Find the error in: 'since

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Invariance, Ch Lie groups and Lie algebras: rotations, SU(2) and the double cover

Problem

Find the error in: 'since (n^⋅σ)2=1(\hat{\mathbf n}\cdot\boldsymbol\sigma)^2=\mathsf 1, exponentiating gives e−iθ n^⋅σ=cos⁡θ 1−isin⁡θ (n^⋅σ)e^{-i\theta\,\hat{\mathbf n}\cdot\boldsymbol\sigma}=\cos\theta\,\mathsf 1-i\sin\theta\,(\hat{\mathbf n}\cdot\boldsymbol\sigma). For a rotation by θ=180∘\theta=180^\circ about zz this gives e−iπσz=cos⁡π 1−isin⁡π σz=−1e^{-i\pi\sigma_z}=\cos\pi\,\mathsf 1-i\sin\pi\,\sigma_z=-\mathsf 1.' ((a)) Identify the error and write the correct SU(2)SU(2) element for an active rotation by θ=180∘\theta=180^\circ about zz. ((b)) Show that the faulty (unhalved) formula, applied at θ=360∘\theta=360^\circ, hides the sign that is the signature of the SU(2)SU(2) double cover, and state what the correct formula gives there instead.

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