Charge conjugation in Dirac equation

[forhad_jnu]

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 .

 

[samalkhaiat]

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

Sam

 

[andrien]

One can see sakurai 'advanced quantum mechanics' for an elegant derivation which describes the relationship between charge conjugated wave function and original one.