Symmetries

The rules that limit the possibility of an initial state transforming into another state in a quantum process (collision or decay) are called conservation laws and are expressed in terms of the quantum numbers of those states. We shall not deal with the invariance under continuum transformations in space-time and the corresponding conservation of energy-momentum and of angular momentum, which are known to the reader. We shall consider the following types of quantum numbers.

Discrete additive If a quantum number is additive, the total quantum number of a system is the sum of the quantum numbers of its components. The ‘charges’ of all fundamental interactions fall into this category, the electric charge, the colour charges and the weak charges. They are conserved absolutely, as far as we know. The conservation of each of them corresponds to the invariance of the Lagrangian of that interaction under the transformations of a unitary group. The group is called the ‘gauge group’ and the invariance of the Lagrangian is called ‘gauge invariance’. The gauge group of the electromagnetic interaction is U(1), that of the strong interaction is SU(3) and that of the electroweak interaction is SU(2) ⊗ U(1). Other quantum numbers in this category are the quark flavours, the baryon number, the lepton flavours and the lepton numbers. They do not correspond to a gauge symmetry and are not necessarily conserved (actually, quark and lepton flavours are not).