PhD thesis - CP3-Origins

Introduction to Elementary Particle Physics 7In order to restore the invariance, we replace the partial derivative with the covariantderivativeD µ = ∂ µ + ieA µ , (2.4)where the gauge field A µ transforms asA µ → A µ − ∂ µ α(x). (2.5)Local phase transformations are called gauge transformations and the invariance underthese transformations is known as gauge invariance. The procedure is called gauging andit fixes the form of interactions. The local, and also global, U(1) symmetry is a continuoussymmetry and the corresponding conserved quantity, in this case, is the electric charge.This can be generalized to non-abelian compact Lie groups. It is important to note therole of internal and space time symmetries.The internal and space-time symmetries are described in terms of Lie groups. There isno non trivial way to combine these symmetries according to Coleman-Mandula theorem[4]. If we consider a more general algebraic structure than a Lie group, namely a gradedLie group, this can be omitted. The Haag-Lopuszánski-Sohnius theorem [5] states thatthe most general graded Lie group of symmetries of a local field theory is the N-extendedsuper Poincaré group. This allows non trivial mixing between internal and space-timesymmetries leading to symmetry relating bosons and fermions.Up to this point, we have considered only exact symmetries. Later, it will be importantto differentiate what actually is symmetric, Lagrangian or the solutions, and at whatscale the symmetry manifests itself and, if broken, how it is broken.Explicit breaking can occur via non-invariant terms in the Lagrangian. This does notmean that the symmetry cannot be used to draw conclusions if the breaking is small.Some of the Lagrangian’s classical symmetries can be spoiled by the quantum effects.This is called an anomalous breaking and the term driving the breaking is called ananomaly. It is important that in the end all the anomalies are cancelled. Otherwisethe renormalizability of the theory is destroyed. If the Lagrangian is more symmetricthan the quantum states, the symmetry is said to be spontaneously broken. Especiallyinteresting is if the vacuum is not invariant under the same symmetries as the Lagrangian.