Symplectic form on scalar-field solutions ($\Omega ({\varphi }_1,{\varphi }_2)$)

For real Klein-Gordon field solutions with suitable support or boundary conditions, $\Omega ({\varphi }_1,{\varphi }_2)={\int }_{\Sigma }({\varphi }_1{n}^a{\nabla }_a{\varphi }_2-{\varphi }_2{n}^a{\nabla }_a{\varphi }_1)d\Sigma$ is a conserved symplectic form. The field equation makes its current divergence-free. Its complexification gives $(u,v{)}_{KG}=i\Omega (u^{\ast },v)$, the Klein-Gordon inner product. A compatible complex structure on the Klein-Gordon solution space is additional data needed for a particle representation.