Why Can't UUU & DDD Show on Spinhalf Diagrams?

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Discussion Overview

The discussion revolves around the inability to represent three up quarks (uuu) or three down quarks (ddd) in spin-half diagrams within the context of elementary particle physics. Participants explore the implications of the Pauli exclusion principle, group theory, and the construction of wave functions in quantum chromodynamics (QCD).

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest that the exclusion principle prevents the formation of a spin 1/2 state for three identical quarks, leading to a violation of the principle.
  • Others mention that understanding the group theory involved, specifically the decomposition of the direct product of SU(3) representations, is crucial to grasping the issue.
  • A participant notes that the wave function must account for color, spin, isospin, and spatial components, complicating the explanation.
  • It is proposed that the combined state of three identical quarks must be completely antisymmetric, which cannot be achieved with the mixed symmetry of the spin addition for three spin-1/2 particles.
  • Some participants argue that the explanation can be simplified beyond group theory, focusing instead on the implications of Fermi-Dirac statistics and the nature of the wave function.

Areas of Agreement / Disagreement

Participants express differing views on the complexity of the explanation, with some advocating for a simpler understanding while others emphasize the necessity of group theory. The discussion remains unresolved regarding the best approach to explain the phenomenon.

Contextual Notes

Limitations include the dependence on understanding advanced concepts in quantum mechanics and group theory, as well as the unresolved nature of how spatial symmetry interacts with the other components of the wave function.

barnflakes
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Can anyone explain to me why you can't have uuu and ddd on the first (spinhalf.gif) diagrams I have uploaded please?
 

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The answers in that thread were completely beyond me. Can somebody explain in a single sentence for me? I'm only studying very elementary particle physics..
 
The short answer is the exclusion principle. But to see that you need to know a bit about groups and their representations. Specifically that the direct product of 3 fundamental '3's of SU(3) decomposes like 3 * 3 * 3 = 10 + 8 + 8 + 1. The decuplet (which contains uuu) is necessarily spin 3/2 and completely symmetric.
 
The attempt to construct a spin 1/2 state |uuu> results in a violation of the Pauli exclusion principle.

As far as I remember from my QCD lectures (~ two decades ago!) the explanation in the other thread misses the spatial part of the wave function. One must take into account
|color> * |spin> * |isospin> * |space>
Ofcourse this doesn't make things easier ...
 
yes the most basic answer is to combine the two of haelfix and tom.stoer
 
The ground state L=0 is space-symmetric in the constituent quark model.
 
The answer is simpler than group theory, but is based on the Pauli principle (Fermi-Dirac statistics) and the addition of spins. The combined state of (color)(space)(spin) must be completely antisymmetric for three identical u quarks. The wave function is antisymmetric in color. The spin addition 1/2+1/2+1/2=1/2 is of mixed symmetry, and so cannot combine with the presumed symmetry spatial ground state and antisymmetric color state to form a completely antisymmetric state. Spin 1/2 for three quarks is only possible if they are not all identical, for instance uud.
 

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