A Problem in Georgi's Lie algebras in particle physics

In summary, the conversation discusses extracting transformation matrices from the trace operator in equation 2.35 on page 49 of the textbook "Lie algebra in particle physics". It is explained that the equation involves vectors of matrices and numbers, and the trace operator can be used to extract the numbers from the matrices. The conversation also touches on using latex and clarifies some concepts for the reader.
  • #1
zahero_2007
75
0
Can some one please explain to me equation 2.35 on page 49 in the textbook "Lie algebra in particle physics " How can he extract the 2 Transformation matrices outside the trace operator ?I think there is something wrong
Sorry I do not know how to use latex
 
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  • #2
in 2.34 Ta and Td are vectors of matrices. And Lad a vector of numbers so Lad Td is not a matrix product but a scalar product. You can put Lad near Td.
It would be the same with another index for Tb. L and L^-1 are the same
You can then multiply, take the traces and extract the numbers (not matrices)

I hope i wrote not too many wrong things!
 
  • #3
I see , Thanks a lot naima .
 
  • #4
When
Ta -> T'a = L [Lac.Tc] L-1
Tb -> T'b = L [Lbd.Td] L-1
then
TaTb -> L [ (Lac.Tc)(Lbd.Td)] L-1
You know that Tr(AB) = Tr (BA) so when you take the trace of the rhs you get
Tr ((Lac.Tc)(Lbd.Td))
Lac and Lbd are numbers and Rc Tc are matrices and as trace is linear so you get 2.35
 

1. What is the significance of Georgi's Lie algebras in particle physics?

Georgi's Lie algebras are a mathematical framework used to study the symmetries and interactions of elementary particles in particle physics. They help to explain the fundamental forces of nature and provide a way to classify and understand the properties of particles.

2. What is the specific problem being addressed in Georgi's Lie algebras in particle physics?

The specific problem being addressed is the incompatibility between the Standard Model of particle physics and the observed masses of certain particles. This is known as the hierarchy problem and it is one of the biggest challenges in modern theoretical physics.

3. What approaches are being taken to solve this problem?

There are several approaches being taken to solve this problem, including supersymmetry, extra dimensions, and composite Higgs models. These theories propose new particles and interactions that could potentially explain the observed mass hierarchy.

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

The hierarchy problem in Georgi's Lie algebras is a major obstacle in our quest to understand the fundamental laws of nature. It challenges our current theories and forces us to look for new explanations and ideas. Solving this problem could lead to a deeper understanding of the universe and its origins.

5. What are the potential implications if this problem is solved?

If the problem in Georgi's Lie algebras is successfully solved, it could lead to a more complete and unified theory of particle physics. This could have a profound impact on our understanding of the universe, as well as potential practical applications in fields such as technology and medicine.

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