Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 304 / 3 / b

Table of contents
- Solution b

Solution

Requiring

D_{μ} ϕ

to transform as

U^{†} (D_{μ} ϕ) U

and using the gauge-field transformation law gives

D_{μ} ϕ = \partial_{μ} ϕ - i g [A_{μ}, ϕ] .

Indeed, differentiating

U^{†} ϕ U

produces two inhomogeneous derivative terms, and those cancel against the inhomogeneous part of the transformed connection.

For

U = 1 + i α^{a} T_{a} + O (α^{2})

,

ϕ^{'} = U^{†} ϕ U = ϕ + i [ϕ, α^{a} T_{a}] + O (α^{2}) .

If

[T_{a}, T_{b}] = i f_{ab}^{c} T_{c}

, then

δ ϕ^{c} = f_{ab}^{c} α^{a} ϕ^{b} .

This is the infinitesimal Adjoint representation of a Lie algebra.

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