ajtribick wrote:You know, I would really like to know just why group theory is useful. The bit of group theory I did in my university course went something along the lines of "This is what a group is. Let's go straight to representation theory!", which left me somewhat confused as to what all the fuss is about.
The Standard Model of elementary particles and many extensions thereof are based on (non-abelian) Lie groups as gauge groups. Doing research in theoretical partical physics without being VERY firm in group theory and representation theory is therefore simply unthinkable. Believe me
. But even on a much more elementary level of Quantum Mechanics, where Clebsch Gordan coefficients due to e.g. angular momentum/spin couplings occur, one needs it heavily. Whenever one has to consider multiparticle quantum states, one has to compose the resulting physical state according to the rules of group/representation theory (using tensor products of representations, followed by a reduction of the result to irreducible representations etc).
In case of the above self-conjugacy argument, the fact may be misleading at first that some members of (non-abelian) gauge boson multiplets do carry some non-vanishing charge. Nevertheless, knowing that gauge bosons always span the self-conjugate
adjoint representation immediately proves the self-conjugacy. The trick is of course that under anti-particle conjugation, some charged members go to other charged members SUCH that the whole multiplet remains invariant.
E.g. the weak SU(2) triplet transforms as follows
(W+, W0, W-) --> (W-, W0, W+), which is unitary equivalent to the original triplet....
Someone without knowledge of group/representation theory might incorrectly conclude that the charged members break the rule...
Incidentally, for a
theoretical physicist knowing group theory is just the "tip of the
mathematical iceberg". A thorough mathematical knowledge is absolutely required in theoretical research. This includes MUCH more of modern math, of course.
Fridger
