How Does Covariant Derivative Relate to Curvature in Nonabelian Gauge Fields?

[wac03]

Dear friends
I am here with mathematical physics question:
we know tha if i have a compact Lie group G with g its Lie algebra, and a connection A on the fibre,
For nonabelain Lie algebra
The relation between covariant derivative and the curvature of A is

[ tex ]\begin{equation*}[D_{m},D_{n}]F_{ab}=[F_{mn},F_{ab}]\end{equation*}[ /tex ]

for any representation of g the Lie algebra
with
D is the covariant derivative
F the curvature of the connection A
my problem:
I will be so grateful if someone could help me to prove that

[ tex ][D_{m},D_{n}]F_{ab}=[F_{mn},F_{ab}][ /tex ]

is valid for any representation of the Lie algebra g especially for the fundamental (defining) representation, because i already did it for the adjoint representation of g.
thank you in advance wissam

 

[dextercioby]

You got me all lost here.Why doesn't the curvature have 4 suffixes...?What are the gauge fields (potentials) and the field tensors...?

Daniel.

 

[wac03]

the gauge field is the yang-mills

You got me all lost here.Why doesn't the curvature have 4 suffixes...?What are the gauge fields (potentials) and the field tensors...?

Daniel.

Dear daniel
Thank u for your interest in my question, the story here, physically talking means that the gauge field is the non-abelain gauge field "A_{mu}, and the field strength is the F_{mu nu}.
the gauge field is not the one of the gravidation
PS: please make difference between abelian( Maxwell theory), and non abelian (yang-Mills).
thank u
wissam

 

[dextercioby]

Do you know how to build any finite representation of the gauge group...?

In the case u do,u'll find that your problem reduces to checking the validity of that equality only in the case of the 2 fundamental (possibly inequivalent) irreps of the gauge group:the fundamental and its contragradient one.

Daniel.

 

[wac03]

Do you know how to build any finite representation of the gauge group...?

In the case u do,u'll find that your problem reduces to checking the validity of that equality only in the case of the 2 fundamental (possibly inequivalent) irreps of the gauge group:the fundamental and its contragradient one.

Daniel.

Dear daniel
we know that
F=dA+1/2[A,A]=dA+A^A (differential forms)
F_{mu nu}=[D_{mu},D_{nu}] in any local basis.
F here is g-valued
F_{mu nu}=F^a T_{a} where T_{a} is the generator of g.


In the irre. adjoint representation of g gives C_{ab}^{c}=(T_{a})^{c}_{b}
where C is the structure constant of g,
in the adjoint irr.rep the covariant derivative is
D=d+[A,-]

So using these facts, the equality in question is feasible.

my problem once again is:
i know that this relation is valid in any irr.rep of the algebra,
i will be so grateful if i can see the proof at least for the irr. fundemantal representation.
PS:
In any arbitrary representation, the covariant derivative written as
D=d+A without the commutator
A, F are G-valued
In components language for any field Q the covariant derivative is
(D^_{mu}Q)^{i}=\partial_{mu}Q^{i}+A^{a}(T_{a})_{j}^{i}Q^{j}
thank you
Wissam

 

[Rogue Physicist]

$$\begin{equation*}[D_{m},D_{n}]F_{ab}=[F_{mn},F_{ab}]\end{equation*}$$ Sorry, I can't view this code, so I posted it to see what it looks like