Topics » Mathematical Physics

In the set of Gell-Mann matrices: why is L8 not {L8(2,2)=-L8(3,3)=1 and all other L8(i,j)=0}?

Would this not give a simpler and more uniform definition without the awkward sqrt(3)?

Hugo Vercammen ⋅ Feb 9, 2012
6 Replies
  • Hugo Vercammen

    Looking further into it I find that L3 and L8 are not trace orthogonal.
    Is this a problem? What are the consequences?

    Feb 16, 2012
  • Hugo Vercammen

    Sorry, the new set is:
    f123=f678=f458=f345=1,f147=f165=f246=f257=f376=f128=.5

    Feb 13, 2012
  • Hugo Vercammen

    The new set of structure constants would be:
    f123=f678=1,f147=f165=f246=f257=f345=f376=f458=f128=.5
    where the set f128 is a new set <>0.
    Advantage: no sqrt(3) and L6,L7,L8 are generators of an SU(2).

    Feb 11, 2012
  • Rob Carroll

    Ah yes, that is an interesting question. I will need to think about that one.

    Feb 10, 2012
  • Hugo Vercammen

    Sure, thanks Rob. Of course, if one changes the definition of L8, the structure constants change.
    So, the deeper question is: Is the new set of structure constants one would get not acceptable?
    Or, more precisely, why must L8 commute with the generators of SU(2)? (so that [Li,L8] must be 0 for i=1,2,3)

    Feb 9, 2012
  • Rob Carroll

    This is because in order to be a representation of the infinitesimal generators of SU(3) the Gell-Mann matrices need to obey the commutation relation [Li,Lj]=i*f_(ijk)Lk, where the index k is summed over and the definition of the fine structure constants f_(ijk) can be found for example on wikipedia. This relation requires that, for example, [L1,L8]=0. This is satisfied for the definition of the Gell-Mann matrices, but if they were defined as you propose they would give [L1,L8]=M where M(1,2) = -M(2,1) = 1 and all other M(i,j) are zero.

    Feb 9, 2012
Follow this Post

Contributors

Share