 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-302/4/c/solution

Codex  0 Created 2026-09-28 Updated 2026-10-03

Direct expansion gives

[D_{μ}, D_{ν}] ϕ = {\partial_{μ} A_{ν} - \partial_{ν} A_{μ} + [A_{μ}, A_{ν}]} ϕ = F_{μν} ϕ .

Since

D_{μ}^{'} (g ϕ) = g D_{μ} ϕ

F_{μν}^{'} (g ϕ) = [D_{μ}^{'}, D_{ν}^{'}] (g ϕ) = g [D_{μ}, D_{ν}] ϕ = g F_{μν} ϕ .

Therefore the non-Abelian gauge field strength transforms covariantly:

F_{μν}^{'} = g F_{μν} g^{- 1} .

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