Autumn · project 11 of 20

Symmetries of the Dirac equation

Topic: Symmetries and Noether's theorem

  1. Using Noether’s theorem, obtain the conserved current for \[\mathcal{L} =i\bar{\psi}\cancel{\partial}\psi - m\bar{\psi}\psi\] under the global gauge transformations \[\psi \rightarrow e^{i \alpha} \psi, \,\, \overline{\psi} \rightarrow e^{-i\alpha} \overline{\psi}.\]

  2. Using Noether’s theorem, obtain the conserved current for \(\mathcal{L} = i\bar{\psi}\cancel{\partial}\psi\) under the chiral transformation \[\psi \rightarrow e^{i \alpha \gamma^5} \psi.\] Find how the field \(\bar{\psi}\) transforms.

  3. Prove that the action of the massless Dirac field is invariant under dilatations: \[x\to x'=e^{-\rho}x, \quad \psi(x)\to \psi'(x') = e^{3\rho/2}\psi(x).\] Find the corresponding Noether current and charge.

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