Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 3 / a

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Solution

The special unitary group is

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

. Differentiating these equations at the identity shows that its Lie algebra is

su (2) = {X : X^{†} = - X, tr X = 0} .

For

T_{i} = - i σ_{i} /2

, the Pauli-matrix identity

[σ_{i}, σ_{j}] = 2 i ϵ_{ijk} σ_{k}

gives

[T_{i}, T_{j}] = ϵ_{ijk} T_{k},

so the structure constants in this basis are

c^{k}_{ij} = ϵ_{ijk}

.

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