Rotating Dirac particle current

[Wiemster]

There is a step that bothers me in my book (Ryder) on QFT and I can't seem to figure it out. It concerns the (spatial) rotation of the spatial part of the Dirac four current:

$$\bar{\psi} \gamma \psi$$

The crucial step here is

$$\frac{1}{4}(\vec{\sigma} \cdot \vec{\theta}) \vec{\sigma}(\vec{\sigma} \cdot \vec{\theta}) = -\vec{\theta} \times \vec{ \sigma}$$

With [itex]\vec{\theta}[/tex] the (infinitesimal) rotation and [itex]\vec{\sigma}[/tex] the vector consiting of the three Pauli matrices.<br /> <br /> I tried writing it in tensor notation and using the commutation reltaions as in the supplied tip, but I can't figure it out... Can anybody show me this?[/itex][/itex]

 

[Epicurus]

What are you trying to prove, invariance of the four current under rotation? This is not that difficult.

 

[Wiemster]

Well, thanks Actually, that the spatial part of the Dirac four-current bahves as a vector under rotations. BUt I finally figured it out myself at last!