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user74200
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I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric.

My idea would be to try to construct the states in the two octuplets and show that they are not antisymmetric and since the singlet should have the same quark content as the "centermost" states in the octuplets(decouplet has no centermost and the "0" state) of the decouplet, the singlet has to be antisymmetric to be distinguishable from the others. But that would mean applying lowering operators at least 68 times(there are 2 states in the center of those octuplets that can only be distinguished by taking different paths to the "center") which would be really time consuming. Is there any quicker way to show it?

I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric.

My idea would be to try to construct the states in the two octuplets and show that they are not antisymmetric and since the singlet should have the same quark content as the "centermost" states in the octuplets(decouplet has no centermost state), the singlet has to be antisymmetric to be distinguishable from the others. But that would mean applying lowering operators at least 6 times(there are 2 states in the center of those octuplets that can only be distinguished by taking different paths to the "center") which would be really time consuming. Is there any quicker way to show it?

I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric.

My idea would be to try to construct the states in the two octuplets and show that they are not antisymmetric and since the singlet should have the same quark content as the "centermost" states in the octuplets and the "0" state of the decouplet, the singlet has to be antisymmetric to be distinguishable from the others. But that would mean applying lowering operators at least 8 times(there are 2 states in the center of those octuplets that can only be distinguished by taking different paths to the "center") which would be really time consuming. Is there any quicker way to show it?

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Qmechanic ♦
Qmechanic ♦
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Prove that the singlet state of 3 quark flavors is fully asymmetricantisymmetric

I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric.

I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric. My idea would be to try to construct the states in the two octuplets and show that they are not antisymmetric and since the singlet should have the same quark content as the "centermost" states in the octuplets(decouplet has no centermost state), the singlet has to be antisymmetric to be distinguishable from the others. But that would mean applying lowering operators at least 6 times(there are 2 states in the center of those octuplets that can only be distinguished by taking different paths to the "center") which would be really time consuming. Is there any quicker way to show it?

Prove that the singlet state of 3 quark flavors is fully asymmetric

I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric. My idea would be to try to construct the states in the two octuplets and show that they are not antisymmetric and since the singlet should have the same quark content as the "centermost" states in the octuplets(decouplet has no centermost state), the singlet has to be antisymmetric to be distinguishable from the others. But that would mean applying lowering operators at least 6 times(there are 2 states in the center of those octuplets that can only be distinguished by taking different paths to the "center") which would be really time consuming. Is there any quicker way to show it?

Prove that the singlet state of 3 quark flavors is fully antisymmetric

I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric.

My idea would be to try to construct the states in the two octuplets and show that they are not antisymmetric and since the singlet should have the same quark content as the "centermost" states in the octuplets(decouplet has no centermost state), the singlet has to be antisymmetric to be distinguishable from the others. But that would mean applying lowering operators at least 6 times(there are 2 states in the center of those octuplets that can only be distinguished by taking different paths to the "center") which would be really time consuming. Is there any quicker way to show it?

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user74200
user74200
  • 269
  • 2
  • 11

Prove that the singlet state of 3 quark flavors is fully asymmetric

I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric. My idea would be to try to construct the states in the two octuplets and show that they are not antisymmetric and since the singlet should have the same quark content as the "centermost" states in the octuplets(decouplet has no centermost state), the singlet has to be antisymmetric to be distinguishable from the others. But that would mean applying lowering operators at least 6 times(there are 2 states in the center of those octuplets that can only be distinguished by taking different paths to the "center") which would be really time consuming. Is there any quicker way to show it?