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Jdeloz828

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- I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles, and have some specific questions related to this which refer to the text.

Hello,

I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles. I have had a fair amount of exposure to elementary group theory, but no representation theory, and have some specific questions related to this which refer to the text.

If anyone can shed some light on these matters it would be much appreciates. Also, if anyone knows of any useful resources that might provide a more detailed discussion of this subject, please let me know

I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles. I have had a fair amount of exposure to elementary group theory, but no representation theory, and have some specific questions related to this which refer to the text.

*Towards the end of section 4.1 Griffith's defines a***Group Representation**as a set a matrices which are homomorphic to the underlying group. He then lists a few examples:*An ordinary scalar belongs to the one-dimensional representation of the rotation group, SO(3)**A three-vector belongs to the three-dimensional representation of SO(3).**Four-vectors belong to the four-dimensional representation of the Lorentz group.**The curious geometrical arrangements of Gell-Mann's Eightfold Way correspond to representations of SU(3).*

*Griffith's will frequently make statements like the following, referring to the space of gluon color states:*

3 ⊗ 3' = 1 ⊕ 8

It seems this is the author's group theoretic way of stating that the color/anti-color states are comprised of a color singlet and color octet. Would a more precise way of stating this be that if you take the direct product of a color state and an anti-color state, you form a nine-dimensional vector space with a one-dimensional subspace and an eight-dimensional subspace?

*The author also makes the following statement referring to the product of two quark color states:*

3 ⊗ 3 = 3' ⊕ 6

I don't quite what would lead to the difference in this case. Why is a product space of two quark color states fundamentally different in structure than the gluon color/anti-color state space?

*The author also seems to suggest that color singlet states are invariant under the underlying symmetry group.*

If anyone can shed some light on these matters it would be much appreciates. Also, if anyone knows of any useful resources that might provide a more detailed discussion of this subject, please let me know

*.*Thanks.