A very important group is SU(3), since it is related to the colour carried by the quarks, the basic building blocks of QCD.
The transformations within SU(3) are all those amongst a vector consisting of three complex objects that conserve the length of the vector. These are all three-by-three unitary matrices, which act on the complex vector by
The complex conjugate vector can be shown to transform as
(8.26) |
with the inverse of the matrix. Clearly the fundamental representation of the group, where the matrices representing the transformation are just the matrix transformations, the vectors have length . The representation is usually labelled by its number of basis elements as . The one the transforms under the inverse matrices is usually denoted by .
What happens if we combine two of these objects, and ? It is easy to see that the inner product of and is scalar,
(8.27) |
where we have used the unitary properties of the matrices the remaining components can all be shown to transform amongst themselves, and we write
(8.28) |
Of further interest is the product of three of these vectors,
(8.29) |