Gauge theory

meaning of the gague invariance

• gauge = measure
• gauge invariance = measurement에 있어서의 invariance를 말함
• Lagrangian should be gauge invariant.

 

 

gauge field

• a gauge field is defined as a four-vector field with the freedom of gauge transformation, and it corresponds to massless particlas of spin one
  

• one example is the electromagnetic field
  

 

 

Gauge invariance of the QED Lagrangian

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

Now we have a Lagrangian with interaction terms.

• local phase transformation of fields
  \(\psi(x) \to e^{i\alpha(x)}\psi(x)\)
  
• gauge transformation of electromagnetic field
  \(A_{\mu}(x) \to A_{\mu}(x)+\frac{1}{e}\partial_{\mu}\alpha(x)}\)
  
• Look at the QED page
  

 

 

gauge field tensor

• electromagnetic field tensor  \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!\)
  
• general gauge fields tensor  \(G_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}-gw^{abc}W_{\mu}^{b}W_{\nu}^{c}\)