Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 4 / b

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- Solution b

Solution

Writing the source interaction as

Tr (J^{μ} A_{μ})

, gauge invariance requires the matrix current to transform in the Adjoint representation,

J^{μ} \mapsto U J^{μ} U^{†} .

The fermion bilinear with this transformation law is

J^{μ a} = g \overset{ˉ}{ψ}_{i} γ^{μ} (T^{a})_{ij} ψ_{j}, J^{μ} = J^{μ a} T_{a},

up to a convention-dependent overall coupling or generator normalization. It is the Noether current that appears when the free derivative is replaced by

D_{μ} = \partial_{μ} - i g A_{μ}

.

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