Rotating Dirac particle current

In summary, there is a step in the book "Ryder" on QFT that the speaker is struggling with. It involves the rotation of the spatial part of the Dirac four current, represented by \bar{\psi} \gamma \psi. The crucial step is proving that \frac{1}{4}(\vec{\sigma} \cdot \vec{\theta}) \vec{\sigma}(\vec{\sigma} \cdot \vec{\theta}) = -\vec{\theta} \times \vec{ \sigma}, with \vec{\theta} being the (infinitesimal) rotation and \vec{\sigma} being the vector consisting of the three Pauli matrices. The speaker has attempted to use tensor notation and
  • #1
Wiemster
72
0
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:

[tex]\bar{\psi} \gamma \psi [/tex]

The crucial step here is

[tex]\frac{1}{4}(\vec{\sigma} \cdot \vec{\theta}) \vec{\sigma}(\vec{\sigma} \cdot \vec{\theta}) = -\vec{\theta} \times \vec{ \sigma} [/tex]

With [itex]\vec{\theta}[/tex] the (infinitesimal) rotation and [itex]\vec{\sigma}[/tex] the vector consiting of the three Pauli matrices.

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?
 
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  • #2
What are you trying to prove, invariance of the four current under rotation? This is not that difficult.
 
  • #3
Well, thanks :wink: 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!
 

Related to Rotating Dirac particle current

1. What is a Rotating Dirac Particle Current?

A Rotating Dirac Particle Current is a phenomenon in which the spin direction of a Dirac particle changes as it moves through space. This rotation can be clockwise or counterclockwise and is caused by the intrinsic spin of the particle.

2. How is the direction of rotation determined?

The direction of rotation is determined by the spin of the particle and the direction of its movement. If the spin is parallel to the direction of movement, the particle will rotate clockwise, while if the spin is antiparallel, it will rotate counterclockwise.

3. What is the significance of a Rotating Dirac Particle Current?

The significance of a Rotating Dirac Particle Current lies in its contribution to understanding the behavior of particles at the quantum level. It also has practical applications in fields such as quantum computing and spintronics.

4. How is a Rotating Dirac Particle Current observed?

A Rotating Dirac Particle Current can be observed through experiments such as the Stern-Gerlach experiment, which uses a magnetic field to deflect particles and reveal their spin direction. It can also be observed indirectly through the effects it has on other particles.

5. Can a Rotating Dirac Particle Current be controlled?

Currently, it is not possible to control the direction of rotation of a Rotating Dirac Particle Current. However, ongoing research in the field of quantum mechanics may lead to new discoveries and methods for controlling this phenomenon in the future.

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