Advanced Quantum Mechanics Schwabl · Problem 6.10

The Coulomb Potential

Physics — Quantum Mechanics · gamma-matrix-basis

Problem

Properties of the matrices

\Gamma^a

. Taking as your starting point the definitions (6.2.50a--e), derive the following properties of these matrices: ((i)) For every

\Gamma^a

(except

\Gamma^S

) there exists a

\Gamma^b

such that

\Gamma^a\Gamma^b = -\Gamma^b\Gamma^a

. ((ii)) For every pair

\Gamma^a

\Gamma^b

(a \neq b)

there exists a

\Gamma^c \neq \mathbb{1}

such that

\Gamma^a\Gamma^b = \beta\Gamma^c

with

\beta = \pm 1, \pm\mathrm{i}