Proofs of Symmetrized Trace of F^4 for Non-Abelian Gauge Field

[wac03]

my question is pretty technical,
in the course of studying the non-abelian Born-Infeld, i have tried to write out the Born-Infeld langrangian written using the symmetrized trace formalism, i have met at the fourth order of field strength the term
sTr(F^4) =F_{mn}F_{rn}F_{ml}F_{rl}+1/2F_{mn}F_{rn}F_{rl}F_{ml}
which looks at leat for me so involved to get proved.


My given is,
F is field strength of nonabelian gauge field
F is g-valued (g is Lie algbra of nonabelian compact gauge group G)
Fi is in the fundamental representation of g.
i will be so grateful if someone could give me hints or something which could hel to verify the relation in question
thx

 

[R0man]

Thank you for your question. The proof for the symmetrized trace of F^4 for non-abelian gauge fields is indeed quite technical and involves multiple steps. I will try to provide some hints and guidance on how to approach this proof.

Firstly, it is important to note that the symmetrized trace is defined as sTr(A) = Tr(PA), where P is the symmetrization operator defined as P(A) = 1/n!∑σ∈Sn Aσ, where Sn is the symmetric group of order n. In the case of F^4, we have n=4, so the symmetrization operator becomes P(A) = 1/4! (A + A12 + A13 + A14 + A23 + A24 + A34 + A123 + A124 + A134 + A234 + A1234), where the subscripts indicate the indices of the matrix elements.

Now, to prove the relation sTr(F^4) = F_{mn}F_{rn}F_{ml}F_{rl}+1/2F_{mn}F_{rn}F_{rl}F_{ml}, we need to expand the symmetrized trace using the definition of the symmetrization operator. This will give us a sum of terms, each of which will have four field strength tensors F_{mn}. We can then use the properties of the trace and the Lie algebra of the non-abelian gauge group to simplify this expression.

One important property to keep in mind is that the trace of a product of matrices is invariant under cyclic permutations. This means that we can rearrange the order of the matrices inside the trace without changing its value. Another useful property is that the trace of a product of matrices is also invariant under conjugation. This means that we can replace the matrices with their conjugates without changing the value of the trace.

Using these properties, we can rearrange and simplify the terms in the expanded expression for sTr(F^4). We will also need to use the Jacobi identity and the commutation relations of the Lie algebra of the non-abelian gauge group to simplify the expression further.

I cannot provide a complete step-by-step proof here, but I hope these hints will help you in verifying the given relation. I would also suggest consulting textbooks or other resources on non-abelian gauge fields for more detailed explanations and examples.