Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 10Continuous Groups and Representations

In ordinary parlance, the word symmetry is usually associated with beautiful. In fact, recent studies have shown that people with more symmetric bodies are more beautiful to humans. This is particularly interesting since the left–right symmetry of the human body is almost undetectable to the naked eye. In architecture and art, symmetry is used to build beautiful structures and patterns by distributing relatively simple building blocks according to a rule. Most of the symmetries around us are due to rotations and reflections. Symmetry in science usually means invariance of a given system under some operation. As in crystals and molecules, symmetries can be discrete, where applications of certain amounts of rotation, translation, or reflection produce the same structure. After the discovery of the Lagrangian formulation of mechanics, it became clear that the conservation laws are also due to the symmetries of the physical system. For example, conservation of angular momentum follows from the symmetry of a given system with respect to rotations, whereas the conservation of linear momentum follows from symmetry with respect to translations. In these symmetries, a system is carried from one identical state to another continuously; hence they are called continuous or Lie symmetries. With the discovery of quantum mechanics, the relation between conservation laws and symmetries has even become more important. Conservation of isospin and ...