QFT — Michael Luke Lectures · Problem IX.2

Spin 1/2 Fields — Michael Luke

Physics — QFT · lorentz-boost-spinor-representation

Problem

Verify by direct computation that a boost in the

\hat{e}

direction is given by

Q = e^{\vec{\sigma}\cdot\hat{e}\,\phi/2},

where

\cosh\phi = \gamma

and

\sinh\phi = -\sqrt{\gamma^2-1}

parameterize the boost, using the matrix representation

X = \begin{pmatrix} t+z & x-iy \\ x+iy & t-z \end{pmatrix}

and the transformation

X' = QXQ^\dagger