Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 3 / d

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- Solution d

Solution

In the rest frame

P^{μ} = (m, 0, 0, 0)

, antisymmetry gives

W_{0} = 0

. Taking

ϵ_{0123} = + 1

and

J_{3} = M^{12}

gives

W_{3} = - m J_{3} .

Hence

W_{0} ∣ j, j_{3} ⟩ = 0, W_{3} ∣ j, j_{3} ⟩ = - m j_{3} ∣ j, j_{3} ⟩ .

The sign of the second eigenvalue reverses if the opposite Levi-Civita convention is chosen.

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