Quantum mechanics and symmetries

Abstract

Symmetries have always played an important role in physics. With quantum mechanics, however, the interplay between physics and symmetries has reached a new dimension. The very structure of quantum mechanics invites the application of group theoretical methods to an extent that physicists would have been led to invent various concepts of group theory, such as Lie groups, by quantum mechanics had they not been known before. Symmetries are also a direct mediator between experimental facts and the theoretical structure of a theory. This is the case because there is a direct connection between symmetries and conservation laws. Space-time symmetries are an obvious example. Conservation of energy, momentum and angular momentum are linked to invariance under time translation, space translation and rotation in space. It was in atomic physics that space-time symmetries became significant, but they are as important in nuclear physics and in all the physics discovered after that. In nuclear physics, however, a new concept of symmetries, symmetries in an internal space, was discovered with the introduction of isospin. All our socalled fundamental models describing what we know about strong-weak and electromagnetic interactions are built on symmetries in space-time and internal spaces. These symmetries are not only used to extract information from a theory, they are also used to construct these theories, and this for good reasons. It turned out that only theories possessing such symmetries make sense as quantum field theories. Thus symmetries are not only a good tool to deal with quantum field theoretical models, they are necessary to define such models. Following this line of thought, a new type of symmetry, the socalled supersymmetry, proved to be extremely successfull. Quantum theory seems to have a very deep relation to supersymmetry. Thus it is not surprising that the most promising fundamental models of physics are based on supersymmetry, even when they go beyond a local quantum field theory, as string theory does.