Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 3 / b / i

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- Solution i

Solution

For

U (x) \in S U (3)

and

A_{μ} = A_{μ}^{a} T^{a}

, choose

D_{μ} = \partial_{μ} - i g A_{μ}

. The Yang-Mills gauge transformation is

A_{μ}^{'} = U A_{μ} U^{- 1} - \frac{i}{g} (\partial_{μ} U) U^{- 1},

which ensures

D_{μ}^{'} (U Φ) = U D_{μ} Φ

. Since

[D_{μ}, D_{ν}] = - i g F_{μν},

conjugating this commutator immediately gives

F_{μν}^{'} = U F_{μν} U^{- 1} .

Thus the gauge field strength is gauge covariant.

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