 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1/1/d/solution

For real

ϵ

, unitarity to first order gives

I = U^{†} U = (I - i ϵ F^{†}) (I + i ϵ F) = I + i ϵ (F - F^{†}) + O (ϵ^{2}) .

Hence the generator of an infinitesimal unitary transformation obeys $F = F^{†}$ : it is Hermitian. More precisely,

U (ϵ) = I + i ϵ F + O (ϵ^{2})

is a differentiable unitary family at the identity. The omitted second-order terms matter if

ϵ

is finite.

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