[NOTES/CM-03007] Symmetries --- Numerous Applications to Different Areas.

1. In case of a system with continuous symmetry, the Noether generators form a basis of Lie algebra of the symmetry group. Group theoretic methods become available for study and many model independent predictions.
2. {\it Symmetry and degeneracy} A symmetry of Lagrangian, continuous as well as discrete ones, lead to degeneracy of quantum energy levels. This can be found without solving the problem by making use of group theoretic methods.
3. {\it Broken symmetry and removal of degeneracy}
4. {\it Broken symmetries and selection rules}\\ When a symmetry is explicitly broken by the interaction terms in the Lagrangian, the corresponding Noether generator is not a constant of motion. The predictions of theory will then satisfy a set of well defined selection rules. An example of this type is Zeeman effect of hydrogen atom. The presence of magnetic field leads to selection rules for observed transitions.
5. In real applications observed selection rules are used to search for symmetries which are in turn used to obtain information about the interactions present. An extremely successful outcome of this process has been the Weinberg Salam Glashow model of electro-weak interactions.

A discussion of these topics is outside scope of this article. For those who are interested, an internet search is suggested for further references.