5.1.	Conformal Symmetry					120
3.5.2.	de Sitter Group					128
3.6.	On the Validity of Special Relativity					131
3.7.	Concluding Remarks and Bibliographical Notes					133
4.	Quantum Mechanics					135
4.1.	Principles of Quantum Mechanics					135
4.1.1.	Hilbert Space					136
4.1.2.	Operators					136
4.1.3.	States, Observables, and Measurements					138
4.1.4.	Time Evolution					140
4.2.	Symmetry Transformations in Quantum Mechanics					142
4.2.1.	Wigner Theorem					143
4.2.2.	Symmetry Transformations and Observables					145
4.2.3.	"Noether Theorem of Quantum Mechanics"					145
4.2.4.	Symmetries and Superselection Rules					146
4.3.	Quantum Physics and Group Representation					148
4.3.1.	Why Group Representation?					148
4.3.2.	Galilei Operators					148
4.3.3.	Bargmann Group					152
4.3.4.	Symmetries of the Schrodinger Equation					154
4.4.	Concluding Remarks and Bibliographical Notes					159
5.	Relativistic Field Theory					161
5.1.	Representations of the Poincare Group					163
5.1.1.	Global Structure of ISO(3,1)					164
5.1.2.	Transformation of the Generators					164
5.1.3.	The "Little Group"					165
5.1.4.	Classification of Particles					167
5.2.	Symmetry and Quantum Field Theory					173
5.2.1.	Lorentz Symmetry Rules Field Variants					173
5.2.2.	Representations of SL(2, C)					174
5.2.3.	Field Variants					175
5.2.4.	Quantum-Field Theoretical Symmetry Transformations					177
5.3.	Actions					178
5.3.1.	Requirements on a QFT Action					178
5.3.2.	Scalar Fields					180
5.3.3.	Spinor Actions					186
5.3.4.	Gauge Vector Fields					194
5.3.5.	Higher-Spin Fields					209
5.4.	Spontaneous Symmetry Breaking					213
5.4.1.	Goldstone Bosons					213
5.4.2.	Nambu-Goldstone Model					216
5.4.3.	Higgs Mechanism					218
5.5.	Discrete Symmetries					222
5.5.1.	General Preliminary Remarks and Definition of Terms					222
5.5.2.	Space Inversion P					223
5.5.3.	Time Reversal T					226
5.5.4.	Charge Conjugation C					228
5.5.5.	CPT Theorem					229
5.6.	Effective Field Theories					231
5.6.1.	EFT: The Very Idea					231
5.6.2.	Historical Examples					232
5.6.3.	Renormalization (Group)					234
5.6.4.	Chain of Effective Theories					240
5.7.	Concluding Remarks and Bibliographical Notes					242
6.	Particle Physics					249
6.1.	Particles and Interactions					249
6.1.1.	Standard Model Constituents					249
6.1.2.	Quarks as Building Blocks of Hadrons					252
6.1.3.	Interaction Processes					260
6.1.4.	Lagrangian of the Standard Model					262
6.2.	Strong Interactions					264
6.2.1.	Lagrangian of Quantum Chromo Dynamics					264
6.2.2.	Symmetries of QCD					265
6.2.3.	Theoretical Consistency and Experimental Support					266
6.3.	Weak and Electromagnetic Interaction					268
6.3.1.	Fermi-Type Model of Weak Interactions					269
6.3.2.	Current Algebra					270
6.3.3.	Glashow-Salam-Weinberg Model					274
6.3.4.	Theoretical Consistency and Experimental Support					279
6.4.	Paralipomena on the Standard Model					279
6.4.1.	Limits of the Standard Model					280
6.4.2.	Massive Neutrinos					282
6.4.3.	Anomalies					285
6.4.4.	Strong CP Problem					288
6.4.5.	Standard Model and Effective Field Theories					290
6.5.	Concluding Remarks and Bibliographical Notes					294
7.	General Relativity and Gravitation					297
7.1.	Introductory Remarks					297
7.2.	Equivalence Principle					299
7.2.1.	Different Versions of the Equivalence Principle					299
7.2.2.	Reference Systems and Gravitation					302
7.2.3.	Geodesies					303
7.2.4.	The "Principle" of General Covariance					307
