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I am pretty confused about how to construct states to make symmetric / anti-symmetric combination so I would like to ask some questions.

For example, for SU(2), states of three spin-half particles can be decomposed as 2 x 2 x 2 = 4 + 2 + 2, 3 irreducible combination with dim 4, 2, 2.

-if I combine three boxes in a row(in young tableaux), making a total symmetric combination,

then [[a, b], c] = (ab + ba) c + c (ab + ba) = abc + bac + bac + cba (right?) (here the large bracket represents symmetric index)

And there should be four in general non-vanishing independent index: 111, 112=211, 221=122, 222(right?)

But as I see, [[1,1],2] != [[1,2],1] for [[1,1],2] = 112 + 112 + 211 + 211 while [[1,2],1] = 121 + 211 + 112 + 121 and [[2,2],1] != [[1,2],2] for similar reason. (it's because c is symmetric only with [a, b], not with the individual a and b). So there seems to be six non-vanishing index.

I wonder what is wrong with my calculation.

-And as I understand, [{a, b}, c] = (ab - ba) c + c(ab - ba) can have two non-vanishing index:

[{1, 2}, 1], [{1, 2}, 2]

And for {[a, b], c} = (ab + ba) c - c (ab + ba), there are two independent non-vanishing index:

{[1,1],2}, {[2,2],1} (or equivalently {[1,2],2})

Is this correct?

-And if I calculate {[1,1],2} of |+>|->|+>, it's 0. But as I calculated directly using CG table, |+>|->|+> is a part of the state whose {[1,1],2} only is 1. So I guess it shouldn't be zero. Is something wrong with this?

Actually there are many other things I get confused but first I would like to know these and then figure out the next.

Thank you very much.

For example, for SU(2), states of three spin-half particles can be decomposed as 2 x 2 x 2 = 4 + 2 + 2, 3 irreducible combination with dim 4, 2, 2.

-if I combine three boxes in a row(in young tableaux), making a total symmetric combination,

then [[a, b], c] = (ab + ba) c + c (ab + ba) = abc + bac + bac + cba (right?) (here the large bracket represents symmetric index)

And there should be four in general non-vanishing independent index: 111, 112=211, 221=122, 222(right?)

But as I see, [[1,1],2] != [[1,2],1] for [[1,1],2] = 112 + 112 + 211 + 211 while [[1,2],1] = 121 + 211 + 112 + 121 and [[2,2],1] != [[1,2],2] for similar reason. (it's because c is symmetric only with [a, b], not with the individual a and b). So there seems to be six non-vanishing index.

I wonder what is wrong with my calculation.

-And as I understand, [{a, b}, c] = (ab - ba) c + c(ab - ba) can have two non-vanishing index:

[{1, 2}, 1], [{1, 2}, 2]

And for {[a, b], c} = (ab + ba) c - c (ab + ba), there are two independent non-vanishing index:

{[1,1],2}, {[2,2],1} (or equivalently {[1,2],2})

Is this correct?

-And if I calculate {[1,1],2} of |+>|->|+>, it's 0. But as I calculated directly using CG table, |+>|->|+> is a part of the state whose {[1,1],2} only is 1. So I guess it shouldn't be zero. Is something wrong with this?

Actually there are many other things I get confused but first I would like to know these and then figure out the next.

Thank you very much.

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