= Reciprocal rescaling symmetry of two cross-coupled scalar fields
{title2=$\phi_1\mapsto e^s\phi_1,\quad\phi_2\mapsto e^{-s}\phi_2$}
For $\mathcal L=\phi_{1,t}\phi_{2,t}-c^2\phi_{1,x}\phi_{2,x}$, the real linear internal transformations preserving the density are $\operatorname{diag}(r,r^{-1})$ and $\begin{pmatrix}0&r\\r^{-1}&0\end{pmatrix}$ with $r\ne0$. There are four <connected components>; the <identity component> is the displayed reciprocal rescaling with $r=e^s>0$. In the sum-and-difference basis it preserves a <bilinear form> of signature $(1,1)$, with the identity component isomorphic to that of the <One-dimensional Lorentz group>. Its <Noether current> in coordinates $(t,x)$ is $j^0=\phi_1\phi_{2,t}-\phi_2\phi_{1,t}$ and $j^x=c^2(\phi_2\phi_{1,x}-\phi_1\phi_{2,x})$. The two wave equations give $\partial_tj^0+\partial_xj^x=0$. The energy quadratic form is indefinite: the fields $\phi_\pm=(\phi_1\pm\phi_2)/\sqrt2$ have equal and opposite kinetic signs.
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