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Greg McLac
#2
Apr19-07, 05:00 AM
P: n/a
Jay R. Yablon:
> For an Abelian Field strength F_uv, one can write the two-form, in polar
> coordinates, as:
>
> F = F_uv dx^u dx^v = - (g/4pi) sin theta d-theta d-phi (1)
>
> For a non-Abelian field strength F_i_uv where i is the index of the Yang
> Mills group being considered, and T^i are the group generators, we have
> (using SU(2) as an example):
>
> F = T^i F_i = T^i F_i_uv dx^u dx^v
>
> / F_3_uv F_1_uv + iF_2_uv \
> = | | dx_u dx_v (2)
> \ F_1_uv - iF_2_uv -F_3_uv /
>
> Thus, F is no longer a simple scalar like - (g/4pi) sin theta d-theta
> d-phi, but it is a two-by-two matrix transforming under SU(2).
>
> I may figure it our by the time someone replies, but how would one form
> (2) with explicit polar coordinates, analogously to (1)?


First note that (1) isn't correct, it should read:

F = F_uv dx^u dx^v = - F_theta_phi (g/4pi) sin theta d^theta d^phi (1)

Then it isn't that difficult. You can write the F_i_uv dx^u dx^nu like in
(1), and put them in the matrix (distributing dx^u dx^nu in the matrix.)
Then -(g/4pi) sin theta d^theta d^phi factorize and get out of the matrix.

GML