Symmetry rules: how science and nature are founded on symmetry

1. The Concept of Symmetry

1.1. The Essence of Symmetry

1.2. Symmetry Implies Asymmetry

1.3. Analogy and Classification Are Symmetry

1.4. Summary

2. Science Is Founded on Symmetry

2.1. Science

2.2. Reduction Is Symmetry

2.2.1. Reduction to Observer and Observed

2.2.2. Reduction to Quasi-Isolated System and Environment

2.2.3. Reduction to Initial State and Evolution

2.3. Reproducibility Is Symmetry

2.4. Predictability Is Symmetry

2.5. Analogy in Science

2.6. Symmetry at the Foundation of Science

2.7. Summary

3. Symmetry in Physics

3.1. Symmetry of Evolution

3.2. Symmetry of States

3.3. Reference Frame

3.4. Global, Inertial, and Local Reference Frames

3.5. Gauge Transformation

3.6. Gauge Symmetry

3.7. Symmetry and Conservation

3.7.1. Conservation of Energy

3.7.2. Conservation of Linear Momentum

3.7.3. Conservation of Angular Momentum

3.8. Symmetry at the Foundation of Physics

3.9. Symmetry at the Foundation of Quantum Theory

3.9.1. Association of a Hilbert Space with a Physical System

3.9.2. Correspondence of Observables to Hermitian Operators

3.9.3. Complete Set of Compatible Observables

3.9.4. Heisenberg Commutation Relations

3.9.5. Operators for Canonical Variables

3.9.6. A Measurement Result Is an Eigenvalue

3.9.7. Expectation Values and Probabilities

3.9.8. The Hamiltonian Operator

3.9.9. Planck's Constant as a Parameter

3.9.10. The Correspondence Principle

3.10. Summary

4. The Symmetry Principle

4.1. Causal Relation

4.2. Equivalence Relation, Equivalence Class

4.3. The Equivalence Principle

4.4. The Symmetry Principle

4.5. Cause and Effect in Quantum Systems

4.6. Summary

5. Application of Symmetry

5.1. Minimalistic Use of the Symmetry Principle

5.2. Maximalistic Use of the Symmetry Principle

5.3. Summary

6. Approximate Symmetry, Spontaneous Symmetry Breaking

6.1. Approximate Symmetry

6.2. Spontaneous Symmetry Breaking

6.3. Summary

7. Cosmic Considerations

7.1. Symmetry of the Laws of Nature

7.2. Symmetry of the Universe

7.3. No Cosmic Symmetry Breaking or Restoration

7.4. The Quantum Era and The Beginning

7.5. Summary

8. The Mathematics of Symmetry: Group Theory

8.1. Group

8.2. Mapping

8.3. Isomorphism

8.4. Homomorphism

8.5. Subgroup

8.6. Summary

9. Group Theory Continued

9.1. Conjugacy, Invariant Subgroup, Kernel

9.2. Coset Decomposition

9.3. Factor Group

9.4. Anatomy of Homomorphism

9.5. Generator

9.6. Direct Product

9.7. Permutation, Symmetric Group

9.8. Cayley's Theorem

9.9. Summary

10. The Formalism of Symmetry

10.1. System, State

10.2. Transformation, Transformation Group

10.3. Transformations in Space, Time, and Space-Time

10.4. State Equivalence

10.5. Symmetry Transformation, Symmetry Group

10.6. Approximate Symmetry Transformation

10.7. Quantification of Symmetry

10.8. Quantum Systems

10.9. Summary

11. Symmetry in Processes

11.1. Symmetry of the Laws of Nature

11.2. Symmetry of Initial and Final States, the General Symmetry Evolution Principle

11.3. The Special Symmetry Evolution Principle and Entropy

11.4. Summary

12. Summary of Principles

12.1. Symmetry and Asymmetry

12.2. Symmetry Implies Asymmetry

12.3. No Exact Symmetry of the Universe

12.4. Cosmological Implications

12.5. The Equivalence Principle

12.6. The Symmetry Principle

12.7. The Equivalence Principle for Processes

12.8. The Symmetry Principle for Processes

12.9. The General Symmetry Evolution Principle

12.10. The Special Symmetry Evolution Principle

References

Further Reading

Index