 Solution

ID: past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2022/6/c/solution

The finite passive dilatation is

x^{'} = e^{- α} x

ϕ^{'} (x^{'}) = e^{α} ϕ (x)

. To compare at the same argument, invert the coordinate change:

ϕ^{'} (x) = e^{α} ϕ (e^{α} x) = ϕ (x) + α ϕ (x) + α x^{μ} \partial_{μ} ϕ (x) + O (α^{2}) .

Consequently the fixed-argument infinitesimal variation is

\overset{ˉ}{δ} ϕ (x) = α (ϕ + x^{μ} \partial_{μ} ϕ) .

This also follows from

δ ϕ = α ϕ

and

δ x^{μ} = - α x^{μ}

. The plus sign on the derivative term is a consequence of this specified contraction of coordinates.

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