Robert Mann - An Introduction to Particle Physics and the Standard Model · Problem 5.6

Accelerators — Robert Mann

Physics — Particle Physics · pauli-matrices-identities

Problem

Verify the following: (a)

\exp\left[\vec{\theta} \cdot \vec{\sigma}\right] = \sum_{n=0}^{\infty} \frac{(\vec{\theta}\cdot\vec{\sigma})^n}{n!} = \cos\theta + i\hat{\theta}\cdot\vec{\sigma}\sin\theta

(b)

(\sigma_1)^2 = (\sigma_2)^2 = (\sigma_3)^2 = 1

\sigma_1\sigma_2 = i\sigma_3 \text{ and cyclic}

[\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k \qquad \{\sigma_i, \sigma_j\} = 2\delta_{ij}