Lie Algebra in Particle Physics simplified

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SUMMARY

The discussion focuses on the application of Lie algebra in particle physics, specifically regarding the manipulation of sums and products using representations. Users highlight the importance of Young tableaux for splitting product representations into irreducible representations (irreps) and emphasize the necessity of understanding the underlying theory, particularly with SU(2) groups. An example provided illustrates the combination of three spin-1/2 particles resulting in specific spin multiplicities.

PREREQUISITES
  • Understanding of Lie algebra concepts
  • Familiarity with SU(2) group theory
  • Knowledge of Young tableaux for representation theory
  • Basic principles of particle spins and their combinations
NEXT STEPS
  • Study the application of Young tableaux in representation theory
  • Learn about SU(2) group representations and their physical implications
  • Explore the mathematical foundations of Lie algebra in quantum mechanics
  • Investigate the process of combining spins in particle physics
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Physicists, mathematicians, and students interested in quantum mechanics and particle physics, particularly those looking to deepen their understanding of Lie algebra and its applications in theoretical frameworks.

Silviu
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Hello! Is there any rule to do sums and products like the one in the attached picture (Lie.png) without going through all the math theory behind? I understand the first (product) and last (sum) terms, but I am not sure I understand how you go from one to another.
Thank you!
 

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You can split product representations into irreps by using Young tableaux, but I do suggest you learn the theory behind it and why it works.
 
especially with SU(2)'s you can just add spins...
2= spin 1/2
3= spin 1
etc
and then seeing the multiplicity of the final spins

Eg your example has 3 spin 1/2 particles, so you'd get something like:
(1/2 1/2) 1/2 = ( 1 0 ) 1/2 = (1 1/2) (0 1/2) = 3/2 1/2 1/2
so a 4 2 2
 

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