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,

$$\beta_\sigma\left(F\right)=\left( \frac{1} {1 -F^2} \right)^{\mu \rho}\partial_\mu F_{\rho\sigma }=0 $$ **Note:** To do this, it will prove very useful if you can ﬁrst show the following results:

$$∂_μ\left[\operatorname{tr} \ln(1 − F^2)\right] = −4 ∂_\rho F_{μ\sigma}\left(\frac{F}{1-F^2}\right)^{\sigma \rho } $$

which requires use of the Bianchi identity for $F_{\mu \nu }$ and

$$ \tag{*} \begin{align} \partial _\mu \left( \frac{ F}{1-F^2} \right)^{\mu\nu} &= \left( \frac{ F}{1-F^2} \right)^{\mu\rho} \partial_\mu F_{\rho\sigma} \left( \frac{ F}{1-F^2} \right)^{\sigma \nu} \\ &\qquad+ \left( \frac{1} {1 -F^2} \right)^{\mu \rho} \partial_\mu F_{\rho\sigma}\left( \frac{ 1 }{1-F^2} \right)^{\sigma \nu} \end{align}$$

In addition, as given in question 5a

$$F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} $$

My attempt to prove the problem:

LHS $$\partial_{\mu} \left( \frac{ F}{1-F^2} \right)^{\mu\nu} = \partial_{\mu} \left[ F^{\mu}_{\alpha} \left( \frac{1}{1-F^2} \right)^{\alpha \nu} \right] = \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + F^{\mu}_{\alpha} \partial_{\mu} \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} \tag{1} $$

Using the formula in matrix cookbook for the derivative of inverse matrix, Eq. (53)

Eq. (1) becomes $$\left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + 2 F^{\mu}_{\alpha} \left[ \frac{1}{1 -F^2} \left( \partial_{\mu} F \right) F \frac{1}{1-F^2} \right]^{\alpha \nu} $$

$$= \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + 2 \left( \frac{F}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{F}{1-F^2}\right)^{\sigma \nu} \tag{2} $$

The second term in Eq.(2) cancels the second term in the RHS in the problem sheet equation. We then need to show $$ \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + \left( \frac{F}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{F}{1-F^2}\right)^{\sigma \nu} - \left( \frac{1}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{1}{1-F^2}\right)^{\sigma \nu} =0 \tag{3} $$

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