= Symplectic form on scalar-field solutions
{title2=$\Omega(\phi_1,\phi_2)$}
For real <Klein-Gordon field> solutions with suitable support or boundary conditions, $\Omega(\phi_1,\phi_2)=\int_\Sigma(\phi_1n^a\nabla_a\phi_2-\phi_2n^a\nabla_a\phi_1)d\Sigma$ is a conserved <symplectic form>. The field equation makes its current divergence-free. Its complexification gives $(u,v)_{KG}=i\Omega(u^*,v)$, the <Klein-Gordon inner product>. A compatible <complex structure on the Klein-Gordon solution space> is additional data needed for a particle representation.
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