A. Zee, Quantum Field Theory in a Nutshell · Problem II.3.3

Field Theory Redux

Physics — QFT

Problem

Show that

e^{-i\phi K}\gamma^0 e^{i\phi K} = \begin{pmatrix} 0 & e^{-\phi\sigma} \\ e^{\phi\sigma} & 0 \end{pmatrix}

and

e^{\phi\sigma} = \cosh \phi + \vec{\sigma} \cdot \hat{\phi} \sinh \phi

with the unit vector

\hat{\phi} \equiv \vec{\phi}/\phi

. Identifying

\vec{p} = m\hat{\phi} \sinh \phi

, derive the Dirac equation. Show that

\gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix}