String action in light-cone coordinates I am going through textbook Einstein Gravity in a Nutshell by A. Zee and I got mathematically stuck at page 147 where he is talking about the classical string action using light cone coordinates. First, we have the action in the $(t,x)$ coordinates given by
$$S=\int \mathrm dt \mathrm dx [(\partial_t \phi)^2-(\partial_x \phi)^2].$$
Now he defines new coordinates in suitable units) by $x^+=t+x$ and $x^-=t-x$ and he arrives at the expression for action
$$S= 2\int \mathrm dx^+ \mathrm dx^- \frac{\partial \phi}{\partial x^+}\frac{\partial \phi}{\partial x^-}.$$
Now, I see that
$$(\partial_t \phi)^2-(\partial_x \phi)^2=[(\partial_t \phi)-(\partial_x \phi)][(\partial_t \phi)-(\partial_x \phi)]=4\frac{\partial \phi}{\partial x^+}\frac{\partial \phi}{\partial x^-}$$
but when I transform the other terms I get
$$\mathrm dt \mathrm dx=\frac 14 \left[(\mathrm dx^+)^2-(\mathrm dx^-)^2 \right]$$
which is apparently not correct (I should get  $\frac 12\mathrm dx^+ \mathrm dx^-$). Can someone please help me with this?
 A: $\newcommand{\d}{\mathrm{d}}\newcommand{\half}{\frac12}\newcommand{\xp}{x^+}\newcommand{\xm}{x^-}$Well, what is really meant by $\d t\ \d x$ is in fact
$$ ``\d t\ \d x\!"\;:= \left|\d t \wedge \d x\right|, $$
since that is the volume form of the surface. The differentials of the coordinates, $\d t$, $\d x$, are one-forms and $\wedge$ is the wedge product which is antisymmetric.
Therefore, since
$$t = \half\left(\xp+\xm\right) \qquad\text{and}\qquad x = \half\left(\xp-\xm\right),$$
you have
\begin{align}\left|\d t\ \wedge \d x\right| &=\left| \left[\half\left(\d\xp+\d\xm\right)\right]\wedge\left[\half\left(\d\xp-\d\xm\right)\right]\right| \\ &= \frac{1}{4}\Big|\smash[b]{\underset{=0}{\underbrace{\d\xp \wedge \d\xp}}}-\d\xp\wedge\d\xm + \d\xm\wedge\d\xp - \smash[b]{\underset{=0}{\underbrace{\d\xm \wedge \d\xm}}}\Big| =  \vphantom{\underset{=0}{\underbrace{\d\xp \wedge \d\xp}}} \\
&= \frac{1}{4}\left|-2\d\xp \wedge\d\xm\right| \\
&= \half\left|\d\xp \wedge\d\xm\right|=:\half ``\d\xp\ \d\xm\!",
\end{align}
using the antisymmetry of the wedge product.
