How can we build the spin 2 graviton representation of little group O(9) in QFT?

In summary, Weinberg states that for d=11, the spin 2 graviton representation of the little group O(9) is a symmetric traceless tensor with 44 independent components. This includes one component with J23=+(-)2, seven components with J23=+(-)1, and twenty eight components with J23=0. These components are labeled by k and l, which run over the values 4, 5, 6, 7, 8, 9, and 10. However, the speaker admits to struggling in building the tensor and is unsure about the reasons for the specific number of components and the tensor's properties.
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In QFT vol 3 of Weinberg write:''For d=11 the spin 2 graviton representation of little group O(9) is a symmetric traceless tensor with 9x10/2-1=44 independent components:there is one
2+(-)i3,2+(-)i3 component with J23=+(-)2, seven2+(-)i3,k components with J23=+(-)1; and twenty eight k,l components with J23=0(here k,l run over the seven values 4,5,6,7,8,9,10)''
I tryed but fail to build the tensor.How can we build the tensor?I do not know why there are the senven and twenty eight components of J23=1or-1 and J23=0?Why does the tensor has rank 10 but not rank 4 because j=2=4x1/2?Why is tensor traceless?
 
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PS:QFT of Weinberg Vol 3 page 395
 

1. What is the question in QFT of Weinberg?

The question in QFT of Weinberg is how to reconcile the principles of quantum mechanics with those of special relativity in order to create a consistent theory of quantum fields.

2. Why is this question important?

This question is important because quantum field theory (QFT) is a fundamental framework for understanding the behavior of particles at the subatomic level. It has been successfully applied in many areas of physics, including particle physics, condensed matter physics, and cosmology. A consistent theory of QFT would help us better understand the fundamental laws of the universe and potentially lead to new discoveries.

3. How has this question been approached?

This question has been approached through various mathematical models and theoretical frameworks, such as the Standard Model of particle physics and the renormalization group. Additionally, experimental results from particle accelerators, such as the Large Hadron Collider, have provided insights into the behavior of particles at high energies and helped refine our understanding of QFT.

4. What are some current theories or ideas related to this question?

Some current theories and ideas related to this question include string theory, supersymmetry, and loop quantum gravity. These theories attempt to unify the principles of quantum mechanics and special relativity in different ways and offer potential solutions to the question in QFT of Weinberg.

5. What are some potential implications or applications of finding a solution to this question?

If a solution to the question in QFT of Weinberg is found, it could have significant implications in many areas of physics, including our understanding of the early universe, the behavior of particles at high energies, and the possibility of new particles or interactions. It could also have practical applications, such as in the development of new technologies and materials. Additionally, a consistent theory of QFT could lead to a deeper understanding of the fundamental laws of nature and potentially open up new avenues for scientific exploration and discovery.

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