The electromagnetic current vanishes identically in the Majorana case

[Urvabara]

The electromagnetic current vanishes identically in the Majorana case: http://users.jyu.fi/~hetahein/tiede/virta.pdf" .

In the case someone cannot open the pdf, here is the formula:
$$\overline{\psi}\gamma^{\mu}\psi = \overline{\psi^{C}}\gamma^{\mu}\psi^{C} = <br /> -\psi^{\text{T}}C^{\dagger}\gamma^{\mu}C\overline{\psi}^{\text{T}} \overset{?}{=} <br /> \overline{\psi}C\gamma^{\mu\text{T}}C^{\dagger}\psi = -\overline{\psi}\gamma^{\mu}\psi = 0.$$

Does anyone know how is that equality under the ?-sign proved? I just don't know how...

Thanks!

 

[akhmeteli]

The electromagnetic current vanishes identically in the Majorana case: http://users.jyu.fi/~hetahein/tiede/virta.pdf" .

In the case someone cannot open the pdf, here is the formula:
$$\overline{\psi}\gamma^{\mu}\psi = \overline{\psi^{C}}\gamma^{\mu}\psi^{C} = <br /> -\psi^{\text{T}}C^{\dagger}\gamma^{\mu}C\overline{\psi}^{\text{T}} \overset{?}{=} <br /> \overline{\psi}C\gamma^{\mu\text{T}}C^{\dagger}\psi = -\overline{\psi}\gamma^{\mu}\psi = 0.$$

Does anyone know how is that equality under the ?-sign proved? I just don't know how...

Thanks!

First off, the electromagnetic current vanishes identically in the Majorana case only if you assume that the components of \psi are anticommuting. If they commute, the transition in question can be performed as follows: the scalar equals its own transposition, C and C-cross change sign under transposition. However, the resulting sign will differ from that in your formula. You obtain the third sign change when you drag anticommuting components of psi through each other.