 Reciprocal rescaling symmetry of two cross-coupled scalar fields

For $\mathcal{L}={\varphi }_{1,t}{\varphi }_{2,t}-c^2{\varphi }_{1,x}{\varphi }_{2,x}$, the real linear internal transformations preserving the density are $\mathrm{diag}(r,r^{-1})$ and $\left(\begin{array}{cc}0& r\\ r^{-1}& 0\end{array}\right)$ with $r\ne 0$. 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 $j^0={\varphi }_1{\varphi }_{2,t}-{\varphi }_2{\varphi }_{1,t}$ and $j^x=c^2({\varphi }_2{\varphi }_{1,x}-{\varphi }_1{\varphi }_{2,x})$. The two wave equations give ${\partial }_t{j}^0+{\partial }_x{j}^x=0$. The energy quadratic form is indefinite: the fields ${\varphi }_{\pm }=({\varphi }_1\pm {\varphi }_2)/\sqrt{2}$ have equal and opposite kinetic signs.