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Problems QFT 2

From IFPA wiki
  1. In the case of two scalar hermitian fields [math]\Phi_a(x),\ a=1,2[/math] with the same mass, show that the lagrangian density [math]{\mathcal L}=(1/2)\sum_a(\partial_\mu \Phi_a)(\partial^\mu\Phi_a)-m^2\Phi_a^2[/math] has a continuous symmetry. Calculate the associated charge. Compare with the case of a complex scalar field.
  2. In the case of two complex scalar fields [math]\Phi_a(x),\ a=1,2[/math] with the same mass, show that the lagrangian density [math]{\mathcal L}=\sum_a(\partial_\mu \Phi_a^*)(\partial^\mu\Phi_a)-m^2|\Phi_a|^2[/math] has several continuous symmetries. Calculatethe corresponding charges.
  3. Establishthe transformation properties of [math]\bar \psi\gamma^\mu\psi[/math] and [math]\bar \psi\gamma_5\gamma^\mu\psi[/math] under the parity transformation.
  4. Using the Dirac algebra, simplify te following: [math]\gamma_\mu\gamma^\mu[/math],[math]\gamma_\mu\gamma_\nu\gamma^\mu[/math], [math]\gamma_\mu\gamma_\nu\gamma_\tau\gamma^\mu[/math], and [math]\gamma_\mu\gamma_\nu\gamma_\tau\gamma_\sigma\gamma^\mu[/math].

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