Advanced Quantum Mechanics Schwabl · Problem 13.11

Quantization of the Radiation Field

Physics — Quantum Mechanics · dirac-field-spin

Problem

Consider an electron at rest and a stationary positron

\left|e^\mp,\mathbf{k}=0,s\right\rangle = \begin{cases} b^\dagger_{s\mathbf{k}=0}|0\rangle \\ d^\dagger_{s\mathbf{k}=0}|0\rangle \end{cases}.

Show that

\mathbf{J}\left|e^\mp,\mathbf{k}=0,s\right\rangle = \sum_{r=1,2}\left|e^\mp,\mathbf{k}=0,r\right\rangle\frac{1}{2}(\boldsymbol{\sigma})_{rs}

and

J^3\left|e^\mp,\mathbf{k}=0,s\right\rangle = \pm\frac{1}{2}\left|e^\mp,\mathbf{k}=0,s\right\rangle\,,

where

(\boldsymbol{\sigma})_{rs}

are the matrix elements of the Pauli matrices in the Pauli spinors

\chi_r

and

\chi_s