Symmetry Rules : How Science and Nature Are Founded on Symmetry

by Rosen, Joe

• ISBN13: 9783540759720
• ISBN10: 3540759727
• Format: Hardcover
• Copyright: 2008-04-07
• Publisher: Springer Verlag

Summary

When we use science to describe and understand the world around us, we are in essence grasping nature through symmetry. In fact, modern theoretical physics suggests that symmetry is a, if not the, foundational principle of nature. Emphasizing the concepts, this book leads the reader coherently and comprehensively into the fertile field of symmetry and its applications. Among the most important applications considered are the fundamental forces of nature and the Universe. It is shown that the Universe cannot possess exact symmetry, which is a principle of fundamental significance. Curie's principle - which states that the symmetry of the effect is at least that of the cause - features prominently. An introduction to group theory, the mathematical language of symmetry, is included. This book will convince all interested readers of the importance of symmetry in science. Furthermore, it will serve as valuable background reading for all students in the physical sciences.

Author Biography

Joe Rosen served as chair of the Department of Physics and Astronomy of the University of Central Arkansas.

Table of Contents

The Concept of Symmetry	p. 1
The Essence of Symmetry	p. 1
Symmetry Implies Asymmetry	p. 8
Analogy and Classification Are Symmetry	p. 10
Summary	p. 15
Science Is Founded on Symmetry	p. 17
Science	p. 17
Reduction Is Symmetry	p. 20
Reduction to Observer and Observed	p. 22
Reduction to Quasi-Isolated System and Environment	p. 25
Reduction to Initial State and Evolution	p. 26
Reproducibility Is Symmetry	p. 29
Predictability Is Symmetry	p. 32
Analogy in Science	p. 35
Symmetry at the Foundation of Science	p. 37
Summary	p. 38
Symmetry in Physics	p. 39
Symmetry of Evolution	p. 40
Symmetry of States	p. 44
Reference Frame	p. 49
Global, Inertial, and Local Reference Frames	p. 53
Gauge Transformation	p. 55
Gauge Symmetry	p. 58
Symmetry and Conservation	p. 65
Conservation of Energy	p. 66
Conservation of Linear Momentum	p. 67
Conservation of Angular Momentum	p. 68
Symmetry at the Foundation of Physics	p. 70
Symmetry at the Foundation of Quantum Theory	p. 71
Association of a Hilbert Space with a Physical System	p. 71
Correspondence of Observables to Hermitian Operators	p. 73
Complete Set of Compatible Observables	p. 74
Heisenberg Commutation Relations	p. 75
Operators for Canonical Variables	p. 75
A Measurement Result Is an Eigenvalue	p. 75
Expectation Values and Probabilities	p. 76
The Hamiltonian Operator	p. 76
Planck's Constant as a Parameter	p. 77
The Correspondence Principle	p. 77
Summary	p. 77
The Symmetry Principle	p. 81
Causal Relation	p. 81
Equivalence Relation, Equivalence Class	p. 86
The Equivalence Principle	p. 89
The Symmetry Principle	p. 97
Cause and Effect in Quantum Systems	p. 102
Summary	p. 104
Application of Symmetry	p. 107
Minimalistic Use of the Symmetry Principle	p. 107
Maximalistic Use of the Symmetry Principle	p. 125
Summary	p. 130
Approximate Symmetry, Spontaneous Symmetry Breaking	p. 131
Approximate Symmetry	p. 131
Spontaneous Symmetry Breaking	p. 135
Summary	p. 140
Cosmic Considerations	p. 141
Symmetry of the Laws of Nature	p. 141
Symmetry of the Universe	p. 144
No Cosmic Symmetry Breaking or Restoration	p. 147
The Quantum Era and The Beginning	p. 155
Summary	p. 159
The Mathematics of Symmetry: Group Theory	p. 161
Group	p. 161
Mapping	p. 176
Isomorphism	p. 180
Homomorphism	p. 186
Subgroup	p. 192
Summary	p. 194
Group Theory Continued	p. 195
Conjugacy, Invariant Subgroup, Kernel	p. 195
Coset Decomposition	p. 203
Factor Group	p. 207
Anatomy of Homomorphism	p. 209
Generator	p. 215
Direct Product	p. 217
Permutation, Symmetric Group	p. 220
Cayley's Theorem	p. 224
Summary	p. 226
The Formalism of Symmetry	p. 227
System, State	p. 227
Transformation, Transformation Group	p. 229
Transformations in Space, Time, and Space-Time	p. 236
State Equivalence	p. 240
Symmetry Transformation, Symmetry Group	p. 243
Approximate Symmetry Transformation	p. 251
Quantification of Symmetry	p. 253
Quantum Systems	p. 255
Summary	p. 258
Symmetry in Processes	p. 261
Symmetry of the Laws of Nature	p. 261
Symmetry of Initial and Final States, the General Symmetry Evolution Principle	p. 270
The Special Symmetry Evolution Principle and Entropy	p. 274
Summary	p. 280
Summary of Principles	p. 283
Symmetry and Asymmetry	p. 283
Symmetry Implies Asymmetry	p. 283
No Exact Symmetry of the Universe	p. 284
Cosmological Implications	p. 285
The Equivalence Principle	p. 285
The Symmetry Principle	p. 285
The Equivalence Principle for Processes	p. 286
The Symmetry Principle for Processes	p. 286
The General Symmetry Evolution Principle	p. 286
The Special Symmetry Evolution Principle	p. 286
References	p. 289
Further Reading	p. 293
Index	p. 297
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