I feel that I am overlooking something, but for the past hour I've been stuck on what should be rather trivial. I have an integral that I need to evaluate and I would like to change the variables I'm integrating over \begin{equation} \int...dk_1dk_2 \rightarrow \int...\text{Jacobian}\cdot dk^+dk^- \end{equation} It doesn't really matter what's inside, as I have denoted with "...". The $k^{\pm}$ are given by \begin{equation} k^{\pm}=k_1\pm k_2. \end{equation}
Now I would assume, what I have to do is simply write $dk_1$ as \begin{equation} dk_1=\frac{\partial k_1}{\partial k^+}dk^++\frac{\partial k_1}{\partial k^-}dk^-, \end{equation} and hence similarly for $dk_2$. Expressing $k^{1,2}$ in terms of $k^{\pm}$ I get \begin{equation} k^{1,2}=\frac{1}{2}(k^+\pm k^-) \end{equation}
Which means that the way I defined $dk_{1,2}$ gives me \begin{align} dk_1&=\frac{1}{2}(dk^++dk^-)\\ dk_2&=\frac{1}{2}(dk^+-dk^-), \end{align} and finally \begin{equation} dk_1dk_2=\frac{1}{4}\left[(dk^+)^2-(dk^-)^2\right], \end{equation} which is not at all what I expected to get. Where do I go wrong?
Hmm. Well, your transformation matrix is $\mathbf{f}(k^+, k^-)=\left[\begin{matrix}k_1+k_2\\ k_1-k_2\end{matrix}\right],$ making your Jacobian equal to $J=\left[\begin{matrix}1 &1\\ 1 &-1\end{matrix}\right],$ with determinant $-2$. Hence, $dk^+\,dk^-=-2\,dk_1\,dk_2.$ Apparently, for your problem, it happens to be the case that, since $dk_1\,dk_2=-\dfrac12\,dk^+\,dk^-,$ that $-\dfrac12\,dk^+\,dk^-=\dfrac14\,[(dk^+)^2-(dk^-)^2].$