I'm reading David Tong's notes on quantum field theory, and I had a question from page 17 (equations 1.54-55), where he is deriving the conserved currents that arise from a symmetry under a Lorentz transformation.
The relevant portion is in this (very brief) screenshot. I understand how he derived equation 1.54, but I'm stumped on how he gets the next equation 1.55. What happened to the omega's?
My guess is that he ignores the omega's as they're constants, and re-writes the J's in a manner that makes them explicitly anti-symmetric (and they have to be antisymmetric because the omega's were antisymmetric). Is that correct?
Any help would be highly appreciated.
We have $$ j^\mu = -\omega^\rho{}_\nu \, T^\mu{}_\rho x^\nu = -\omega_\rho{}_\nu \, T^\mu{}^\rho x^\nu \\ = -\frac12 \left( \omega_\rho{}_\nu \, T^\mu{}^\rho x^\nu + \omega_\nu{}_\rho \, T^\mu{}^\nu x^\rho \right) \\ = -\frac12 \omega_\rho{}_\nu \left( T^\mu{}^\rho x^\nu - T^\mu{}^\nu x^\rho \right) $$ where I first used the "see-saw rule" to make both indices of $\omega$ be lower, then added a term with $\rho$ and $\nu$ swapped, and finally extracted $\omega_{\rho\nu}$ using its antisymmetry.
Thus we have $$j^\mu = -\frac12 \omega_\rho{}_\nu \mathcal J^{\mu\rho\nu},$$ where $$\mathcal J^{\mu\rho\nu} = T^\mu{}^\rho x^\nu - T^\mu{}^\nu x^\rho.$$