It is natural to group the eight lightest mesons into 4 irreps of isospin as follows:
|(complexified adjoint rep)|
|(dual of defining rep)|
(The dual of the defining rep of is isomorphic to the defining rep, but it's always nice to think of antiparticles as living in the dual of the rep that the corresponding particles live in, so above I have said that the antikaons live in the dual of the defining rep.)
In his theory called
the Eightfold Way, Gell-Mann showed that these eight
mesons could be thought of as a basis for the
the complexified adjoint rep of -- that is,
its rep on the 8-dimensional complex Hilbert space
1. Determine in a similar way how the other mesons are built from quarks and antiquarks, and fill in this chart:
2. In the Eightfold Way, whenever we have particles living
in some rep of , the third component of weak
isospin corresponds to the operator
3. Figure out the same information for the antiquarks:
Hint: Unlike for ,
the defining representation of is not isomorphic
to its dual. So, you should go back to the formula for the
dual of a Lie algebra rep. If we have a rep of some Lie
4. By filling out the following chart, check that you can compute , or for any meson simply by adding these quantities for the quarks and antiquarks it is built from. For example, the is built from a and a . The has and the also has ; adding these we get for the .
In terms of the mathematics of representation theory, why should we be able to compute , or for mesons as a sum of this sort?
If we had more time, we would now go on to explain the baryons -- like the proton and neutron, but also other particles -- in terms of quarks. Then we would explain how the Eightfold Way was eventually incorporated into the Standard Model, in which quarks are held together by the `strong force' to form mesons and baryons. Ironically, while the strong force is described by a theory with symmetry group , this symmetry group has nothing to do with Gell-Mann's symmetry! Gell-Mann's is now seen to be just an approximate symmetry coming from the fact that the up, down and strange quarks all act roughly the same, though they have different masses and charges. The symmetry of the strong force describes how quarks of any sort come in three `colors'.
But alas, summer is fast approaching, and there is no time to continue our adventure into particle physics. If you want to learn more, try reading these books, in approximately increasing order of difficulty and detail:
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© 2003 John Baez - all rights reserved.