I have a question in David Tong's Example Sheet 4 Problem 5b, how to verify the last equation (*) on p.2? (There is a solution for example sheet 3, but seems to be no solution for example sheet 4.)
Problem 5b:
Show that the equations of motion arising from the Born-Infeld action are equivalent to the beta function condition for the open string,
βσ(F)=(11−F2)μρ∂μFρσ=0
Note: To do this, it will prove very useful if you can first show the following results:
∂μ[trln(1−F2)]=−4∂ρFμσ(F1−F2)σρ
which requires use of the Bianchi identity for Fμν and
∂μ(F1−F2)μν=(F1−F2)μρ∂μFρσ(F1−F2)σν+(11−F2)μρ∂μFρσ(11−F2)σν
In addition, as given in question 5a
Fμν=∂μAν−∂νAμ
My attempt to prove the problem:
LHS ∂μ(F1−F2)μν=∂μ[Fμα(11−F2)αν]=(∂μFμα)(11−F2)αν+Fμα∂μ(11−F2)αν
Using the formula in matrix cookbook for the derivative of inverse matrix, Eq. (53)
Eq. (1) becomes (∂μFμα)(11−F2)αν+2Fμα[11−F2(∂μF)F11−F2]αν
=(∂μFμα)(11−F2)αν+2(F1−F2)μρ(∂μF)ρσ(F1−F2)σν
The second term in Eq.(2) cancels the second term in the RHS in the problem sheet equation. We then need to show (∂μFμα)(11−F2)αν+(F1−F2)μρ(∂μF)ρσ(F1−F2)σν−(11−F2)μρ(∂μF)ρσ(11−F2)σν=0
then I didn't find a way to show Eq. (3) hold. I tried to combine the second and third term, and rearrange them, but didn't got a simple expression.
This post imported from StackExchange Physics at 2014-03-07 16:42 (UCT), posted by SE-user user26143