Symmetry Rules: How Science and Nature Are Founded on Symmetry (The Frontiers Collection)

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. 1 The Concept of Symmetry -- 1 -- 1.1 The Essence of Symmetry -- 1 -- 1.2 Symmetry Implies Asymmetry -- 8 -- 1.3 Analogy and Classification Are Symmetry -- 10 -- 2 Science Is Founded on Symmetry -- 17 -- 2.1 Science -- 17 -- 2.2 Reduction Is Symmetry -- 20 -- 2.2.1 Reduction to Observer and Observed -- 22 -- 2.2.2 Reduction to Quasi-Isolated System and Environment -- 25 -- 2.2.3 Reduction to Initial State and Evolution -- 26 -- 2.3 Reproducibility Is Symmetry -- 29 -- 2.4 Predictability Is Symmetry -- 32 -- 2.5 Analogy in Science -- 35 -- 2.6 Symmetry at the Foundation of Science -- 37 -- 3 Symmetry in Physics -- 39 -- 3.1 Symmetry of Evolution -- 40 -- 3.2 Symmetry of States -- 44 -- 3.3 Reference Frame -- 49 -- 3.4 Global, Inertial, and Local Reference Frames -- 53 -- 3.5 Gauge Transformation -- 55 -- 3.6 Gauge Symmetry -- 58 -- 3.7 Symmetry and Conservation -- 65 -- 3.7.1 Conservation of Energy -- 66 -- 3.7.2 Conservation of Linear Momentum -- 67 -- 3.7.3 Conservation of Angular Momentum -- 68 -- 3.8 Symmetry at the Foundation of Physics -- 70 -- 3.9 Symmetry at the Foundation of Quantum Theory -- 71 -- 3.9.1 Association of a Hilbert Space with a Physical System -- 71 -- 3.9.2 Correspondence of Observables to Hermitian Operators -- 73 -- 3.9.3 Complete Set of Compatible Observables -- 74 -- 3.9.4 Heisenberg Commutation Relations -- 75 -- 3.9.5 Operators for Canonical Variables -- 75 -- 3.9.6 A Measurement Result Is an Eigenvalue -- 75 -- 3.9.7 Expectation Values and Probabilities -- 76 -- 3.9.8 The Hamiltonian Operator -- 76 -- 3.9.9 Planck's Constant as a Parameter -- 77 -- 3.9.10 The Correspondence Principle -- 77 -- 4 The Symmetry Principle -- 81 -- 4.1 Causal Relation -- 81 -- 4.2 Equivalence Relation, Equivalence Class -- 86 -- 4.3 The Equivalence Principle -- 89 -- 4.4 The Symmetry Principle -- 97 -- 4.5 Cause and Effect in Quantum Systems -- 102 -- 5 Application of Symmetry -- 107 -- 5.1 Minimalistic Use of the Symmetry Principle -- 107 -- 5.2 Maximalistic Use of the Symmetry Principle -- 125 -- 6 Approximate Symmetry, Spontaneous Symmetry Breaking -- 131 -- 6.1 Approximate Symmetry -- 131 -- 6.2 Spontaneous Symmetry Breaking -- 135 -- 7 Cosmic Considerations -- 141 -- 7.1 Symmetry of the Laws of Nature -- 141 -- 7.2 Symmetry of the Universe -- 144 -- 7.3 No Cosmic Symmetry Breaking or Restoration -- 147 -- 7.4 The Quantum Era and The Beginning -- 155 -- 8 The Mathematics of Symmetry: Group Theory -- 161 -- 8.1 Group -- 161 -- 8.2 Mapping -- 176 -- 8.3 Isomorphism -- 180 -- 8.4 Homomorphism -- 186 -- 8.5 Subgroup -- 192 -- 9 Group Theory Continued -- 195 -- 9.1 Conjugacy, Invariant Subgroup, Kernel -- 195 -- 9.2 Coset Decomposition -- 203 -- 9.3 Factor Group -- 207 -- 9.4 Anatomy of Homomorphism -- 209 -- 9.5 Generator -- 215 -- 9.6 Direct Product -- 217 -- 9.7 Permutation, Symmetric Group -- 220 -- 9.8 Cayley's Theorem -- 224 -- 10 The Formalism of Symmetry -- 227 -- 10.1 System, State -- 227 -- 10.2 Transformation, Transformation Group -- 229 -- 10.3 Transformations in Space, Time, and Space-Time -- 236 -- 10.4 State Equivalence -- 240 -- 10.5 Symmetry Transformation, Symmetry Group -- 243 -- 10.6 Approximate