The bilinear symbol and its counter-propagating values

Since every monomial of $J_3$ has exactly two derivatives, $\mathsf{q}$ is real, symmetric under $1\leftrightarrow 2$, polynomial, and homogeneous of degree two in $(k_1,k_2)$ jointly: $\mathsf{q}({ϵ}_1,\lambda k_1;{ϵ}_2,\lambda k_2)={\lambda }^2\mathsf{q}({ϵ}_1,k_1;{ϵ}_2,k_2)$. The operator $Q$ of Lemma lem:cauchyred is $\mathsf{q}$ with $k_a\mapsto -i\partial$ acting on ${\gamma }_a$, followed by ${{\partial }_t}^2\mapsto \Delta$.