(adsbygoogle = window.adsbygoogle || []).push({}); What is the meaning of the local gauge transformation exactly??

These days I'm studying. [D.J. Griffiths, Introduction to Elementary Particles 2nd Edition, Chapter 10. Gauge Theories] Here the Section 3. Local Gauge Invariance, the author gives the Dirac Lagrangian, [itex]\mathcal{L}=i \hbar c \bar{\psi} \gamma^{\mu} \partial_{\mu} \psi - mc^{2} \bar{\psi} \psi[/itex] and show this is invariant under the global phase(gauge) transformation which is defined by [itex]\psi \rightarrow e^{i \theta} \psi[/itex]. As auther mentioned, I already awared about this from quantum mechanics. After that, the auther introduce a local gauge transformation. At first I just passed this. I just regarded the auther asking to me just like "then, what about [itex]\theta[/itex] is not a constant??". Then, he abruptly put [itex]\lambda (x) = -\frac{\hbar c}{q} \theta(x) [/itex].

Now I confused. What the hell this 'charge' q come from? How can someone arrive that conclusion that he should introduce 'charge' here? So I visited my professor and he answered just "this is the 'photon'."

I confused AGAIN. What the?? Where is the right position should I begin my study step by step?

I thought this [itex]\lambda(x)[/itex] is just a gauge function that makes the Lagrangian invariant under the given local gauge transformation. In fact, after few lines the author indeed mention about a new field that transform under the local gauge transformation as [itex]A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \lambda [/itex] But he said this contents talking about a 'particle'. How can I understand this?

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# What is the meaning of the local gauge transformation exactly?

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