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Gell-Mann matrices

  1. Jul 23, 2011 #1
    This is not a homework, only something embarrasing..

    [T_8, T_4 + i T_5] = (3^(1/2) / 2) T_4 + i T_5

    from http://phys.columbia.edu/~cyr/notes/QFT_3/lecture3.pdf" [Broken]

    I can't see how to get the structure constant (3^(1/2) / 2).

    T_4 + i T_5 is a 3x3 matrix with a one at (2,3), the rest zeroes. I multiply T_8 with T_4 + i T_5, then T_4 + i T_5 with T_8, then substract.

    I don't get (3^(1/2) / 2) times T_4 + i T_5.

    I get (3/ 4x3^(1/2)) times T_4 + i T_5.


    thanks for any help
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jul 24, 2011 #2

    jambaugh

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    Hmmm....
    [itex] [T_8, T_4 + i T_5] = \frac{1}{4}[\lambda^8,\lambda^4 +i\lambda^5] = \frac{1}{4}\left( [\lambda^8,\lambda^4] + i[\lambda^8,\lambda^5]\right)[/itex]

    I get [itex][\lambda^8,\lambda^4] = \frac{3i}{\sqrt{3}}\lambda^5=i\sqrt{3}\lambda^5[/itex]
    and [itex][\lambda^8,\lambda^5] = -i\sqrt{3}\lambda^4[/itex]
    [itex][T_8, T_4 + i T_5] = \frac{\sqrt{3}}{4}\left(i\lambda^5 + \lambda^4\right)=\frac{\sqrt{3}}{2}\left(T_4 + i T_5\right)[/itex]

    It looks to me like you forgot to double when going back to the T form from the lambda form. Also note that [itex]\frac{a}{\sqrt{a}} = a^1 a^{-1/2} = a^{1-1/2} = a^{+1/2} = \sqrt{a}[/itex].
     
  4. Jul 24, 2011 #3
    Ahh, of course! As I said, it is rather embarassing...

    many thanks, Jambaugh
     
  5. Jul 25, 2011 #4

    jambaugh

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    You shouldn't be embarrassed about making a mistake... (we all make them)... only about refusing to acknowledge your mistakes.
     
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