Standard Model and Beyond · Problem 3.6

Gauge Theories

Physics — Particle Physics · determinant-exponential-traceless

Problem

Prove that

\det[e^{\alpha A}] \equiv \sum_k (\alpha A)^k/k! = e^{\alpha \mathrm{Tr} A}

for any Hermitian matrix

A

. This implies that the condition for a unitary or orthogonal matrix to be special (unit determinant) is that the generators be traceless.