Parenthesis on Global and Local Symmetries

Starting from our consideration of symmetry as an aesthetic cultural necessity, we are going to try now to think of symmetry as a source of the fundamental interactions. Let us remember that the existence of a symmetry implies the existence of a conservation law. In fact, given a Lagrangian ${\cal L}(\psi,\partial_{\mu}\psi)$ which verifies

$$\frac{\partial{\cal L}}{\partial\psi} - \partial_{\mu} \frac {\partial {\cal L}}{\partial (\partial_{\mu}\psi)} = 0$$ (145)

when a change $\delta\psi$ due to variations of continuous parameters is performed,it suffer a variation:

$$\delta {\cal L} = \partial_{\mu} \left[\frac{\partial {\cal L}}{\partial (\partial_{\mu}\psi)} \delta \psi\right]$$ (146)

now, if $L$ is symmetric (invariant) with respect to the change $\delta\psi$, that is

$$\delta {\cal L} = 0$$ (147)

there results the Noether theorem: a current is conserved

$$\partial_{\mu}J^{\mu} = \partial_{\mu}\left[\frac{\partial {\cal L}}{\partial (\partial_{\mu}\psi)} \delta\psi\right] = 0$$ (148)

For example, the invariance of $L$ with respect to traslations implies the conservation of the momentum and of the energy. The invariance in front of rotations gives place to the conservation of the angular momentum, etc. We are then in the presence of continuous groups of symmetry. In the first case we have a four parameter group: $\delta x^{\alpha}$ , the displacements. In the case of the rotations is a three parameter group, the angles, etc. In the general case we will be leading with a continuous group of symmetries of the Lagrangian which will depend on $n$ parameters $\alpha_{k}\;(k=1,2,..., n)$. The general case can go beyond the space-time symmetries of the previous examples and can refer to symmetries linked to the formalism own structure: the internal symmetries.