Local Phase Invariance

We are trying to impose the invariance to the theory even when $\alpha$ in (152) (and $\theta_{a}$ in (164)) is a function of the space-time position $x$.

This is the idea leading to the gauge field theories. As a first motivation for doing this step forward, the Yang-Mills words can be used:

``The concept of field and the concept of local interactions imply a spreading of information to neighbouring points and eliminate the action at a distance. Besides, in the Lagrangian there are products of fields (and of their derivatives in the same point. It is then understood that the global phase invariance - the same in every $x$ point - seems to contradict the generalized idea of locality and it is worth investigating the invariance in front of different rotations in different space-time points".

In other words, it is worth to investigate what happens when one allows $\alpha =\alpha(x)$ ( $\theta_{a} = \theta_{a}(x)$) and consequently, transformations of the form

$$\phi(x) \rightarrow \phi'(x) = e^{i \alpha(x)} \phi(x)$$ (163)

$$\psi(x) \rightarrow \psi'(x) = e^{i \theta_{a}(x) t_{a}} \psi(x)$$ (164)

requiring then the presence of a local symmetry in the theory. In this way one goes from the study of gauge transformations of the first class (global) to the study of second class (local) ones.