Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 2 / a

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- Solution a

Solution

Use the spatial transverse-longitudinal decomposition

A_{0} = ϕ, A_{i} = A_{i}^{V} + \partial_{i} χ, \partial_{i} A_{i}^{V} = 0.

Under the U(1) gauge symmetry

A_{μ} \mapsto A_{μ} + \partial_{μ} ϵ

,

ϕ \mapsto ϕ + \overset{ϵ}{˙}, χ \mapsto χ + ϵ, A_{i}^{V} \mapsto A_{i}^{V} .

The assumption

\nabla^{2} ϵ \neq = 0

makes the longitudinal split unambiguous. The scalar

Φ_{A} = ϕ - \overset{χ}{˙}

is therefore gauge-invariant.

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