Symmetry Transformation -- 251 -- 10.7 Quantification of Symmetry -- 253 -- 10.8 Quantum Systems -- 255 -- 11 Symmetry in Processes -- 261 -- 11.1 Symmetry of the Laws of Nature -- 261 -- 11.2 Symmetry of Initial and Final States, the General Symmetry Evolution Principle -- 270 -- 11.3 The Special Symmetry Evolution Principle and Entropy -- 274 -- 12 Summary of Principles -- 283 -- 12.1 Symmetry and Asymmetry -- 283 -- 12.2 Symmetry Implies Asymmetry -- 283 -- 12.3 No Exact Symmetry of the Universe -- 284 -- 12.4 Cosmological Implications -- 285 -- 12.5 The Equivalence Principle -- 285 -- 12.6 The Symmetry Principle -- 285 -- 12.7 The Equivalence Principle for Processes -- 286 -- 12.8 The Symmetry Principle for Processes -- 286 -- 12.9 The General Symmetry Evolution Principle -- 286 -- 12.10 The Special Symmetry Evolution Principle-- 286. Ernest Rutherford (New Zealand–British physicist, 1871–1937), the 1908 Nobel Laureate who discovered the existence of atomic nuclei, is famously quoted as having said: “Physics is the only real science. All the rest is butter?y collecting.” Or something to that e?ect. I like to include this quote in my introductory remarks at the ?rst class meetings of the physics courses I teach. I have seen that there are those whointerpret this as a put-down of amateurs (butter?y collectors) in science. However, my own interp- tation of Rutherford’s statement is that he is claiming that, except for physics, all of the rest of science is involved merely in collecting facts and classifying them (butter?y collecting). It is physics, uniqueamong the sciences, that is attempting to ?nd explanations for the classi?ed data. The periodic table of the chemical elements, originally proposed by DmitriIvanovichMendeleev(Russianchemist,1834–1907), presentsan example of this. Chemists toiled to discover the chemical elements and their properties and then classi?ed the elements in the scheme that is expressed by the periodic table. Here was the chemists’ butter?y collecting. It took physicists to explaintheperiodictablebymeansof quantum theory. Front Matter....Pages I-XIV The Concept of Symmetry....Pages 1-15 Science Is Founded on Symmetry....Pages 17-38 Symmetry in Physics....Pages 39-79 The Symmetry Principle....Pages 81-105 Application of Symmetry....Pages 107-130 Approximate Symmetry, Spontaneous Symmetry Breaking....Pages 131-140 Cosmic Considerations....Pages 141-160 The Mathematics of Symmetry: Group Theory....Pages 161-194 Group Theory Continued....Pages 195-226 The Formalism of Symmetry....Pages 227-259 Symmetry in Processes....Pages 261-281 Summary of Principles....Pages 283-287 Back Matter....Pages 289-305 When we use science to describe and understand the world around us, we are in essence grasping nature through symmetry. 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. Written by a renowned expert, this book will convince all interested readers of the importance of symmetry in science. Modern theoretical physics suggests that symmetry is a, if not the, foundational principle of nature. Emphasizing the concepts, this book introduces symmetry and its applications. It is shown that the universe cannot possess exact symmetry, which is a principle of fundamental significance. Joe Rosen. Includes Bibliographical References (p. [289]-295) And Index.