Problem in Weinberg III

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In summary, the speaker is struggling to understand the concept that for a given $Z_{nm}$ to be non-zero, all of the $\sigma$s in Eq. (32.1.1) must have opposite values for $Q_n$ and $Q_m$. They attempt to prove this by calculating the commutator of $\{Q_n,Q_m\}$ with $J_d1$ and eliminating $\{Q_n,Q_m\}$. However, they encounter a problem with the rotation of the 1-direction and are seeking assistance in understanding this concept.
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Si
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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.
 
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Maybe I wrote too much... quite simply, can anyone explain the first sentence of the paragraph after equation 32.1.5 in Weinberg III?
 

1. What is the Weinberg III problem?

The Weinberg III problem is a theoretical issue in particle physics that arises from the Standard Model, specifically in the electroweak sector. It refers to the fact that the Higgs field, which is responsible for giving particles their mass, is extremely sensitive to quantum corrections and can lead to very large values for the mass of the Higgs boson, making the theory unstable and unreliable.

2. How does the Higgs field contribute to the Weinberg III problem?

The Higgs field is a fundamental part of the Standard Model and is responsible for giving particles their mass. However, it is also highly sensitive to quantum corrections, which can lead to large fluctuations in the predicted mass of the Higgs boson. This can make the theory inconsistent and unreliable.

3. What are some proposed solutions to the Weinberg III problem?

There are several proposed solutions to the Weinberg III problem, including supersymmetry, which introduces new particles to balance out the quantum corrections, and technicolor, which suggests that the Higgs field is not fundamental but is composed of other particles. Another solution is the idea of a composite Higgs, where the Higgs boson is made up of smaller particles.

4. How does the Weinberg III problem impact our understanding of the universe?

The Weinberg III problem is a fundamental issue in our current understanding of the universe and the laws that govern it. If left unresolved, it could lead to inconsistencies and inaccuracies in our predictions and models of particle physics, which could have a major impact on our understanding of the origins and behavior of the universe.

5. What current research is being done to address the Weinberg III problem?

Scientists are actively researching and developing new theories and models to address the Weinberg III problem. This includes experiments at the Large Hadron Collider, as well as theoretical investigations into alternative models such as supersymmetry, technicolor, and composite Higgs. Additionally, new data and observations from other fields of physics, such as cosmology and astrophysics, may also provide valuable insights into this problem.

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