7.3.	Riemann-Cartan Geometry					307
7.3.1.	Tensors					308
7.3.2.	Affine Connection and Covariant Derivative					309
7.3.3.	Torsion and Curvature					312
7.3.4.	Metric					313
7.3.5.	Tetrads and Spin Connections					317
7.4.	Physics in Curved Spacetime					322
7.4.1.	Mechanics, Hydrodynamics, Electrodynamics					322
7.4.2.	Coupling Relativistic Fields to Gravity					325
7.5.	Geometrodynamics					327
7.5.1.	Field Equations					327
7.5.2.	Action Functionals for General Relativity					331
7.5.3.	Covariance, Invariance, and Symmetries					343
7.5.4.	Noether Identities and Conservation Laws					348
7.6.	*Modifications and Extensions of/to General Relativity					361
7.6.1.	Interpreting GR as a Spin-2 Field Theory					362
7.6.2.	Altering the Geometry					364
7.6.3.	Gravitation as a Gauge Theory					368
7.6.4.	Changing Structures and Modifying Principles					378
7.7.	Concluding Remarks and Bibliographical Notes					385
8.	*Unified Field Theories					387
8.1.	Grand Unified Theories					387
8.1.1.	Motivation and Basic Concepts					387
8.1.2.	SU(5) Grand Unification					390
8.1.3.	SO(10) Grand Unification					394
8.1.4.	Instead of a Conclusion					395
8.2.	Kaluza-Klein Theory					395
8.2.1.	Kaluza's and Klein's Contributions to the KK Theory					395
8.2.2.	The 5D Model					398
8.2.3.	Beyond Five Dimensions: Einstein-Yang-Mills Theory					405
8.2.4.	Instead of a Conclusion					417
8.3.	Supersymmetry					418
8.3.1.	Why Supersymmetry?					419
8.3.2.	Compelling Consequences of Fermi-Bose Symmetry					421
8.3.3.	Global Supersymmetry					422
8.3.4.	Local Supersymmetry and Supergravity					437
8.3.5.	Instead of a Conclusion					444
8.4.	Further Speculations					445
8.4.1.	Compositeness and Technicolor					445
8.4.2.	Strings and Branes					446
8.4.3.	Gauge/Gravity Duality Conjecture					453
9.	Conclusion					455
9.1.	Symmetries: The Road to Reality					455
9.1.1.	Symmetry: The Golden Thread					455
9.1.2.	The "Weltgesetze" and Their Symmetries					459
9.1.3.	History of Symmetry Considerations					462
9.2.	Are Symmetries a Principle of Nature?					472
9.2.1.	... and Other Philosophical Questions					472
9.2.2.	Symmetries and the Unification of Physics					474
9.2.3.	Laws of Nature and Principles of Physics					479
9.2.4.	Origin of Symmetries					486
9.3.	Physics Beyond Symmetries					486
9.3.1.	Prominent Non-Symmetries					487
9.3.2.	Other Notions of Fundamental Physics					488
9.3.3.	Are we Biased, or Haughty, or Simply in a Specific World?					491
Appendix A	Group Theory					493
A.1.	Basics					493
A.1.1.	Definitions: Algebraic Structures					493
A.1.2.	Mapping of Groups					497
A.1.3.	Simple Groups					498
A.2.	Lie Groups					501
A.2.1.	Definitions and Examples					501
A.2.2.	Generators of a Lie Group					505
A.2.3.	Lie Algebra Associated to a Lie Group					506
A.2.4.	Inonu-Wigner Contraction of Lie Groups					510
A.2.5.	Classification of Lie Groups					512
A.2.6.	Infinite-Dimensional Lie Groups					515
A.3.	Representation of Groups					516
A.3.1.	Definitions and Examples					516
A.3.2.	Representations of Finite Groups					517
A.3.3.	Representation of Continuous Groups					520
A.3.4.	Examples: Representations of SO(2), SO(3), SU(3)					523
A.3.5.	Projective Representations and Central Charges					530
Appendix B	Spinors, Z2-gradings, and Supergeometry					535
B.1.	Spinors					535
B.1.1.	Pauli and Dirac Matrices					536
B.1.2.	Weyl Spinors					538
B.1.3.	Spinors and Tensors					541
B.1.4.	Dirac and Majorana Spinors					542
B.2.	*Z2 Gradings					544
