Broken symmetry revisited

In general, a physical system consists of a finite or infinite number of degrees of freedom which may or may not interact. The dynamics is prescribed by a set of evolution equations which follow from varying the action with respect to the different degrees of freedom. A symmetry then corresponds to a group of transformations on the space time coordinates and/or the degrees of freedom that leave an action and therefore also the evolution equations invariant. External symmetries have to do with invariances (e.g. Lorentz invariance) under transformations on the space time coordinates. Symmetries not related to transformations of space time coordinates are called internal symmetries. We also discriminate between global symmetries and local symmetries. A global or rigid symmetry transformation is the same throughout space time and usually leads to a conserved quantity. Turning a global symmetry into a local symmetry, i.e. allowing the symmetry transformations to vary continuously from one point in space time to another, requires the introduction of additional gauge degrees of freedom mediating a force. It is this so-called gauge principle that has eventually led to the extremely successful standard model of the strong and electro-weak interactions between the elementary particles based on the local gauge group SU(3)× SU(2)× U(1).