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I Symmetrized Lagrangian (second quantization)

  1. Feb 15, 2017 #1
    Hello,

    I need some help to find the correct symmetrized Lagrangian for the field operators. After some work I guess that

    $$\mathcal{L} = i[\overline{\psi}_a,({\partial_\mu}\gamma^\mu \psi)^a] -m[\overline{\psi}_a,\psi^a ]$$

    should be the correct Lagrangian but I'm not sure with this.

    I'm also intrested in the question of reordinger this Lagrangian in such a way that all [itex]\overline{\psi}[/itex] are on the left and all [itex]\psi[/itex] are on the right side. My problem: I don't know how to deal with products like [itex](\partial_\mu \gamma^\mu \psi) \overline{\psi}[/itex]

    Thanks for your help.
    Neutrino
     
  2. jcsd
  3. Feb 16, 2017 #2

    haushofer

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    How do you define e.g. the second term?
     
  4. Feb 16, 2017 #3
    It should be [itex] [\overline{\psi}_a, \psi^a] := \sum_a \overline{\psi}_a \cdot \psi^a - \psi^a \cdot \overline{\psi}_a[/itex]..
    To ensure that the Lagrangian is hermitian we may add an aditional four divergence.
     
  5. Feb 18, 2017 #4

    haushofer

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    I'm sorry, I only understand the first term where the barred psi comes first.
     
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