QFT — Michael Luke Lectures · Problem IX.1

Spin 1/2 Fields — Michael Luke

Physics — QFT · spinor-bilinears-four-vector

Problem

Show that the three components of

u^\dagger_+ \vec{\sigma} u_+ \equiv (u^\dagger_+ \sigma_x u_+, u^\dagger_+ \sigma_y u_+, u^\dagger_+ \sigma_z u_+)

form a three-vector under rotations. Then show that the four components of

V^\mu = (u^\dagger_+ u_+,\, u^\dagger_+ \vec{\sigma} u_+)

form a four-vector under proper Lorentz transformations.