B.2.1.	Definitions					545
B.2.2.	Supertrace and Superdeterminant					545
B.2.3.	Differentiation and Integration					546
B.2.4.	Pseudo-Classical Mechanics					549
B.3.	*Supergeometry					550
B.3.1.	Superspace					550
B.3.2.	Superfields					552
B.3.3.	Superactions					558
B.4.	*Supergroups					563
B.4.1.	OSp(N/M) and the Super-Poincare Algebra					563
B.4.2.	SU(N/M) and the Super-Conformal Algebra					564
Appendix C	Symmetries and Constrained Dynamics					565
C.1.	Constrained Dynamics					565
C.1.1.	Singular Lagrangians					565
C.1.2.	Constraints as a Consequence of Local Symmetries					567
C.1.3.	Rosenfeld-Dirac-Bergmann Algorithm					568
C.1.4.	First-Class Constraints and Symmetries					574
C.1.5.	Second-Class Constraints and Gauge Conditions					581
C.1.6.	Constraints in Field Theories: Some Remarks					583
C.1.7.	Quantization of Constrained Systems					584
C.2.	Yang-Mills Type Theories					586
C.2.1.	Electrodynamics					586
C.2.2.	Maxwell-Dirac Theory					588
C.2.3.	Non-Abelian Gauge Theories					590
C.3.	Reparametrization-Invariant Theories					592
C.3.1.	Immediate Consequences of Reparametrization Invariance					592
C.3.2.	Free Relativistic Particle					594
C.3.3.	Metric Gravity					604
C.3.4.	Tetrad Gravity					619
C.3.5.	Einstein-Dirac-Yang-Mills-Higgs Theory					622
C.4.	Alternative Approaches					624
C.5.	Constraints and Presymplectic Geometry					626
C.5.1.	Legendre Projectability					626
C.5.2.	Symmetry Transformations in the Tangent and Cotangent Bundle					628
C.5.3.	Constraint Stabilization					631
Appendix D	*Symmetries in Path-Integral and BRST Quantization					633
D.1.	Basics					634
D.1.1.	Path-integral Formulation of Quantum Mechanics					634
D.1.2.	Functional Integrals in Field Theory					638
D.1.3.	Faddeev-Popov Ghost Fields in Theories with Local Symmetries					641
D.2.	Noether and Functional Integrals					645
D.2.1.	Noether Currents and Ward-Takahashi-Slavnov-Taylor Identities					646
D.2.2.	Quantum Action and its Symmetries					647
D.2.3.	BRST Symmetries					649
D.2.4.	Fujikawa: Fermionic Path Integrals and Anomalies					655
Appendix E	*Differential Geometry					657
E.1.	Differentiable Manifolds					658
E.1.1.	From Topological Spaces to Differentable Manifolds					658
E.1.2.	Tensor Bundles					659
E.1.3.	Flows and the Lie Derivative					662
E.1.4.	Symplectic Manifolds					663
E.2.	Cartan Calculus					663
E.2.1.	Differential Forms					663
E.2.2.	Differentiation with Respect to a Form					665
E.2.3.	Hodge Duality*					665
E.2.4.	Integration of Differential Forms and Stokes's Theorem					667
E.2.5.	Poincare Lemma and de Rham Cohomology					669
E.3.	Manifolds with Connection					670
E.3.1.	Linear Connection on Tensor Fields					671
E.3.2.	Covariant Derivative					672
E.3.3.	Torsion and Curvature					672
E.4.	Lie Groups					674
E.4.1.	Lie Algebra					675
E.4.2.	Group Covariant Derivative					676
E.4.3.	Group Curvature					677
E.4.4.	Isometries and Coset Manifolds					678
E.5.	Fibre Bundles					682
E.5.1.	Definition, Various Types, and Examples of Fibre Bundles					682
E.5.2.	Connections in Fibre Bundles					685
E.5.3.	Yang-Mills Gauge Field Theory in Fibre Bundle Language					687
E.5.4.	Metric and Tetrad Gravity on a Bundle					690
Appendix F	*Symmetries in Terms of Differential Forms					695
F.1.	Actions and Field Equations					695
F.1.1.	The "World" Action					695
F.1.2.	Field Equations					700
F.2.	Symmetries					706
F.2.1.	Generic Variational Symmetries					706
F.2.2.	Lorentz Transformations					709
F.2.3.	Diffeomorphisms					711
