Lorentz invariance gives the <Lorentz current>
$$
M^{\mu\rho\sigma}
=x^\rho T^{\mu\sigma}-x^\sigma T^{\mu\rho},
\qquad \partial_\mu M^{\mu\rho\sigma}=0.
$$
An infinitesimal rotation through angle $\theta$ around the $z$-axis has
$$
\delta x^1=-\theta x^2,\qquad
\delta x^2=\theta x^1,\qquad
\delta x^0=\delta x^3=0,
$$
corresponding, up to the stated index convention, to the only nonzero components $\omega^{12}=-\omega^{21}=\theta$. Its conserved angular momentum is
$$
\boxed{J_z=\int d^3x\,M^{012}
=\int d^3x\,(x^1T^{02}-x^2T^{01})}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-301/1/ii/solution