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Jay R. Yablon
Dec7-08, 05:00 AM
P: n/a

Looking for Feedback on Yang-Mills "Exercise"

"Henrique de Andrade Gomes" <> wrote in message
.. . .
> I haven't read the whole paper with care, but your main equation
> stating that the curvature is the anti-symmetrized gauge exterior
> derivative of the connection seems wrong. For the gauge exterior
> derivative is already anti-symmetric on space-time indices, hence
> anti-
> symmetrization does not affect it. I.e. DA= Antisymmetrized (DA).
> Note that DA is NOT equal to F (even though the variation of F is
> equal to D of the variation of A). Basically, there is a 1/2 on the
> commutator of the connection forms that is not there in DA.
> Cheers,
> Henrique
> Jay R. Yablon wrote:
>> I have completed the "exercise" I have been working on the past few
>> weeks, on integrating-by-parts, the action in Yang-Mills Theory, and
>> posted this at:
>> I would like feedback on whether I have done this exercise correctly,
>> and to facilitate the learning that I am trying to do by way of this
>> exercise, of course would like to know if there are any erroneous
>> calculations and / or statements. . . .

It would help if you used an equation number rather than refer to "your
main equation."

But from what I can parse out of what you are saying, you are correct
that F=DA includes the antisymmetry in the spacetime indexes. But when
you "unpack" this to show explicit spacetime indexes, the antisymmetry
returns (see, e.g., my (2.1) and discussion in section 2). I believe I
have addressed this in the footnote at the bottom of page 2 including
the factor of 1/2. What I should have put in also for clarity, which I
will in the next draft, is that the differentiation of any p-form H is
specified by:

dH=(1/p!)d_vH_u1 u2 u3...dx^v dx^u1 dx^u2 dx^u3...

with dx^u dx^v = -dx^v dx^u.

The differentials are anticommuting Grassman variables (sometime wedge
products are shown but they just clutter up the page) and they naturally
carry the asymmetries you speak of, while the 1/p! =1/2 for p=2 is the
source of the 1/2 factor you speak of for the two-form F.

All of this, I believe, is correctly used in the "exercise" paper.