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Mar2-05, 05:38 AM
marlon's Avatar
P: 4,006
Quote Quote by da_willem
1)how do these particle structures realte to the mathematics of group theory? (note that my knowledge of group theory is very limited)
The physics of color is not understandable if all one knows is that there are 3 colors. One must really understand something about SU(3). SU(3) is the group of 3 x 3 unitary matrices with determinant 1. This is the symmetry group of the strong force. What this means is that, as far as the strong force is concerned, the state of a particle is given by a vector in some vector space on which elements of SU(3) act as linear (in fact unitary) operators. We say the particle "transforms under some representation of SU(3)". For example, since elements of SU(3) are 3 x 3 matrices (and these matrices can be constructed using the given generators in your formula), they can act on column vectors by matrix multiplication. This gives a 3-dimensional representation of SU(3). The quarks are represented by this 3*1-matrix. The antiquarks can be represented by row vectors because we can multiply a 3*3-matrix with a row vector on the LEFT side of the matrix.

The gluons are represented by the socalled adjoint representation which consits out of traceless 3*3matrices. It can be seen that a row of such a matrix represents one quark colour and a colom of such a matrix represents a anti-colour. each gluon is therefore constructed out of a colour-anticolour combination. Given that there are 3 such colours and anticolours, you would expect 9 gluons. However there are only eight . Can you see why ???

ps : you know that the colours are red green and blue and it is the postulate of QCD that the sum of these three represents colour-neutrality !!! This is the main law that needs to be respected : in interactions : the sum of all involved colours must be WHITE

2) Has the symmetry breaking in gauge theories have anything to do with the broken symmetries (eg the masses of the particles in the multplets differ, the symmetries are not complete) in particle multiplets?
First of all, mass itself breaks symmetry because it mixes the two types of chirality which are fundamentally different in nature. So every elementary particle is massless. Read more on this in my journal. I also suggest the text i wrote on the Higgsfield, which is the system that accounts for generated mass after symmetry breaking in QFT. Just look it up in my journal
ps : the text on elementary particles is on page 3 or 4


here you can find a nice course on Lie Groups IN DUTCH