Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 1 / ii

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- Solution ii

Solution

Lorentz invariance gives the Lorentz current

M^{μ ρ σ} = x^{ρ} T^{μ σ} - x^{σ} T^{μ ρ}, \partial_{μ} M^{μ ρ σ} = 0.

An infinitesimal rotation through angle

θ

around the

z

-axis has

δ x^{1} = - θ x^{2}, δ x^{2} = θ x^{1}, δ x^{0} = δ x^{3} = 0,

corresponding, up to the stated index convention, to the only nonzero components

ω^{12} = - ω^{21} = θ

. Its conserved angular momentum is

J_{z} = \int d^{3} x M^{012} = \int d^{3} x (x^{1} T^{02} - x^{2} T^{01}) .

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