I am reading Weinberg's book Quantum theory of fields.

Could you explain to me the following things? Vol.1, page 60 (transcribed from this image):

To first order in $\omega$ and $\epsilon$, we have then \begin{align} U(\Lambda,a) \left[\tfrac12\omega_{\rho\sigma}J^{\rho\sigma} -\epsilon_\rho P^\rho\right] U^{-1}(\Lambda,a) & = \tfrac12(\Lambda\omega\Lambda^{-1})_{\mu\nu} J^{\mu\nu} \\ & \quad - (\Lambda\epsilon - \Lambda\omega\Lambda^{-1}a)_\mu P^\mu \tag{2.4.7} \end{align} Equating coefficients of $\omega_{\rho\sigma}$ and $\epsilon_\rho$ on both sides of this equation (and using (2.3.10)) we find \begin{align} U(\Lambda,a) J^{\rho\sigma} U^{-1}(\Lambda,a) & = \Lambda_\mu^{\ \rho}\Lambda_\nu^{\ \sigma} (J^{\mu\nu} - a^\mu P^\nu + a^\nu P^\mu) \tag{2.4.8}\\ U(\Lambda,a) P^\rho U^{-1}(\Lambda,a) & = \Lambda_\mu^{\ \rho} P^\mu \tag{2.4.9} \end{align}

How are we equating the coefficients? How we find the formula (2.4.8)?

The previous formula that the chapter refers to is

$$ (\Lambda^{-1})^{\rho}_{\nu} = \Lambda_{\nu}^{\rho} = \eta_{\mu\nu}\eta_{\rho\sigma}\Lambda_{\sigma}^{\nu} \tag{2.3.10}$$

  • $\begingroup$ Welcome, by the way! What are the coefficients of ω and ϵ on the LHS of equation (2.4.7)? $\endgroup$ – Deschele Schilder Apr 20 '19 at 11:10
  • $\begingroup$ In equation (2.3.10),$ (\Lambda^{-1})^{\rho}_{\nu} = \Lambda_{\nu}^{\rho} = \eta_{\mu\nu}\eta_{\rho\sigma}\Lambda_{\sigma}^{\nu}$ , shouldn't the first $\Lambda$ and the second ${\Lambda}^{-1}$ have a $\rho$ and $\eta$ as indices instead of a $\rho$ and $\nu$? $\endgroup$ – Deschele Schilder Apr 20 '19 at 11:36

Equation (2.4.9) follows easy:

The coefficient of $\eta$ on the LHS of (2.4.7) is $-U P U^{-1}$. And on the RHS it is $-\Lambda P$ (here we don't have to worry about the indices). Set these two equal and (2.4.9) follows.

The coefficient of $\omega$ on the LHS of (2.4.7) is $U J U^{-1}$. The coefficient on the RHS is $(\frac 1 2 \Lambda {\Lambda}^{-1}J-\Lambda {\Lambda}^{-1} a P)$.

I'll leave it to you to put in the indices after which you can use (2.4.10) and obtain (2.4.8), though I have the feeling that this is where you are stuck. Try harder! I'm not supposed to give you the full answer.


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