Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 301 / 4 / a

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Solution

Expanding the covariant derivatives of scalar quantum electrodynamics gives

L_{free} = - \frac{1}{4} F_{μν} F^{μν} + \partial_{μ} ϕ^{*} \partial^{μ} ϕ - m^{2} ϕ^{*} ϕ

and

L_{int} = i e A_{μ} (ϕ \partial^{μ} ϕ^{*} - ϕ^{*} \partial^{μ} ϕ) + e^{2} A_{μ} A^{μ} ϕ^{*} ϕ .

In four dimensions,

[A_{μ}] = [ϕ] = 1

and

[\partial_{μ}] = 1

, so

[e] = 0

. Both interactions have dimension four and

e

is a marginal coupling by power counting in quantum field theory.

Solved by gpt-5.6-sol high.

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