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A Manipulation with the Dirac equation

  1. Mar 23, 2017 #1
    I know that the Dirac equation is ##i\gamma^{\mu}\partial_{\mu}\psi=m\psi##.

    How do I use this to show that ##(\partial_{\mu}\bar{\psi})\gamma^{\mu}=im\bar{\psi}##?
     
  2. jcsd
  3. Mar 23, 2017 #2

    vanhees71

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    First of all you have (in the usual standard representations of the Dirac matrices)
    $$\gamma^{\mu \dagger}=\gamma^0 \gamma^{\mu} \gamma^0,$$
    the "pseudo-hermitecity relation" and the definition
    $$\overline{\psi}=\psi^{\dagger} \gamma^0.$$
    So now take the Dirac equation and first apply Hermitean conjugation:
    $$-\mathrm{i} \partial_{\mu} \psi^{\dagger} \gamma^{\mu \dagger}=m\psi^{\dagger}.$$
    Now use ##(\gamma^0)^2=1## to first write on the left-hand side
    $$-\mathrm{i} \partial_{\mu} \overline{\psi} \gamma^0 \gamma^{\mu \dagger} = m \psi^{\dagger}.$$
    Finally multiply this equation with ##\gamma^0## and use the pseudo-hermitecity relation of the Dirac matrices to finally get the claimed equation:
    $$-\mathrm{i} \partial_{\mu} \overline{\psi} \gamma^{\mu}=m \overline{\psi}.$$
     
  4. Mar 23, 2017 #3

    dextercioby

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    How do you get from Psi to Psi-bar in the absence of a symmetric Lagrangian? @vanhees71 this smells like a homework problem (incorrectly placed outside the HW section), so I wouldn't throw in the solution.
     
  5. Mar 23, 2017 #4

    vanhees71

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    What has the Lagrangian to do with basic definitions of the ##\gamma## matrices and bispinors?
     
  6. Mar 23, 2017 #5

    dextercioby

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    Well, the Dirac equation in line 1 is one of the two Euler-Lagrange equations for the symmetrized Lagrangian density. The other Euler-Lagrange equation is the one whose derivation he sought. Your solution is direct, I was trying to lead him there.
     
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