- #1
Si
- 20
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Hi, and thanks in advance for reading this. I've been bashing my head on my desk for days on this now.
My problem is the first sentence of the paragraph after equation 32.1.5 in Weinberg III:
"... note that for a given $Z_{nm}$ to be non-zero, since it is a scalar all of the $\sigma$s in Eq. (32.1.1) must be opposite for $Q_n$ and $Q_m$."
This is not obvious to me. To prove it I calculate the commutator of $\{Q_n,Q_m\}$ with e.g. $J_d1$ using Eq. (32.1.5) and the Poincare algebra, then use Eq. (32.1.5) to eliminate $\{Q_n,Q_m\}$. Then from the resulting equation I extract the weight $w=0$ terms (where $w$ is defined in Eq. (32.1.2)) to get $(-w(Q_n)+w(Q_m))[Z_{nm} +\sum_{i=2}^{d-1} \Gamma^i_{nm}P_i]=0$. The second term vanishes if I rotate the 1-direction to point in the spatial part of $P$. If this was OK, the rest is easy: $(-w(Q_n)+w(Q_m))Z_{nm}=0$ so for $Z_{nm}\neq 0$ to be possible we need $w(Q_n)=-w(Q_m)$. But this is not OK, because a rotation of the 1-direction changes the definition of $J_{d1}$, the $Q_n$ and the $w(Q_n)$.
So how do I prove that for $Z_{nm}\neq 0$ to be possible we need $w(Q_n)=-w(Q_m)$?
Thanks in advance for any help.
My problem is the first sentence of the paragraph after equation 32.1.5 in Weinberg III:
"... note that for a given $Z_{nm}$ to be non-zero, since it is a scalar all of the $\sigma$s in Eq. (32.1.1) must be opposite for $Q_n$ and $Q_m$."
This is not obvious to me. To prove it I calculate the commutator of $\{Q_n,Q_m\}$ with e.g. $J_d1$ using Eq. (32.1.5) and the Poincare algebra, then use Eq. (32.1.5) to eliminate $\{Q_n,Q_m\}$. Then from the resulting equation I extract the weight $w=0$ terms (where $w$ is defined in Eq. (32.1.2)) to get $(-w(Q_n)+w(Q_m))[Z_{nm} +\sum_{i=2}^{d-1} \Gamma^i_{nm}P_i]=0$. The second term vanishes if I rotate the 1-direction to point in the spatial part of $P$. If this was OK, the rest is easy: $(-w(Q_n)+w(Q_m))Z_{nm}=0$ so for $Z_{nm}\neq 0$ to be possible we need $w(Q_n)=-w(Q_m)$. But this is not OK, because a rotation of the 1-direction changes the definition of $J_{d1}$, the $Q_n$ and the $w(Q_n)$.
So how do I prove that for $Z_{nm}\neq 0$ to be possible we need $w(Q_n)=-w(Q_m)$?
Thanks in advance for any help.