F.3.	Gravitational Theories					718
F.3.1.	Energy-Momentum Conservation					718
F.3.2.	Gravitational Theories Beyond Einstein					723
F.3.3.	Topological Terms					726
F.4.	Gauge Theories					729
F.4.1.	Global Symmetry					730
F.4.2.	Local Gauge Transformations					730
	References					735
	Index					757

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Description 1 online resource (763 pages) Series Fundamental theories of physics ; v. 176 Fundamental theories of physics ; volume 176. Contents Preface -- Notation -- Introduction -- Symmetry: Argument, Principle, and Leitmotif -- Operations and Invariants -- "Symmetries" in "Fundamental Physics" -- The Scope of Symmetries -- Bibliographical Notes -- Classical Mechanics -- Newtonian and Analytical Mechanics -- Symmetries and Conservation Laws -- Galilei Group -- Electrodynamics and Special Relativity -- Electrodynamics à la Maxwell -- Special Relativity -- Noether Theorems -- Poincaré Transformations -- Generalizations of Poincaré Symmetry -- On the Validity of Special Relativity -- Quantum Mechanics -- Principles of Quantum Mechanics -- Symmetry Transformations in Quantum Mechanics -- Quantum Physics and Group Representation -- Relativistic Field Theory -- Representations of the Poincaré Group -- Symmetry and Quantum Field Theory -- Actions -- Spontaneous Symmetry Breaking -- Discrete Symmetries -- Effective field theories -- Particle Physics -- Particles and Interactions -- Strong Interactions -- Weak and Electromagnetic Interaction -- Paralipomena on the Standard Model -- General Relativity and Gravitation -- Introductory Remarks -- Equivalence Principle -- Riemann-Cartan Geometry -- Physics in Curved Spacetime -- Geometrodynamics -- Modifications and Extensions of/to General Relativity -- Unified Field Theories -- Grand Unified Theories -- Kaluza-Klein Theory -- Supersymmetry -- Further Speculations -- Conclusion -- Symmetries: The Road to Reality -- Are Symmetries a Principle of Nature? -- Physics Beyond Symmetries -- Appendices -- A Group Theory -- B Spinors, Z2-gradings, and Supergeometry -- C Symmetries and Constrained Dynamics -- D Symmetries in Path-Integral and BRST Quantization -- E Differential Geometry -- F Symmetries in Terms of Differential Forms -- Bibliography Summary "Over the course of the last century it has become clear that both elementary particle physics and relativity theories are based on the notion of symmetries. These symmetries become manifest in that the 'laws of nature' are invariant under spacetime transformations and/or gauge transformations. The consequences of these symmetries were analyzed as early as in 1918 by Emmy Noether on the level of action functionals. Her work did not receive due recognition for nearly half a century, but can today be understood as a recurring theme in classical mechanics, electrodynamics and special relativity, Yang-Mills type quantum field theories, and in general relativity. As a matter of fact, as shown in this monograph, many aspects of physics can be derived solely from symmetry considerations"--Provided by publisher Bibliography Includes bibliographical references and index Notes Online resource; title from PDF title page (ebrary, viewed January 13, 2014) Subject Symmetry (Physics) SCIENCE -- Energy. SCIENCE -- Mechanics -- General. SCIENCE -- Physics -- General. Physique. Astronomie. Symmetry (Physics) Form Electronic book ISBN 9789400776425 940077642X

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