2.3. Gauge Symmetries

2.3.1. U(1) Global Gauge Invariance

Assuming the Lagrangian is invariant under U(1) global gauge transformation

\[\psi(x) \to e^{i\alpha} \psi(x),\]

we obtain the Noether current

\[j^\mu = - e \bar \psi \gamma^\mu \psi,\]

which is conserved

\[\partial_\mu j^\mu = 0.\]

Derivation of Noether Current

QED

2.3.2. U(1) Local Gauge Invariance

Introduce the U(1) local gauge transformation

\[\psi(x) \to e^{i\alpha(x) } \psi(x)\]

to the Lagrangian, we notice that the Lagragian is generally not invariant. However, if we require it to be invariant, a new field \(A_\mu\) should be introduced, so that

\[A_\mu \to A_\mu + \frac{1}{2} \partial_\mu \alpha(x).\]

The way this new field comes into the Lagriangian is to replace all the derivatives \(\parital_\mu\) with \(\mathrm D_\mu\),

\[\mathrm D_\mu = \partial_\mu - i e A_\mu.\]

What about the kinetic term for this new field? It is constructed from the field strength tensor

\[F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,\]

so that the kinetic term is

\[- \frac{1}{4} F_{\mu\nu}F^{\mu\nu}.\]

Finally we reach the new (QED) Lagrangian that is invariant under U(1) gauge transformation

\[\mathcal L = \bar \psi ( i\gamma^\nu \partial_\mu - m )\psi + e\bar \psi \gamma^\mu \partial_\mu \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} .\]

2.3.3. Non-Abelian

Introduce the non-abelian local gauge transformation

\[q(x) \to e^{i\alpha_a (x) T_a} q(x).\]

The Lagrangian that is invariant under this transformation is

\[\mathcal = \bar q (i\gamma^\mu \partial_\mu - m) q - g(\bar q \gamma^\mu T_a q) G_{\mu}^a - \frac{1}{4} G_{\mu\nu}^a G^{\mu\nu}_a.\]