Why do probabilities distinguish between multiplet states?

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SUMMARY

The discussion centers on the role of isospin invariance in strong interactions and its implications for probabilities associated with multiplet states. It is established that the ratios of cross sections for strong interactions are determined by SU(2) Clebsch-Gordan coefficients, which are influenced by the third component of isospin. This raises the question of why probabilities for strong processes vary among multiplet members if isospin symmetry implies indistinguishability. The conclusion drawn is that while isospin symmetry is a guiding principle, conservation laws, such as electric charge, necessitate variations in probabilities based on the specific members of the multiplet involved.

PREREQUISITES
  • Understanding of isospin invariance in particle physics
  • Familiarity with SU(2) symmetry and Clebsch-Gordan coefficients
  • Knowledge of strong interactions and their properties
  • Basic principles of conservation laws in physics, particularly electric charge conservation
NEXT STEPS
  • Study the implications of isospin invariance in quantum chromodynamics (QCD)
  • Explore the derivation and applications of SU(2) Clebsch-Gordan coefficients
  • Investigate the role of conservation laws in particle interactions
  • Examine case studies of strong interaction processes and their cross-section calculations
USEFUL FOR

This discussion is beneficial for particle physicists, researchers in quantum field theory, and students studying the fundamentals of strong interactions and symmetries in physics.

metroplex021
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I'm pretty sure the answer to this will be more than obvious, but I've been wondering about the following.

Isospin invariance is assumed to be an (approximate) symmetry of the strong interactions. Ratios of cross sections for strong interactions are then given by the appropriate SU(2) Clebsch-Gordan coefficients. These coefficients depend on the third component of isospin of the particles involved; hence the probabilities and cross sections for strong interactions do too. But if isospin is a symmetry of the strong interactions (and let's assume for simplicity it is an exact symmetry), why should probabilities of strong processes depend on the third component of isospin? Since isospin symmetry means (roughly!) that the strong interaction doesn't 'see' any differences between the members of an isospin multiplet, why should the probabilities of strong processes vary depending on which members of a given multiplet are involved?
 
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Because electric charge needs to be conserved as well.
 

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