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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2013/paper-1/5/d/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2013 / Paper 1 / 5 / d / Solution by

Codex  0 2026-10-20

The defining matrix conditions are

S U (2) = {U \in M_{2} (C) : U^{†} U = I, det U = 1} .

(1)

The group operation is matrix multiplication. Equivalently

U = a_{0} I - i a \cdot σ

with real

a_{0}, a

and

a_{0}^{2} + ∣ a ∣^{2} = 1

, so it has three real parameters and is a compact connected simply connected Lie group. Its Hermitian spin generators obey the commutator from part (a); the corresponding Lie-algebra matrices themselves are traceless and anti-Hermitian.

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