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Charge conjugation in Dirac equation

  1. Nov 5, 2012 #1
    I need to know the mathematical argument that how the relation is true $(C^{-1})^T\gamma ^ \mu C^T = - \gamma ^{\mu T} $ .
    Where $C$ is defined by $U=C \gamma^0$ ; $U$= non singular matrix and $T$= transposition.


    I need to know the significance of these equation in charge conjuration .
     
  2. jcsd
  3. Nov 6, 2012 #2

    samalkhaiat

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    Start with the Dirac equation
    [tex]( i \gamma^{\mu}\partial_{\mu} + e \gamma^{\mu}A_{\mu} - m) \psi = 0[/tex]
    Now take the complex conjugate of that and multiply from the left by some non-singular matrix [itex]\mathcal{C}[/itex], you then can write
    [tex][(i \partial_{\mu} - e A_{\mu})\ \mathcal{C}\ (\gamma^{\mu})^{*} \mathcal{C}^{-1} + m ] \ \mathcal{C}\psi = 0[/tex]
    Thus, if
    [tex]\mathcal{C}\ (\gamma^{\mu})^{*} \mathcal{C}^{-1} = - \gamma^{\mu},[/tex]
    then the field
    [tex]\psi_{c}\equiv \mathcal{C}\psi[/tex]
    describes another Dirac particle with opposite charge.
    Now, if you write
    [tex]\mathcal{C} = C \gamma^{0}[/tex]
    then your relation follows in the representation where
    [tex]\gamma^{0} = ( \gamma^{0})^{T} = ( \gamma^{0})^{-1}.[/tex]

    Sam
     
  4. Nov 7, 2012 #3
    One can see sakurai 'advanced quantum mechanics' for an elegant derivation which describes the relationship between charge conjugated wave function and original one.
     
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